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yields a function with positive real part and the second of Yabuta's criteria applies. 2.3. Strongly exposed points. Whereas it seems impossible to give a characterization of exposed points that allows to check if a given function in aB(HI (II)))) is an exposed point, it is possible to characterize the strongly exposed points. This is the content of work of Nakazi and, independently, the first author. Let us describe strongly exposed points in terms of properties of the exposing functional. THEOREM 2.12 ([2,4]). II alunction IE 8B(H 1 (1I)))) is strongly exposed, then d( Tfr, HOO) < 1. (d = Loo-distance). PROOF. Suppose that the LOO-distance of
O. This calculation serves to illustrate that if for a given extreme point f the distance of 7Ilfl to H oo + C is less than 1, the ("only") thing that can prevent f from being exposed (and hence strongly exposed) is the divisibility of f in Hi by functions of the form (1 - U)2 with u(z) = >.z (>. E 11') a particularly simple inner function. Functions in Hi that lack this divisibility property are called strong outer functions. We see that for any strong outer function that is not exposed, like Inoue's example, [22], it is true that the distance of 7/1fl to H OO +C is 1. Alternatively, the strongly exposed points are the (normalized) strong outer functions f that satisfy distallfl, H oo + C) < 1.
3. The boundary of B(Hi(O)) Having studied the sets of exposed and strongly exposed points in the unit ball of the classical Hardy space Hi of the unit disc, we will investigate how these results hold up for the Hardy space Hi of a domain of finite connectivity ("finite domain") 0 c C. Such domains are e.g. conformally equivalent to domains with a smooth boundary consisting of finitely many components or to a disc from which a finite number of disjoint slits are deleted, cf. [33]. We defined Hi (0) in Section 1.2. There are two important differences with the classical Hardy space that make the analysis very different. On domains of finite connectivity, Hi-functions may not allow a classical factorization using Blaschke products, (singular) inner functions and outer functions. Indeed, the argument principle prevents the existence of a holomorphic function f on the annulus A = {I < Izl < 2}, such that If I = 1 a.e. on 8A and f has exactly one zero on A. Secondly, and in a way related, there now exist extreme points in the unit ball with (finitely many) zeros (Section 3.1). While such zero sets are somewhat generic for extreme points, their location plays a surprisingly crucial role in its being a (strongly) exposed point (Section 3.2). As in Section 2.1 we call a function I E HOO(O) an inner function if 11*1 = 1 almost everywhere with respect to arc length (dO') on 80. Consequently, II(z)1 ::; 1 for all z E O. Let w(·, z) denote harmonic measure on 80 at z. We call a function F E Hi(O) an outer function if for every z in 0 (equivalently, for at least one
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
73
zEn): log IF(z)1 = (
lao.
log IF*(~)I dw(~, z).
An outer function is zero free on n. Green's function G(z; zo) and the Blaschke factor Bzo(z) = t:z:oz for are related by G(z; zo) = log IBzo(z)l. This suggests the following definition.
-1:°1
]I))
DEFINITION 3.1. Let G(z; zo) be Green's function on a finite domain n with pole at Zo and let G(z; zo) denote its (multiple valued) harmonic conjugate. Suppose that Zl, Z2, . .. satisfy the Blaschke condition En G(z; Zn) < 00. Then the Blaschke product with zeros at Zl, Z2, ... is the multiple valued function
(3.1)
B(z) = exp( - L G(z; zn)"- iL G(z; Zn». n
n
DEFINITION 3.2. A multiple valued function F on n is modulus automorphic if • IF(z)1 is well-defined; • locally on n, IFI coincides with the absolute value of a holomorphic function. If in addition, IF(z)IP admits a harmonic majorant on n (0 < p < oo), then we say that F E MHP(n}. If IFI is bounded, we say that F E MHOO(n}. I E MHOO(n} is called innerif 11*1 = 1 a.e on r. I is singular inner if I is in addition zero-free. Blaschke products are modulus holomorphic inner functions. One can show that if FE MHP(n}, then IFI has non-tangential limits (denoted IF*!) a.e. on an, IF*I E p(an,da}, and unless F == 0, 10glF*1 E LI(an,da) and (3.2)
log IF*(z)l:::; (
lao.
log IF*(~}ldw(~, z),
for all zEn.
If equality holds in (3.2) at all points zEn (equivalently, for at least one zEn), we call F an outer function in MHI(n). While a single valued inner-outer decomposition is impossible in HP(n), a decomposition in modulus automorphic inner and outer factors is possible.
THEOREM 3.3 ([48]). Let f E HP(n) be not identically zero. Then there exist a Blaschke product BE MHOO(n), a singular inner function 8 E MHOO(n}and an outer function F E MHP (n) such that for all zEn
If(z}1 = IB(z)I·18(z}I·IF(z}l· This factorization is unique in the following sense: if B 1 , 8 1 and FI are a Blaschke product, a singular inner function and an outer function on n for which If(z)1 = IB1 (z}I·181 (z}I·IFI(z)l, then IBI = IBII, 181 = 181 1 and IFII = IFI· 3.1. Extreme points in HI(n}. The question arises which functions are extreme in the unit ball of Hl(n). After the deLeeuw-Rudin theorem 2.3, the following result is elementary: LEMMA 3.4. Iff E HI(n} is an outer function of unit norm, then f is extreme in the unit ball of HI(n}. Any attempt to copy the proof of the deLeeuw-Rudin theorem in the other direction will break down. For suppose f = I· F where I is a non-trivial inner function (in MHOO(n)}. Then deLeeuw and Rudin look at the function 9 = (1+12)F
74
PAUL BENEKER AND JAN WIEGERINCK
and show that IIf ± gill = 1. However, unless I is a single valued inner function, 1 + 12 is not a well-defined modulus automorphic function. There is no remedy for this problem, because as we have already mentioned, when m ~ 2 there exist extreme points with a non-trivial inner part. The following theorem of F. Forelli is crucial in understanding which inner functions can appear in the inner-outer factorizations of an extreme points. THEOREM 3.5 ([14]). Let f be an extreme point of the unit ball of H1(n). Then the codimension of the H 1-closure of f . Hoo in H1 is at most T' when n is bounded by m + 1 closed smooth curves. All functions in the H 1-closure of f . HOO(n) inherit the zeros of f, hence Forelli's theorem implies that an extreme point can only have a limited number of zeros. In fact one has: COROLLARY 3.6 ([14]). If f is an extreme point of the unit ball of H1 (n), then zeros. the inner part of f is a finite Blaschke product with at most
T
Inspection of the proof of the deLeeuw-Rudin theorem gives us the following criterion for extremity (where we identify H 1-functions with their boundary values): LEMMA 3.7. Let f E H1(n) be of unit norm. Then f is not extreme if and only if there exists a non-constant real function k E LOO(an) for which kf E H1(n). Suppose f = I . F is of unit norm where I is a finite Blaschke product, and F outer. Let us suppose that f is not an extreme point of the unit ball of H1(n). Then let k be as in the lemma, and let 9 E H1(n) have boundary values kf. Because F is an outer function, for all zEn: Ig(z)1 $ IIkll oo ·1F(z)l. Hence, because also III = 1 everywhere on an, the meromorphic function h = g/ f is bounded near an, real-valued (a.e.) on an, and has its poles in the zeros of f (with corresponding mUltiplicities). Conversely, if h is a meromorphic function on n with these three properties, then with k := h on an and 9 := hf E H 1(n), we have 9 = kf on an, so by the previous lemma, f is not an extreme point. We come to the following definition. DEFINITION 3.8. Let I be a finite Blaschke product. We say that I is an extremal Blaschke product if there exists no meromorphic function h on n that is bounded near an, real-valued on an and has its poles in the zeros of I, with no greater multiplicity than the zeros of I. The conclusions of the previous paragraphs may thus be summarized as follows: PROPOSITION 3.9. Let the norm of f E H1(n) be 1. Then f is an extreme point of the unit ball of H1(n) if and only if the inner part of f is an extremal Blaschke product. We wish to stress that Forelli's theorem also gives us an upper bound for the number of zeros of an extremal Blaschke product on n. Also, by the previous proposition we see that it is only the location of the zeros of a function in H1 (n) and not so much the outer factor that decides whether or not the function is extreme in the unit ball (after normalization). The problem of determining the extreme points of H1 (n) has thus been reduced to a problem on meromorphic functions on n with pre-described poles, that is: a problem concerning meromorphic divisors on n.
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
75
Let I be a finite Blaschke product with zeros Zl, Z2, ... , Zn repeated according to multiplicity. Thus I has n zeros on O. Let 8 := 1'ZI +1'Z2+' -+1,zn be the divisor on 0 associated with I. If 8' = L:zEf! d'(z) . Z is another divisor on 0 we say that 8' ~ 8 if at every Z E 0: 8'(z) ~ 8(z). The space of all meromorphic differentials w on 0 that are real-valued on 00 and for which the associated divisor (w) satisfies (w) ~ 8 is a real linear space of dimension MD(8). Using a theorem of H.L. Royden [39], based on the Riemann-Roch theorem, T.W. Gamelin & M. Voichick proved the following result: THEOREM 3.10 ([15]). The Blaschke product I with zeros Zl, Z2, ... , Zn and associated divisor 8 is extremal if and only if MD(8) + 2n = m. In particular, using only the fact that the inner factor of an extreme point is a finite Blaschke product (as shown by Forelli), Gamelin & Voichick also arrived at Forelli's upper bound (W-) for the number of zeros of an extremal Blaschke product. They proved that this upper bound is also sharp. THEOREM 3.11 ([15]). The HI-closure of the set of extreme points in the unit ball of HI(O) is the collection of all functions in HI(O) that have unit norm and no more than W- zeros. There is a special type of finite domains where Gamelin and Voichick described the zero sets of extreme points explicitly, namely the so called real slit domains. These are usually defined as the extended complex plane with a finite number of intervals deleted. In our situation we prefer a (conformly equivalent) definition. DEFINITION 3.12. We will call any domain n of the form JDl \ (It u ... U 1m), where It, 12', • .• ,Im are disjoint, bounded and closed intervals in (-1, 1) a real slit domain.
n
THEOREM 3.13 ([15]). Let be a real slit domain and let Zl, Z2, ... , Zn be points of (not necessarily distinct). Then the Blaschke product with zero set Zl, Z2, ... ,Zn is extremal if and only if:
n
• n::; W- and • for all i, j: Zi =I- Zj. (In particular, none of the Zi is real.)
We omit most of the proof, and only observe that because the meromorphic function Z~Zi + z~z; + l-~iz + l-kz is bounded and real-valued on an c JR., no zeros of an extremal Blaschke product are conjugated.
3.2. Strong exposedness and the location of zeros. In this section we investigate exposed and strongly exposed points in the unit ball of HI(O). We give several examples and criteria for (strongly) exposed points. Also we show that nontrivial properties of strongly exposed points in HI (JDl) (for example: Ll-invertibility on the boundary) have no analogue for finite domains. Finally, we again look at the zero sets of extreme points and the question of divisibility of extreme functions by functions of the form (1 + U)2, where u is a non-constant inner function. The Hahn-Banach theorem again gives that for f an exposed point of the unit ball of HI(O) the exposing functional L for f is unique and given by;
L : 9 E HI
1-+
fan 9 I~I da.
PAUL BENEKER AND JAN WIEGERINCK
76
Hence, like H1(1I))), a function I in the boundary of the unit ball of H1(n) is exposed if and only if it is rigid: apart from (positive) constant multiples of I there is no H 1-function with the same argument a.e. (dO") on an. The following criteria for rigidity of H 1 (1I)))-functions carryover to finite domains word for word: • If IE H1 and 1/1 E Hl, then I is rigid (Theorem 2.9. 1). • If there is agE Hoo such that Re(fg) > 0 a.e. on an, then I is rigid (Theorem 2.9. 2). • If u is a non-constant inner function such that 1/(1 + u)2 is in Hl, then I is not rigid (or I == 0). A priori the first two conditions can only be used to demonstrate rigidity of outer functions. In both cases III cannot be too small near the boundary of 0.: if I satisfies the second condition, then 1/1 E H1 -"'(an) for all E: > 0, so 1/1 is "nearly" in H1. The first condition can be modified to allow for exposed points with zeros on n. PROPOSITION
3.14 ([4]). II I is extreme in H1(n) and 1/1/1 E L1(an) then I
is exposed.
Similar to Theorem 2.13 is: THEOREM 3.15. Let I be a lunction in H1(n). Then I is strongly exposed in the unit ball 01 H1 il and only il I is exposed and L oo-dist(7 /1/1, Hoo+C(an)) < 1. Throughout the remainder of this section the domain R will be a real slit domain with m slits that contains the origin. Note that on 11'\ {i} the function (z+i)2 has the same argument as iz so (z+i)2 is not rigid in H1 (II))). LEMMA 3.16 ([4]). For all m ~ 2, the normalized lunction I(z) = c(z + i)2 is strongly exposed in the unit ball 01 H 1(R). The proof is an amusing exercise in elementary function theory. One supposes that 9 E H1 has the same argument as I a.e. on oR and sets h = g/ f. Schwarz's reflection principle eventually shows that h extends to a rational function which turns out to have no poles at all. Hence I is exposed and by Theorem 3.15 strongly exposed. REMARK 3.17. Similarly, if m > k + 1, then the normalized function hk(Z) = c(z + i)2k is strongly exposed in the unit ball of H 1 (R). In particular we see that there exist strongly exposed points in the unit ball of H1 (R) that are "small" on the boundary: 1/lhkl rt L 1 / 2k (OR). We recall that for I = I . F to be an extreme point the only requirement is that the inner part I of I is an extremal Blaschke product - a generic zero set; "most" of the properties of the function I then follow from its outer factor, i.e., the size of Ilion the boundary an. It is reasonable to ask whether exposedness is also essentially a property of the outer factor. We make this question precise in the following sense: if I E H1 is a rigid outer function, I is an extremal Blaschke product on 0., and 9 is invertible in MHOO(n) and such that Ig E HOO(n) (a single valued function), is the extreme point Ig· 1/IIIg . 1111 also exposed? (Compare
THE BOUNDARY OF THE UNIT BALL IN Hl_TYPE SPACES
77
with the first example in Section 2.2.) Proposition 3.14 tells us the answer is yes if 11f E Ll(80). Theorem 3.18 below shows that in general the answer is no. We mentioned in Section 2.2 Helson's criterium for exposedness of outer functions, 2.8. It fails for finite domains: THEOREM
3.18 ([4)). For m = 3, there exists ~
f(z)
=
E
R such that the function
c(z - ~)(z + i)2
is extreme in the unit ball of Hl(R), but not exposed.
We will only describe a non-trivial HI (R)-function 9 with the same argument
as
f a.e. on an, for suitable ~ E R \ lR and refer to [4J for details.
Suppose the three slits are the intervals [Xl,YlJ, [X2,Y2J, [X3,Y3J. Let k(z) be a rational function with poles of order 2 in ~,~, 1/~, 1/~, and poles of order 4 in ±i l , and double zeros at ±1, while k(oo) = -1. and zeros at the 12 points Next, let q be the (well-defined) square root of k on R with q(oo) = -i. Now with a suitable choice of ~ and an appropriate polynomial p of degree 8 that is positive on lR U '][', one can show that the function
xtl, yt
g(z)
p(z) + q(z) (z - ~)(z - i)2
1 (1- ~z)(l- ~z) belongs to HOO(R) and has the same argument as f on 8R \ {-i}, which implies that f is not rigid. Because the inner part of f is the extremal Blaschke product with zero at ~, f is extreme, however. By Lemma 3.16, (z + i)2 is rigid. We conclude that if Helson's criterion were valid, for any two inner functions u, v not both constant on R, (z + i)2/(u(z) + v(z))2 (j. Hl(R), hence also (z-~)(z+i)2/(u(z)+v(z))2 (j. Hl(R), a contradiction! =
4. The Bergman space AI(JD)) Recall from Section 1.2 the definition of the Bergman space
AP(JD)) = H(JD)) n LP(JD)) In the Bergman space extreme and exposed points are extremely simple. Indeed, every f E 8B(Al(JD))) is exposed, and (hence) extreme. The exposing functional is
Lf :
gl---+
kgl~ldA(Z).
If Lfg = 1 for some 9 E 8B(Al(JD))), the argument of 9 would be equal to that of f, (a.e) and this clearly implies f = g. However, to study strongly exposed points we need some machinery.
4.1. Bergman projection and Bloch space. An important tool for the study of strongly exposed points will be the Bergman projection; this is the orthogonal projection It is given by an integral operator (4.1)
(4.2)
Pcp(z)
= =
(f, kz)
[
=
f(w)
iD (1 _ zw)2 dA(w)
fUn+ 1)· 1m f(w)'W'dA(w))zn. n=O
D
PAUL BENEKER AND JAN WIEGERINCK
78
Elementary properties are, see e.g. [18], • If cp has compact support in]l)), then Pcp is holomorphic on a neighborhood of Jij. • If cp is Coo on Jij, then Pcp is Coo on Jij. The effect of the Bergman projection on Loo(]I))) and on C(Jij) will be very important for us. We need the Bloch spaces. DEFINITION 4.1. The Bloch space B consists of all holomorphic functions f on ]I)) with the property that (1-lzI2)lf'(z)1 is bounded on]l)). Equipped with the norm
(4.3)
IlfllB := If(OI
+ sup (1 -lzI 2)1f'(z)l, zED
B becomes a Banach space. The set of all functions f in B for which the expression (1-lzI2)1f'(z)1 Izl --> 1 is a closed subspace of B, called the little Bloch space Bo.
-->
0 as
Let Co denote the continuous functions on Jij that are zero on T. We have the following theorem of R. Coifman, R. Rochberg and G. Weiss: THEOREM 4.2 ([8]). The Bergman projection P maps Loo(.]I))) boundedlyonto B. Furthermore, P maps both C(D) and Co boundedlyonto Bo. The result proved in [8] is much more general than Theorem 4.2. As we state it, the theorem may be found in [18], Theorem 1.12, with an elementary proof. There we also find the following results THEOREM 4.3. (1) The dual space of Al is the Bloch space B under the following pairing:
(4.4)
g E B : f E Al !--+lim ( fr(z)g(z)dA(z). rTl
JD
(2) The dual space of the little Bloch space Bo is the Bergman space A l under the pairing: f
E
Al : g
E
Bo
1--+
lim ( f(z)gr(z)dA(z). rTl
JD
REMARK 4.4. When one identifies (A1)* with the Bloch space in Theorem 1 B, the dual norm on B yields a norm that is equivalent with, but not equal to the norm 11.1113 that we have previously defined on B. Hence, there exists a norm 11.11. on the Bergman space A I that is equivalent to 11.111 and is such that the dual norm of 9 E B = (AI)* equals IIgIiB. The strongly exposed points in the unit ball of Al with the norm 11.11. have been described by C. Nara ([32]), who also showed that up to isometrical isomorphisms, Al with the 11.11. norm is the unique pre-dual of B. 4.2. Strongly exposed points of AI(.]I))). The following theorem is a consequence of Theorems 5.3 and 5.2, the fact that Al(]I))) may be identified with the subspace of HI (lI£2) consisting of functions that depend only on one variable, and the fact that these functions are exposed in B(HI(lI£2). We set
(4.5)
THE BOUNDARY OF THE UNIT BALL IN Hl-TYPE SPACES
79
THEOREM 4.5. Let I E Al be 01 unit norm. Then I is strongly exposed in Ball(AI) il and only il the LOO-distance 01 filII to the space (AI)1. + C(D) is less than one. Henceforth we will simply write (AI)1.
+C
instead of (AI)1.
+ C{D).
The question now is: how can we estimate the distance in Loo of cp = 1/111 to (AI)1. +C, where I is a given function in AI? (Clearly the distance cannot exceed one. ) Let us first look at polynomials of a particularly simple form: I (z) = c( z - 0:) n , where c is normalizing. We will assume n 2: 1 because the constant functions are strongly exposed by Theorem 4.5. We distinguish three cases in order of increasing difficulty: 10:1 > 1, 10:1 < 1 and 10:1 = 1. The case where 10:1 > 1 is very easy: 1/111 is continuous on D, so I is strongly exposed. In fact, we may even take non-integer powers n and products of such functions and we always obtain strongly exposed points after normalization. When 10:1 < 1, let us write cp = 1/111 = 'l/Jo + 'l/JI, where 'l/JI is compactly supported in 1D> and cp == 'l/JI on a neighborhood of 0:, while 'l/Jo is smooth on D. From (4.1) we see that P'l/JI is holomorphic across the unit circle because 'l/JI is compactly supported in 1D>. Next, because 'l/Jo is smooth on D, also P'l/Jo is smooth on D. Hence Pcp is continuous on D. Now cp - Pcp is bounded, so cp = (cp - Pcp) + Pcp is contained in (AI)1. + C. By Theorem 4.5 I is strongly exposed. Again, our reasoning readily shows that the normalized product I of functions (z - O:i)n;, for all ni and all O:i ft 'll', is strongly exposed. Now suppose 10:1 = 1; we may take 0: = 1. Let us write In{z) = cn (1- z)n and CPn = In/l/nl· Introducing polar coordinates and applying Cauchy's theorem one finds that the exposing functional L for h is given by
L(g) = [ g(z) ~1 -
J"D
1-
z~: dA{z) = Z
[ Ig{z) (1 -
JD -
z
z) dA{z) =
Cog{O)
+ CIg'{O).
But then there exists a polynomial P2 such that L(g) = fD gP2 dA. Therefore CP2 - P2 is contained in (A I) 1., hence CP2 E (A I) 1. + C so h is strongly exposed. Similarly, for all even n, CPn is contained in (A I) 1. + C and In is strongly exposed in AI. Again, we may introduce non-integer exponents. Let 1f3 = cf3(1 - z)f3, where (3 > -2 to ensure that I f3 E A I; the constant Cf3 > 0 is normalizing. Set
cpf3 = 1f3/I/f3I· PROPOSITION 4.6 ([5]). For all (3 > -1, the Loo-distance 01 CPf3 to (AI)1. + C is at most Isin( f327i") I· In particular, lor all (3 > -1, (3 =I- 1,3,5, ... , the lunction I f3
is strongly exposed in the unit ball 01 A I. The proof consists of observing that if 1(3 - 2nl < 1, then IIcpf3 - CP2nli00
=
IsinCf)1 < 1. We will come back to odd exponents in Section 4.4 and end this section with and example of a boundary points of B (A I (1D>)) that is not strongly exposed. 2
EXAMPLE 4.7 ([5]). The normalized function I(z) = (I-z)2~~g2(I--cz) is not strongly exposed in the unit ball of A I. The functions I f3 tend pointwise to O. However, limf3!-2 fD /{3"(j5 dA = 1.
PAUL BENEKER AND JAN WIEGERINCK
80
·4.3. The space (AI).1 + C. We have observed that (AI).1 + C plays the same role in Theorem 4.5 with respect to the Bergman space as (HI).1 + C('1I') = Hoo +C('1I') with respect to the Hardy space HI(JI))) (Theorem 2.13). We mentioned already in Section 2.3 that Hoo + C is closed in L oo . From this then it followed relatively easily that Hoo + C('1I') is in fact an algebra, cf. [41], Theorem 6.5.5. How far do these results extend to the space (A I ).1 + C? From [5] we quote (1) Al (JI))).1 + c(il~) is a closed subspace of LOO(JI))). (It equals P-I(80 )!) (2) AI(JI))).1 + C(ii)) is a C(ii))- module. (3) A I (JI))).1 + C(ii)) is not an algebra. (4) The space (A 1).1 + C is invariant under composition with holomorphic automorphisms of JI)). EXAMPLE 4.8. [5] As for (3), let J{3 = (1- z){3 and let 'P{3 = J{3/IJ{31 for {3 E R Then 'P2 and 'P-4 E (AI).1, but 'P-2 is not contained in (AI).1 + C. The space (A I ).1 + C is not an algebra. The properties of (A 1).1 proposition.
+C
lead in a straightforward way to the following
PROPOSITION 4.9. Let J be a strongly exposed point in AI. Then (a) iJu is an automorphism oJJI)), then the normalized Junction FI = CI(fou) is strongly exposed; (b) iJ v E A(JI))) is zero-Jree on the circle, then the normalized Junction F2 = C2!v is strongly exposed. Furthermore, the functions J IIJI, FdlFII and F2/1F21 have the same Loo-distance to (AI).1 + C. 4.4. Strong exposedness of (1 - z){3. We saw in Section 4.2 that the functions J{3 = c{3(1 - z){3 are strongly exposed in the unit ball of Al for all {3 > -1 except possibly when {3 = 1,3,5, .... This was deduced from rather straightforward estimates of the L 00 -distances of the functions 'P = ff3 I IJ{31 to the space (A 1).1 + C (Proposition 4.6). In [5] a much sharper result is proved. THEOREM 4.10. For all {3 ::::: 0, the Bloch distance oj the Junction P'P{3 to 8 0 4I sin({h)1
equals 1r
{3+T .
SKETCH OF PROOF. It is convenient to rewrite 'P{3 as 'P{3(w) = (1- w){3/2/(1w){3/2. Using the series expansions for the Bergman kernel 1/(1- zw)2 (see (4.2)), as well as for (1 - W){3/2, and 1/(1 - w){3/2, we evaluate the Bergman projection P'P{3' One obtains P'P{3 = E~=o c{3,nzn, where
c{3,n =
-(n + 1) sin(~) ~ r(m + ~)r(m + n - ~) 271" ~ m!(m + n + I)! .
It is proved in [5] that for fixed (3 > 0:
(4.6)
~r(m+~)r(m+n-~) =:0 m!(m + n + I)!
where the o(I)-term tends to zero as n
4
= n2{3({3 + 2) (1 + 0(1)),
-+ 00.
-2sin(~)
This implies that
c{3,n = 7I"({3 + 2)n (1
+ 0(1)),
THE BOUNDARY OF THE UNIT BALL IN where the o{I)-term vanishes as n
--> 00.
!::tv
(3,n
SPACES
81
But then,
so the Bloch distance of PCP(3 to Bo is at least large N, I{ ~ c
HI-TYPE
41:~~~1I.
On the other hand, for
zn)'1 < ~ nlc 1.l z ln-1 < 21 sin{~)ll + 0(1) - n~ (3,n - 7r(,B + 2) 1 - Izl '
where the o{I)-term tends to zero as N increases. Using the fact that the polynomials are contained in Bo it follows that the Bloch distance of PCP(3 to Bo is at most 4I sin (¥)1 o ... ((3+2) . COROLLARY 4.11. Let d(cp(3, (Al)1. (Al)1. + C. Then for all,B 2: 0, (4. 7)
+ C)
denote the LOO-distance of CP(3 to
~ 1sin( !?f) 1 < d( (A 1) 1. + C) < ~ 1sin( ~) 1 < ~ 2 ,B + 2 CP(3, - 7r ,B + 2 - 7r'
In particular, all f(3 are strongly exposed for ,B 2: O. SKETCH OF PROOF. Let q: B --> B/Bo be the quotient map. By Theorem 4.2, the map q 0 P : L OO --> B/Bo is continuous and surjective. The kernel of the map q 0 P is the space (Al)1. + C, cpo Property (I) from Section 4.3. Hence the derived map
P* : L oo / ((Al)1.
+ C)
-->
B /Bo
is bijective and bounded (by ~ as follows from the proof in [18] of Theorem 4.2). This gives the lower bound for d(cp(3, (Al)1.
+ C), because
IIP*cp(311 =
~ ISi~~y)l.
By the closed graph theorem, the inverse P* -1 of P* is also bounded. Actually, one can show directly that II P* -111 ::; 1, which in turn yields the upper bound for d{cp(3, (Al)1.+C). Supposing that FE B/Bo has norm 1, we will show that P*-I(F) has norm at most 1 in L oo /((Al )1.+C). For any c > 0, we can find a representative fEB of the coset F such that IIfl18 < 1 + c. In the proof of Theorem 4.2 in [18] one finds that
'¢(w) = (I -IWJ2) . f'(w) ~ f'(O)
E
L oo
w satisfies f(z) - P'¢{z) = f(O) + j'(O)z E Bo. Thus '¢ is a representative of P*-I(F) in LOO. As a consequence IIp*-l{F)IILOO/((AI)J.+C) ::; d(,¢, (Al)1.
=
+ C)
::; lim esssuPr K2 gives rise to the unital *-homomorphism C(K2 ) -> C(Kd of 'composition with T', and conversely it is not much harder to see that any unital *-homomorphism C(K2 ) -> C(Kd comes from a continuous T in this way. Moreover such 7r is 1-1 (resp. onto) if and only if the corresponding T is onto (resp. 1-1). Thus the noncommutative version of a homeomorphism between compact spaces is a (surjective 1-1) *-isomorphism between unital C*-algebras. Coming back to 'noncommutative functional analysis', it is convenient for some purposes (but admittedly not for others) to view 'complete isometries' as the noncommutative version of isometries. It is very important in what follows that a 1-1 *-homomorphism 7r : A - B between C*-algebras, is by a simple and well known spectral theory argument, automatically an isometry, and consequently (by the same principle applied to 7r n ), a complete isometry. Similarly, a *-homomorphism 7r : A - B (which is not a priori assumed continuous) is aut.omatically completely contractive, and has a closed range which is a C* -algebra *-isomorphic to the C* -algebra quotient of A by the obvious two-sided ideal, namely the kernel of the *-homomorphism. The entries we have just described in this 'dictionary' are all easily justified by well known theorems (for example Gelfand's characterization of commutative C*algebras). That is, if one applies the noncommutative definition in the commutative world, one recovers exactly the classical object. Similarly one sometimes finds oneself in the very nice 'ideal situation' where one can prove a theorem or establish a theory in the noncommutative world (i.e. about operator spaces or operator algebras), which when one applies the theorem/theory to objects which are Banach spaces or function algebras, one recovers exactly the classical theorem/theory. An illustration of this point is the Banach-Stone theorem. The following is a much simpler form of Kadison's characterization of isometries between C*-algebras [17]:
COMPLETE ISOMETRIES - AN ILLUSTRATION
87
THEOREM 1.1. (Folklore) A surjective linear map T : A --+ B between unital C* -algebras is a complete isometry if and only if T = U7r('), for a unitary u E B and a *-isomorphism 7r : A --+ B. PROOF. (Sketch.) The easy direction is essentially just the fact mentioned earlier that 1-1 *-homomorphisms are completely isometric. The other direction can be proved by first showing (as with Kadison's theorem) that T(I) is unitary, so that without loss of generality T(I) = 1. The well known Stinespring theorem has as a simple consequence the Kadison-Schwarz inequality T(a)*T(a) ~ T(a*a). Applying this to T- 1 too yields T(a)*T(a) = T(a*a), and now the result follows immediately from the 'polarization identity' a*b = ~ E~=o(a + ikb)*(a + ikb). 0 Note that if one takes A = C(Kd and B = C(K2 ) in Theorem 1.1, and consults the 'dictionary' above, then one recovers exactly the classical Banach-Stone theorem. Indeed as we remarked earlier, in this case complete isometries are the same thing as isometries, unit aries are unimodular functions, and a *-isomorphism is induced by a homeomorphism between the underlying compact spaces. Indeed consider the following generalization of the Banach-Stone theorem: THEOREM 1.2. [15,22, 1, 20] Let fl be compact and Hausdorff, and A a unital function algebra. A linear contraction T : A --+ C(fl) is an isometry if and only if there exists a closed subset E of fl, and two continuous functions "f : E --+ '][' and r.p: E --+ 8A, with r.p surjective, such that for all y E E T(f)(y) = "f(y)f(r.p(y))·
Here 8A is the Shilov boundary of A (see Section 2). We have supposed that. T maps into a 'selfadjoint function algebra' C(fl); however since any function algebra is a unital subalgebra of a 'selfadjoint.' one, the theorem also applies to isometries between unital function algebras. If A is a C(K) space too, then 8A = K and then t.he theorem above is called Holsztynski's theorem. We refer the reader to [16] for a survey of such variants on the classical Banach-Stone theorem. Often the transition from the 'classical' to the 'noncommutative' involves the introduction of much more algebra. Next we appeal to our dictionary above to give an equivalent restatement of Theorem 1.2 in more algebraic language. THEOREM 1.3. (Restatement of Theorem 1.2) Let A, B be unital function algebras, with B selfadjoint. A linear contraction T : A --+ B is an isometry if and only if (A) there exists a closed ideal I of B, a unitary u in the quotient C*-algebra B / I, and a unital 1-1 *-homomorphism 7r : A --+ B / I, such that qI (T( a)) = u7r(a) for all a E A. Here qI is the canonical quotient *-homomorphism B
--+
B/I.
In light of Theorems 1.1 and 1.3 one would imagine that for any complete isometry T : A --+ B between unital operator algebras, the condition (A) above should hold verbatim. This would give a pretty noncommutative generalization of Theorem 1.3. Indeed if Ran T is also a unital operator algebra, then this is true (see ego B.l in [3]). However, it is quite easily seen that such a result cannot hold generally. For example, let Mn = Mn(C); for any x E Mn of norm 1, the map A 1---4 Ax is a complete isometry from C into Mn. Now Mn is simple (Le. has no
88
DAVID P. BLECHER AND DAMON M. HAY
nontrivial two-sided ideals), and so if the result above was valid then it follows immediately that x = u. This is obviously not satisfactory. To resolve the dilemma presented in the last paragraph, we have offered in [5] several alternatives. For example, one may replace the quotient B / I by a quotient of a certain *-subalgebra of B. The desired relation qJ(T(a)) = U7r(a) then requires u to be a unitary in a certain C* -triple system (by which we mean a subspace X of a C* -algebra A with X X* X c X). Or, one may replace the quotient B / I by a quotient B / (J + J*), where J is a one-sided ideal of B. Such a quotient is not an algebra, but is an 'operator system' (such spaces have been important in the deep work of Kirchberg (see [18, 19] and references therein). Alternatively, one may replace such quotients altogether, with certain subspaces of the second dual B** defined in terms of certain orthogonal projections of 'topological significance' (Le. correspond to characteristic functions of closed sets in K if B = C(K)) in the second dual B** (which is a von Neumann algebra [25]). The key point of all these arguments, and indeed a key approach to Banach-Stone theorems for linear maps between function algebras, C* -algebras or operator algebra.-" is the basic theory of C* -triple systems and triple morphisms, and the basic properties of the noncommutative Shilov boundary or triple envelope of an operator space. These important and beautiful ideas originate in the work of Arveson, Choi and Effros, Hamana, Harris, Kadison, Kirchberg, Paulsen, Ruan, and others. Indeed our talk at the conference spelled out these ideas and their connection with the Banach-Stone theorem; and the background ideas are developed at length in a book the first author is currently writing with Christian Le Merdy [7] (although we do not characterize non-surjective complete isometries there). Moreover, a description of our work from this perspective, together with many related results, may be found in [12]. Thus we will content ourselves here with a survey of some related and interesting topics, and with a new and self-contained proof of some characterizations of complete isometries between unital operator algebras which do not appear elsewhere. This proof has several advantages, for example the projections arising naturally with this approach seem to be more useful for some purposes. Also it will allow us to avoid any explicit mention of the theory of triple systems (although this is playing a silent role nonetheless). We also show how such noncommutative results are generalizations of the older characterizations of into isometries between function algebras or C(K) spaces. We thank A. Matheson for telling us about these results. In the final section we present some evidence towards the claim that (general) isometries between operator algebras are not the correct noncommutative generalization of isometries between function algebras. For the reader who wants to learn more operator space theory we have listed some general texts in our bibliography.
2. The noncommutative Shilov boundary At the present time the appropriate 'extreme point' theory is not sufficiently developed to be extensively used in noncommutative functional analysis. Although several major and beautiful pieces are now in place, this is perhaps one of the most urgent needs in the subject. However there are good substitutes for 'extreme point' arguments. One such is the noncommutative Shilov boundary of an operator space. Recall that if X is a closed subspace of C(K) containing the identity fUIlction lK on
COMPLETE ISOMETRIES - AN ILLUSTRATION
89
K and separating points of K, then the classical Shilov boundary may be defined to be the smallest closed subset E of K such that all functions J E X attain their norm, or equivalently such that the restriction map J f-+ JIE on X is an isometry. This boundary is often defined independently of K, for example if A is a unital function algebra then we may define the Shilov boundary as we just did, but with K replaced by the maximal ideal space of A. In fact we prefer to think of the classical Shilov boundary of X as a pair (aX, i) consisting of an abstract compact Hausdorff space ax, together with an isometry j : X -+ C(aX) such that j(IK) = lax and such that j(X) separates points of ax, with the following universal property: For any other pair (0, i) consisting of a compact Hausdorff space 0 and a complete isometry i : X -+ C(O) which is unital (i.e. i(IK) = IA), and such that i(X) separates points of 0, there exists a (necessarily unique) continuous injection r : ax -+ 0 such that i(x)(r(w)) = j(x)(w) for all x E X, w E ax. Such a pair (aX, i) is easily seen to be unique up to an appropriate homeomorphism. The fact that such ax exists is the difficult part, and proofs may be found in books on function algebras (using extreme point arguments). Consulting our 'noncommutative dictionary' in Section 1, and thinking a little about the various correspondences there, it will be seen that the noncommutative version of this universal property above should read as follows. Or at any rate, the following noncommutative statements, when applied to a unital subspace X c C(K), will imply the universal property of the classical Shilov boundary discussed above. Firstly, a unital operator space is a pair (X, e) consisting of an operator space X with fixed element e EX, such that there exists a linear complete isometry Il, from X into a unital C*-algebra C with Il,(e) = Ie. A 'noncommutative Shilov boundary' would correspond to a pair (B,j) consisting of a unital C*-algebra B and a complete isometry j : X -+ B with j(e) = IB, and whose range generates B as a C* -algebra, with the following universal property: For any other pair (A, i) consisting of a unital C* -algebra and a complete isometry i : X -+ A which is unital (Le. i(e) = IA), and whose range generates A as a C* -algebra, there exists a (necessarily unique, unital, and surjective) *-homomorphism IT : A -+ B such that IT 0 i = j. Happily, this turns out to be true. The existence for any unital operator space (X, e) of a pair (B,j) with the universal property above is of course a theorem, which we call the Arveson-Hamana theorem [2, 13] (see [3] for complete details). As is customary we write C;(X) for B or (B,j), this is the 'C*-envelope of X'. It is essentially unique, by the universal property. If X = A is a unital operator algebra (see Section 1 for the definition of this), then j above is forced to be a homomorphism (to see this, choose an i which is a homomorphism, and use the universal property). Thus A may be considered a unital subalgebra of C;(A). If A is already a unital C* -algebra, then of course we can take C; (A) = A. To help the reader get a little more comfortable with these concepts, we compute the 'noncommutative Shilov boundary' in a few simple examples. Example 1. Let Tn be the upper triangular n x n matrices. This is a unital subspace of M n , and no proper *-subalgebra of Mn contains Tn. Let (B,j) be the C* -envelope of Tn. By the universal property of the C* -envelope, there is a surjective *-homomorphism IT : Mn -+ B such that IT(a) = j(a) for a E Tn. The kernel of IT is a two-sided ideal of Mn. However Mn has no nontrivial two-sided ideals. Hence IT is 1-1, and is consequently a *-isomorphism, and we can thus identify Mn with B. Thus Mn is a C*-envelope of Tn.
90
DAVID P. BLECHER AND DAMON M. HAY
Example 2. Consider the linear subspace X of M3 with zeroes in the 1-3, 2-3, 2-1, 3-1 and 3-2 entries, and with arbitrary entries elsewhere except for the 3-3 entry, which is the average of the 1-1 and 2-2 entries. It is easy to see that the C* -algebra generated by X inside M3 is M2 EB C. However this is not the C*-envelope. Indeed the 3-3 entry here is redundant, since the norm of x E X is the norm of the upper left 2 x 2 block of x. The canonical *-homomorphism from M2 EB C onto Nh when restricted to X is a unital complete isometry from X onto T2 (see Example 1). Thus if one takes the quotient of M2 EB C by the kernel of this homomorphism, namely the ideal 02 EB C, then one obtains M 2 , which by Example 1 is the C* -envelope. Indeed this is typical when calculating the C* -envelope of a unital subspace X of Mn. The C*-algebra generated by X is a finite dimensional unital C*-algebra. However such a C* -algebras is *-isomorphic to a finite direct sum B of full 'matrix blocks' M nk • Some of these blocks are redundant. That is, if p is the central projection in B corresponding to the identity matrix of this block, then x t-+ x(IBp) is completely isometric. If one eliminates such blocks then the remaining direct sum of blocks is the C* -envelope. Example 3. Let B be a unital C*-algebra. Consider the unital subspace S(B) of the C*-algebra M 2 (B) consisting of matrices
[~;
:1]
for all x,y E B and >.,Ji, complex scalars. We claim that M2(B) is the C*-envelope C of S(B), and we will prove this using a similar idea to Example 1 above. Namely, first note that M 2(B) has no proper C*-subalgebra containing S(B), Thus by the Arveson-Hamana theorem there exists a *-homomorphism 11' : M 2 (B) ---+ C which possesses a property which we will not repeat, except to say that it certainly ensures that 11' applied to a matrix with zero entries except for a nonzero entry in the 1-2 position, is nonzero. It suffices as in Example 1 to show that Ker 11' = {a}. Suppose that 11'(x) = 0 for a 2 x 2 matrix x E M 2(B). Let Eij be the four canonical basis matrices for M 2, thought of as inside M2(B). Then 11'(E1i XEj2 ) = 11'(Eli)11'{X)11'(Ej2) = 0 for i,j = 1,2. Thus by the fact mentioned above about the 1-2 position, we must have EliXEj2 = O. Thus x = O. In fact a variant of the C*-envelope or 'noncommutative Shilov boundary' can be defined for any operator space X. This is the triple envelope of Hamana (see [14]). This is explained in much greater detail in [3], together with many applications. For example it is intimately connected to the 'noncommutative .M-ideals' recently introduced in [4]. This 'noncommutative Shilov boundary' is, as we mentioned in Section 1, a key tool for proving various Banach-Stone type theorems. However in the present article we shall only need the variant described earlier in this section. 3. Complete isometries between operator algebras We begin this section with a collection of very well known and simple facts about closed two-sided ideals] in a C* -algebra A, and about the quotient C* -algebra AI]. We have that ]1.1. is a weak* closed two-sided ideal in the von Neumann algebra A**, and there exists a unique orthogonal projection e in the center of A** with ]1.1. = A**(1 - e). The projection 1 - e is called the support projection for I, and
COMPLETE ISOMETRIES - AN ILLUSTRATION
91
1 - e may be taken to be the weak* limit in A ** of any contractive approximate identity for I. Thus it follows that A ** / I J..J.. ~ A ** e as C* -algebras. Therefore also A/Ie (A/ 1)** ~ A** / IJ..J.. ~ A**e
as C*-algebras. Explicitly, the composition of all these identifications is a 1-1 *homomorphism taking an a + I in A/I, to ae = eae in A **. Here' is the canonical embedding A -+ A** (which we will sometimes suppress mention of). Thus A/I may be regarded as a C*-subalgebra of A**, or of the C*-algebra eA**e. We next illustrate the main idea of our theorem with a simple special case. (The following appeared as part of Corollary 3.2 in the original version of [5], with the proof left as an exercise). Suppose that T : A -+ B is a complete isometry between unital C*algebras, and suppose that T is unital too, that is T(I) = 1. Let C be the C*-subalgebra of B generated by T(A). Applying the Arveson-Hamana theorem 1 we obtain a surjective *-homomorphism () : C -+ A such that (}(T(a)) = a for all a E A. If I is the kernel of the mapping (), then C / I is a unital C* -algebra *-isomorphic to A. Indeed there is the canonical *-isomorphism 'Y : A -+ C / I induced by (), taking a to T(a) + I. The next point is that C/I may be viewed as we mentioned a few paragraphs back, as a C* -subalgebra of C** , and therefore also of B**. Indeed if e is the central projection in C** mentioned there, then C / I may be viewed as a C* -subalgebra of eC** e C eB** e C B**. In view of the last fact, the map 'Y induces an 1-1 *-homomorphism 7r : A -+ B** taking an element a E A to the element of B** which equals (1)
-- -- --
T(a)e = eT(a)
eT(a)e
--
(these are equal because e is central in C**). Conversely, if T : A -+ B is a complete contraction for which there exists a projection e E B** such that eT(a)e is a 1-1 *-homomorphism 7r, then for all a E A,
IIT(a)1I
~ Ile~ell
=
117r(a)11 = Iiall
using the fact mentioned earlier that 1-1 *-homomorphisms are necessarily isometric. Thus T is an isometry, and a similar argument shows that it is a complete isometry. Thus we have characterized unital complete isometries T : A -+ B. If H is a Hilbert space on which we have represented the von Neumann algebra B** as a weak* closed unital *-subalgebra, then B may be viewed also as a unital C*-subalgebra of B(H), whose weak* closure in B(H) is (the copy of) B**. In this case we shall say that B is represented on H universally. (The explanation for this term is that the well-known 'universal representation' tru of a C*-algebra is 'universal' in our sense, and conversely if 7r is a representation which is 'universal' in our sense then 7r(B)" is isomorphic to 7ru(B)" ~ B**. See [27] Section 1.) If, further, e E B** is a projection for which (1) holds, then with respect to the splitting H = eH EEl (1- e)H we may write T(a) =
[7ro(·)
0]
SO '
for all a E A. We will see that this is essentially true even if T(IA) -lIB: 1We remark in passing that one does not need the full strength of the Arveson-Hamana theorem here, one may use the much simpler [8] Theorem 4.1.
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DAVID P. BLECHER AND DAMON M. HAY
THEOREM 3.1. Let T : A --> B be a completely contractive linear map from a unital operator algebra into a unital C* -algebra. Then the following are equivalent:
(i) T is a complete isometry, (ii) There is a partial isometry u E B** with initial projection e E B**, and a (completely isometric) 1-1 *-homomorphism 1r : C;(A) 1r(1) = e, such that for all a E A T(a)e
= tL1r(a)
and 1r(a)
-->
eB**e with
-
= u*T(a).
Moreover e may be taken to be a 'closed projection' (see [25] 3.11, and the discussion towards the end of our proof). (iii) If H is a Hilbert space on which B is represented universally, then there exist two closed subspaces E, P of the Hilbert space H, a 1-1 *-homomorphism 1r : C;(A) --> B(E) with 1r(1) = IE, and a unitary u : E --> P, such that T(a)IE = u1r(a), and T(a)IE.L C p.l., for all a E A. Here E.l. for example is the orthocomplement of E in H. (iv) If H is as in (iii), then there exists two closed subspaces E, P of H, a unital 1-1 *-homomorphism 1r : C;(A) --> B(E), a complete contraction S : C; (A) --> B( E.l. , p.l.), and unitary operators U : E EEl p.l. --> Hand V : H --> E EEl E.l., such that T( ) - U [ 1r(a)
a -
0
0
S(a)
] V
for all a E A.
(v) There is a left ideal J of B, a 1-1 *-homomorphism 1r from C;(A) into
a unital subspace of B I (J + J*) which is a C* -algebra, and a 'partial isometry' u in B I J such that qJ(T(a)) = u1r(a)
&
for all a E A, where qJ is the canonical quotient map B
-->
BIJ.
Before we prove the theorem, we make several remarks. First, we have taken B to be a C* -algebra; however since any unital operator algebra is a unital subalgebra of a unital C* -algebra this is not a severe restriction. We also remark that there are several other items that one might add to such a list of equivalent conditions. See [5, 6]. Items (ii)-(iv), and the proof given below of their equivalence with (i), are new. We acknowledge that we have benefitted from a suggestion that we use the Paulsen system to prove the result. This approach is an obvious one to those working in this area (Ruan and Hamana used a variant of it in their work in the '80's on complete isometries and triple morphisms [28, 14]). However we had not pushed through this approach in the original version of [5] because this method does not give several of the results there as immediately. Statement (v) above has been simply copied from [5, 6] without proof or explanation. We have listed it here simply because Theorem 1.3 may be particularly easily derived from it as the special case when A and B are commutative (see comments below). Note that (iii) above resembles Theorem 1.2 superficially.
COMPLETE ISOMETRIES - AN ILLUSTRATION
93
PROOF. The fact that the other conditions all imply (i) is easy, following the idea in the paragraph above the theorem, namely by using the fact that a 1-1 *-homomorphism is completely isometric. In the remainder of the proof we suppose that T is a complete isometry. We view A as a unital subalgebra of C;(A) as outlined in Section 3. We define a subset S(B) of M 2(B) as in Example 3 in Section 2. Similarly define a subset S(T(A)) of S(B) using a similar formula (note that S(T(A)) has 1-2 entries taken from T(A) and 2-1 entries taken from T(A)*). Similarly we define the subset S(A) of the CO-algebra M2(C;(A)) (Le. S(A) has scalar diagonal entries and off diagonal entries from A and A*). We write 1 EEl 0 for the matrix in S(A) with 1 as the 1-2 entry and zeroes elsewhere. Similarly for 0 EEl 1. We also use these expressions for the analogous matrices in S(B). The map q, : S(A) -+ S(T(A)) c M 2(B) taking
[~!
:1] ~
[T~)* ~~~)]
is well known to be a unital complete isometry (this is the well known Paulsen lemma, see the proof of 7.1 in [23]). Let C be the C' -subalgebra of M2 (B) generated by S(T(A)). The CO-envelope of S(A) is well known to be M2(C;(A)) (see Example 3 in Section 2 where we proved this in the case that A is already a C* -algebra, or for example [3] Proposition 4.3 or [30]). Thus by the Arveson-Hamana theorem we obtain a surjective *-homomorphism () : C -+ 1V/z(C;(A)) such that () 0 q, is simply the canonical embedding of S(A) into M2(C;(A)). As in the special case considered above the theorem, we let 10 be the kernel of the mapping (), then C /10 is a unital CO-algebra *-isomorphic to M2(C;(A)). Indeed there is the canonical *-isomorphism "(: M2(C;(A)) -+ Clio induced by (), taking
[~!
:1] ~
[T~:)* T~~)]
+
10 ,
As in the simple case above the theorem, C /10 may be viewed as a C* -subalgebra of PoC"Po, for a central projection Po E C** (namely, the complementary projection to the support projection of 10)' Now PoC**Po C C** C M2(B)**, and it is well known that M2(B)** ~ M2(B**) as CO-algebras. Thus we may think of C** as a C* -subalgebra of .1112 (B**). Also, C'* contains C as a C* -subalgebra, and the projections 1 EEl 0 and 0 EEl 1 in C correspond to the matching diagonal projections 1 EEl 0 and 0 EEl 1 in M2(B**). These last projections therefore commute with Po, since Po is central in C**, which immediately implies that Po is a diagonal sum fEEl e of two orthogonal projections e, f E B**. Thus we may write the C* -algebra POM2(B**)po as the C*-subalgebra [ fB** f eB** f
f B**e ] eB*'e
of M2(B**). We said above that Clio may be regarded as a C*-subalgebra of the subalgebrapoM2(B**)po of M2(B*'). Thus the map "( induces a 1-1 *-homomorphism III : M2(C;(A)) -+ .!I1z(B**). It is easy to check that 1lI(1 EEl 0) = fEEl 0 and III (0 EEl 1) = 0 EEl e. Since III is a *-homomorphism it follows that III maps each of the four corners of M 2(C;(A)) to the corresponding corner of POM2(B**)po C M2(B**). We let R: C;(A) -+ fB**e be the restriction of III to the '1-2-corner'. Since III is 1-1, it follows that R is 1-1. If 7r is the restriction of III to the '2-2-corner', then 7r is a *-homomorphism C;(A) -+ eB**e taking lA to e. Applying the *-homomorphism
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DAVID P. BLECHER AND DAMON M. HAY
111 to the identity
[~ ~][~ ~]=[~ ~] we obtain that u = R(I) is a partial isometry, with u*u = 71"(1) = e. Similarly uu* = f. A similar argument shows that R(a) = R(I)7I"(a) for all a E C;(A). Thus u* R(a) = u*u7l"(a) = 7I"(a) for all a E C;(A). Next, we observe that 111 takes the matrix z which is zero except for an a from A in the 1-2-corner, to the matrix w = Pocp(z)Po. Since cp(z) E C** and Po is in the center of that algebra, we also have w = cp(z)Po = Pocp(z). Also w viewed as a matrix in M2(B**) has zero entries except in the .1-2-corner, which (by the last sentence) equals
---- ----
ff(;;) e = f(;;) e = ff(;;). Also using these facts and a fact from the end of the last paragraph we have --...
----* -
---*-
u*T(·) = R(I)*T(·) = (fT(I)e)*T(·) = eT(I) fT(·) = eT(I) T(·)e = u* R(·) = 71". Thus
-
-
-
T(·)e = fT(·) = uu*T(·) = U7l"(')' We have now also established most of (ii). One may deduce (iii) from (ii) by viewing B c B** c B(H), and setting E = eH, and F = (uu*)H. We also need to use facts from the proof above such as u*u = e. Clearly (iv) follows from (iii). As we said above, we will not prove (v) here. Claim: if e is the projection in (ii) above, then 1 - e is the support projection for a closed ideal I of a unital *-subalgebra D of B. Equivalently (as stated at the start of this section), there is a (positive increasing) contractive approximate identity (bd for I, with bt ~ 1 - e in the weak* topology. This claim shows that 1 - e is an 'open projection' in B**, so that e is a closed projection, as will be obvious to operator algebraists from [25] section 3.11 say. For our other readers we note that for what comes later in our paper, one can replace the assertion about closed projections in the statement of Theorem 3.1 (ii) with the statement in the Claim above. To prove the Claim, recall from our proof that Po = fEB e = Ie - Pi, where Pi is the support projection for a closed ideal 10 of C. Thus Pi = (1 - f) EB (1 - e). As stated at the start of Section 3, Pi is the weak* limit in C**, and hence also in M 2(B**), of a contractive approximate identity (et) of 10, By the separate weak* continuity of the product in a von Neumann algebra, it follows that the net bt = (OEB l)et(OEB 1) has weak* limit (OEB I)Pl (OEB 1) = OEB (1- e). Viewing these as expressions in B, the above says that bt ~ 1 - e weak* in B* *. View (0 EB 1) C (0 EB 1) as a *-subalgebra D of B, and view (0 EB 1)10(0 EB 1) as a two sided ideal I in D. It is easy to see that (bd is a contractive approximate identity of I. Thus it follows that 1 - e is the support projection of the ideal I. 0 Some applications of results such as Theorem 3.1 may be found in [6]. Next we discuss briefly the relation between our noncommutative characterization of complete isometries (for example Theorem 3.1 above), and Theorem 1.3. Our point is not to provide another proof for Theorem 1.3 - the best existing proof is certainly short and elegant. Rather we simply wish to show that the noncommutative result contains 1.3. Indeed Theorem 1.3 quite easily follows from Theorem
COMPLETE ISOMETRIES - AN ILLUSTRATION
95
3.1 (v). Since however we did not prove Theorem 3.1 (v), we give an alternate proof. COROLLARY 3.2. Let A, B be a unital function algebras, with B selfadjoint. Then condition (ii) in Theorem 3.1 implies condition (A) in Theorem 1.3. PROOF. By hypothesis, T(·)e = U1l"('), and u*u = e = 11"(1) so that u = u1l"(1) = T(I)e. Thus eT(I)*T(·)e = u*U1l"(') = 11"(1)11"(') = 11", so that Ran 11" C eBe = Be (note B** is commutative in this case). From [25] 3.11.10 for example, the 'closed projection' e in B** corresponds to a closed ideal J in B whose support projection is 1 - e. Alternatively, to avoid quoting facts from [25], we will also deduce this from the 'Claim' towards the end of the proof of Theorem 3.1. If I is the ideal in that Claim, let J be the closed ideal in B generated by I. Since J = Bl, the contractive approximate identity of I is a right contractive approximate identity of J. Thus J has support projection 1 - e too, by the first paragraph of Section 3 above. By facts in the just quoted paragraph, we have a canonical unital 1-1 map '11 : B / J ~ B** taking the equivalence class b + J of b E B to ebe. Indeed in this commutative case we see by inspection that '11 is a *-homomorphism from the C*algebra B/J onto the C*-subalgebra M = eBe of B**. Define O(a) = '11-1(1I"(a)), this is a 1-1 *-homomorphism A ~ B/J. Since 11"(1) = e, 0 is a unital map too. Since uu* = u*u = e, u is unitary in M, and so 'Y = '11-1(U) is unitary in B/J. Note also that T(a)e = '11(T(a) + J). Applying '11- 1 to the equation T(·)e = U1l"('), we obtain qJ(T(a)) = 'Y O(a), that is, condition (A) in Theorem 1.3. D If one attempts to use the ideas above to find a characterization analogous to condition (A) from Theorem 1.3 but in the noncommutative case, it seems to us that one is inevitably led to a condition such as (v) in Theorem 3.1. We address a paragraph to experts, on generalizations of the proof of Theorem 3.1. Consider a complete isometry between possibly non-unital C* -algebras. Or much more generally, suppose that T is a complete isometry from an operator space X into a C* -triple system W. One may form the so called 'linking C* -algebra' of W, with the identities of the 'left and right algebras of W' adjoined. Call this C'(W). As in the proof of Theorem 3.1 we think of S(W) c .c'(W). Similarly, if Z is the 'triple envelope' of X (or if X = Z is already a C*-algebra or C*-triple system), then we may consider S(X) C S(Z) c .c'(Z). As in the proof of Theorem 3.1 we obtain firstly a unital complete isometry : S(X) ~ S(T(X)) c .c'(Z), and then a unital 1-1 *-homomorphism 11" : .c'(Z) ~ .c'(W)**. By looking at the 'corners' of 11" we obtain projections e, f in certain second dual von Neumann algebras, so that fT( ·)e is (the restriction to X of a completely isometric) a 1-1 triple morphism into W**. In fact we have precisely such a result in [5] (see Section 2 there), but the key point is that the new proof gives different projections e, f, which are more useful for some purposes.
4. Complete isometries versus isometries Finally, as promised we discuss why we believe that in this setting of nonsurjective maps between C* -algebras say, general isometries are not the 'noncommutative analogue' of isometries between function algebras. The point is simply this. In the function algebra case we can say thanks to Holsztynski's theorem that the isometries are essentially the maps composed of two disjoint pieces Rand S, where R is
96
DAVID P. BLECHER AND DAMON M. HAY
isometric and 'nice', and S is contractive and irrelevant. However at the present time it looks to us unlikely that there ever will be such a result valid for general nonsurjective isometries between general C* -algebras. The chief evidence we present for this assertion is the very nice complementary work of Chu and Wong [9] on isometries (as opposed to complete isometries) T : A -+ B between CO-algebras. They show that for such T there is a largest projection p E B** such that T(·)p is some kind of Jordan triple morphism. This appears to be the correct 'structure theorem', or version of Kadison's theorem [11], for nonsurjective isometries. However as they show, the 'nice piece' R = T(·)p is very often trivial (Le. zero), and is thus certainly not isometric. Thus this approach is unlikely to ever yield a characterization of isometries. A good example is A = M 2 , the smallest noncommutative C* -algebra. Simply because A is a Banach space there exists, as in the discussion in the first paragraph of our paper, a linear isometry of A into a C (K) space. However it is easy to see that there is no nontrivial *-homomorphism or Jordan homomorphism from A into a commutative C* -algebra. Such an isometry is uninteresting, and this is perhaps because the interesting 'nice part' is zero. Thus we imagine that the 'good noncommutative notions of isometry' are either complete isometries or the closely related class of maps for which the piece T(·)p from [9] is an isometry. This leads to three questions. Firstly, can one independently characterize the last mentioned class? Secondly, if T is a complete isometry, then is the projection p in the last paragraph equal (or closely related) to our projection e above? Finally, H. Pfitzner has remarked to us, there is already a gap between the isometry and the 2-isometry cases (not only isometries and complete isometries). It would be interesting if there were a characterization of 2-isometries.
References [1] J. Araujo and J. J. Font, Linear isometries between subspaces of continuous functions, Trans. Amer. Math. Soc. 349 (1997), 413-428. [2] W. B. Arveson, Subalgebms ofC*-algebms, Acta Math. 123 (1969), 141-224; II, 128 (1972), 271-308. [3] D. P. Blecher, The Shilov boundary of an opemtor space, and the chamcterization theorems, J. Funct. An. 182 (2001),280-343. [4] D. P. Blecher, E. G. Effros and V. Zarikian, One-sided M-Ideals and multipliers in opemtor spaces. 1. To appear Pacific J. Math. [5] D. P. Blecher and D. Hay, Complete isometries into C* -algebms, http://front.math.ucdavis.edu/math.OA/0203182, Preprint (March '02). [6] D. P. Blecher and L. E. Labuschagne, Logmodularity and isometries of opemtor algebms, To appear, Trans. Amer. Math. Soc .. [7] D. P. Blecher and C. Le Merdy, Opemtor algebms and their modules - an opemtor space approach, To appear, Oxford Univ. Press. [8] M. D. Choi and E.G. Effros, The completely positive lifting problem for C·-algebms, Ann. Math. 104 (1976), 585-609. [9] C-H. Chu and N-C. Wong, Isometries between C· -algebms, Preprint, to appear Revista Matematica Iberoamericana. [10] J. B. Conway, A Course in Opemtor Theory, AMS, Providence, 2000. [11] E. G. Effros and Z. J. Ruan, Opemtor Spaces, Oxford University Press, Oxford (2000). [12] R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces, Book to appear, CRC press. [13] M. Hamana, Injective envelopes of opemtor systems, Pub!. R.I.M.S. Kyoto Univ. 15 (1979), 773-785.
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[14] M. Hamana, Triple envelopes and Silov boundaries of operator spaces, Math. J. Toyama University 22 (1999), 77-93. [15] W. Holsztynski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math. 24 (1966), 133-136. [16] K. Jarosz and V. Pathak, Isometries and small bound peturbations of function spaces, In "Function Spaces", Lecture Notes in Pnre and Applied Math. Vol. 136, Marcel Dekker (1992). [17] R. V. Kadison, Isometries of operator algebras, Ann. of Math. 54 (1951), 325-338. [18] E. Kirchberg, On restricted peturbations in inverse images and a description of normalizer algebras in C*-algebras, J. Funct. An. 129 (1995), 1-34. [19] E. Kirchberg and S. Wassermann, C*-algebras generated by operator systems, J. Funct. Analysis 155 (1998), 324-351. [20] A. Matheson, Isometries into function algebras, To appear. [21] M. Nagasawa, Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodia Math. Sem. Rep. 11 (1959), 182-188. [22] W. P. Novinger, Linear isometries of subspaces of continuous functions. Studia Math. 53 (1975), 273-276. [23] V. I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Math., Longman, London, 1986. [24] V. I. Paulsen, Completely bounded maps and operator algebras, To appear Cambridge University Press. [25] G. Pedersen, C*-algebras and their automorphism groups, Academic Press (1979). [26] G. Pisier, Introduction to operator space theory, To appear Camb. Univ. Press. [27] M. Rieffel, Morita equivalence for C*-algebras and W*-algebras, J. Pure Appl. Algebra 5 (1974), 51-96. [28] Z. J. Ruan, Subspaces ofC*-algebras, Ph. D. thesis, U.C.L.A., 1987. [29] E. L. Stout, The theory of uniform algebras, Bogden and Quigley (1971). [30] C. Zhang, Representations of operator spaces, J. Oper. Th. 33 (1995), 327-351. DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON.
E-mail address, David P. Blecher: dblecher~ath. uh. edu E-mail address, Damon Hay: dhayOmath.uh.edu
TX 77204-3008
Contemporary Mathematics Volume 328, 2003
Some Recent Trends and Advances in Certain Lattice Ordered Algebras Karim Boulabiar, Gerard Buskes, and Abdelmajid Triki ABSTRACT. In this paper we give a survey, intended as both a supplement as well as an update to a survey by Huijsmans [57], with results that have been obtained in the last ten years on Archimedean lattice ordered algebras. Special attention is paid to I-algebras, almost I-algebras and d-algebras and problems that were posed in the survey by Huijsmans about these special classes of lattice ordered algebras.
CONTENTS
l. Introduction 2. Definitions and elementary properties 3. i-algebra multiplications in C (X) 4. Multiplication by an element as an operator 5. Uniform completion and Dedekind completion 6. Powers in i-algebras 7. Functional Calculus on f-algebras 8. Relationships between i-algebra multiplications 9. Connection between algebra and Riesz homomorphisms 10. Positive derivations 11. Cauchy-Schwarz inequalities 12. Order biduals 13. Ideal theory 14. Representation of f-algebras 15. Linear biseparating maps on f-algebras References 1991 Mathematics Subject Classification. 06F25, 13J25, 16W80, 46A40, 46840, 46842, 46E25, 47L07, 47847, 47865. Key words and phrases. almost I-algebra, algebra homomorphism, I-algebra, d-algebra, lattice ordered algebra, order ideal, orthomorphism, representation theory, Riesz homomorphism, ring ideal, space of continuous functions, uniformly complete Riesz space. The second named author gratefully acknowledges support from an Office of Naval Research Grant with number N00014-01-1-0322. Part of this survey was written while the first named author was visiting the University of Mississippi in the Spring of 2002. © 2003 American Mathematical Society
99
~!3QtJLA8lAR,
GERARD BUSKES, AND ABDELMAJID TRIKI
1. Introduction
ftehistory of lattice ordered vector spaces (so called Riesz spaces or vector
lattices) goes back to Riesz and the International Congress of Mathematicians in Bologna in 1928. A study of the most important class of lattice ordered algebras (but not their name), J-algebras, was initiated by Nakano in [75] for a-Dedekind complete ordered vector space in 1951, subsequently in 1953 by Amemiya in [3], and finally with its present definition and name in 1956 by Birkhoff and Pierce in [22]. A precise date for the very first definition of lattice ordered algebras in general is very hard to pinpoint, but originated at around the same time as the previous three references. Indeed, in his review [44] of Birkhoff's 1950 address to the International Congress of Mathematicians in Cambridge, Massachusetts, Fl.·ink observed that a general study of lattice ordered rings seems to be needed to study what are now called averaging or Reynolds operators. A call for lattice ordered rings also had gone out by Birkhoff himself in the form of a listed problem at the end of his seminal 1942 paper [20] on lattice ordered groups. Thus lattice ordered algebras and J-algebras seem to have multiple origins, including a study of averaging operators, which themselves sprang forth from problems in fluid mechanics. An appearance at about the same time of J-rings and J-algebras has not resulted in an historical development on complete common ground for these objects. This is not unlike the development of lattice ordered groups versus the development of Riesz spaces. Where the latter have attracted attention from researchers in analysis, the former have been more widely investigated by algebraists. A similar divided attention from analysis versus algebra seems to underlie the connected but somewhat separate tracks of lattice ordered rings versus lattice ordered algebras. Though this separation of tracks is to some extent unavoidable, where each track does have ground that is truly its own, some overlap in results does exist, resulting in difficulties making accurate literature attributions in a survey like ours. We are grateful to two referees for pointing at some references that were missing in our manuscript, though we take full responsibility for possible remaining omissions in the reference list. This survey places itself almost completely on the track of lattice ordered algebras and our only apology for not linking algebra facts in a systematic way to ring results is that all three of us authors were trained as analysts. There is a natural back and forth between the two theories, in one direction by forgetting some of the structure, and in the other by finding, so to speak adjointly, an enveloping algebra. A nice survey on J-rings was written by Henriksen in 1995 (see [50]), to which we refer the interested reader for linkage to some of what follows in this survey. Historically, a lot of the credit for a revival of the theory of J-algebras points to the highly motivating Arkansas Lecture Notes by Luxemburg [67] and the 1982 Ph.D. thesis of de Pagter [76], who systematically explored both the existing literature as well as new directions. Another impetus to research in the area of J-algebras derived from the desire of Zaanen, who in the late seventies started to develop a program to prove many of the elementary results in the theory of Riesz spaces without using representation theorems for vector lattices. This desire is directly linked with a preference not to use the Axiom of Choice unnecessarily. The present survey is intended as an update to the one by Huijsmans [57]. We hasten to point out that we do not intend this survey to replace the one by Huijsmans, but rather that we think of it as augmenting part of it. Since [57] appeared, much progress has been made and several of the problems explicitly phrased in
LATTICE ORDERED ALGEBRAS
101
[57] have been solved. At the same time, some topics like ideal theory, connections between Riesz homomorphisms and algebra homomorphisms, and representations of f-algebras were absent in [57]. Thus an update as well as a supplement was needed. However, some sections of [57] receive no attention at all in this survey. We have not included important topics like the role of f-algebras in positive operator theory (e.g., we do not even include results on the previously mentioned averaging operators) and probability theory. We are rather focused on placing this update, as much as possible, in the setting of spaces of functions, hoping to interest as large an audience as possible via this approach. Moreover, we feel that the great source of inspiration was and continues to be the beautiful book by Gillman and Jerison [47], which certainly inspired and continues to inspire a large part of the research in lattice ordered algebras and rings. Last but not least we focus on what could be called distortions of f-algebras, in particular on Archimedean almost falgebras and d-algebras. Though these distortions have the potential to be seen as aberrations by some, we believe they point the way to techniques that are needed for the broader theory of lattice ordered algebras, as well as for illumination of various aspects in the theory of f-algebras. Note that the distortions disappear if the lattice ordered algebra under consideration has a multiplicative identity which is a weak order unit. Indeed, such algebras are automatically f-algebras. It should be mentioned that classes of lattice ordered algebras other than the ones that appear in this survey have been studied. Notably, the papers [85], [86], [87], and [88] by Steinberg discuss lattice ordered algebras in which every square is positive, a class of algebras that includes all almost f-algebras. We also pay no attention at all to non-Archimedean lattice ordered algebras. Finally, we point out that several results in this survey rely heavily on the (relative) uniform topology on Riesz spaces. In particular, since in Archimedean Riesz spaces uniform limits are unique, we shall include the 'Archimedean' property in the definition of uniformly complete Riesz spaces. A complete investigation of that topology can be found in Sections 16 and 63 of [69]. For terminology and concepts not explained or proved in this survey we refer the reader to the standards books [2], [47], [69], [72], [92] and [93].
2. Definitions and elementary properties A (real) Riesz space A is called a lattice ordered algebra (briefly, an i-algebra) if A also is an algebra and the positive cone A+ = {J E A : f ~ O} is closed under multiplication, that is, if f,g ~ 0 then fg ~ 0 (equivalently, if Ifgl :::; Ifllgl for all f,g E A).
We make the following blanket assumption: all Riesz spaces under consideration in this paper are assumed to be Archimedean (however, the latter blanket assumption has not stopped us to explicitly add the word Archimedean to the list of conditions in various results below). After B.irkhoff and Pierce (see [22, p. 55]), we define an i-algebra A to be an f-algebra if for every f, 9 E A, the condition f /\g = 0 implies Uh) /\g = (hi) /\g = 0 for all h E A+
102
KARIM BOULABIAR. GERARD BUSKES, AND ABDELMA.lID TRIKI
holds. We call the E-algebra A an almost f-algebra after Birkhoff in [21, Section 6] if f /\ g
= 0 ill A implies
fg
= O.
An E-algebra A for which f /\ 9 = 0 in A and h E A+ imply (.fh) /\ (gh)
=
(hf) /\ (hg) = 0
is called a d-algebra. The notion of d-algebra goes back to Kudlacek in [65]. Our focus in this survey on E-algebras is almost exclusively on f -algebras, almost falgebras and d-algebras. In this paragraph, we recall some properties of f-algebras. Using the Axiom of Choice, Birkhoff and Pierce in [22, Theorem 13] proved that any f-algebra is commutative. A constructive proof of this fact, due to Zaanen, can be found in [61, Theorem 2.1] or [92, Theorem 140.10]. All squares in an f-algebra are positive. Also, Ifgl = Ifllgl for all f,g in an f-algebra A. The multiplication by an element in the f -algebra A is order continuous, i.e., if inf {fT : T} = 0 in A then inf {g fT : T} = o for all 9 E A +. Phrased more generally, the multiplication 7rf by an element f E A (7rf (g) = f 9 for all 9 E A) is an orthomorphism and all orthomorphisms are order continuous. Recall that an orthomorphism on a Riesz space L is an order bounded linear operator 7r such that 17r (f)I/\ Igl = 0 whenever Ifl/\ Igl = 0 in L (the reader is referred to [2] or [92] for elementary properties of orthomorphisms). There is another important relationship between orthomorphisms and f-algebras, which we mention next. Indeed, let Orth (L) be the set of all orthomorphisms on a Riesz space L. Under the operations and the ordering inherited from £b (L), the ordered algebra of all order bounded operators on L, and under composition as multiplication, Orth (L) is an Archimedean f-algebra with the identity map h on L as unit element. The details of the facts recalled above can all be found in [2], [76] or [92]. Next we present some properties of almost f-algebras. Almost f-algebras, like f-algebras, are commutative too. The latter fundamental property was first established by Scheffold in [80, Theorem 2.1] for almost f-algebras that are Banach lattices. Using both Scheffold's result and the Axiom of Choice, Basly and Triki were the first to prove commutativity for arbitrary almost f-algebras [10, Thorme 1.1]. The first proof of the commutativity for almost f-algebras within ZermeloFraenkel set theory was given by Bernau and Huijsmans in their paper [13, Theorem 2.15]. Recently, a shorter constructive proof was published in [34, Corollary 3] by Buskes and van Rooij. Another property of f-algebras holds for almost f-algebras, namely the positivity of squares. Also, if A is an almost f-algebra then f2 = Ifl2 for all f E A. However, contrary to the order continuity of the multiplication in falgebra..'l, the multiplication by a fixed element in an almost f-algebra is not always order continuous as is shown in the following example. EXAMPLE 2.1. Write A = C ([0,1]), the vector space of all real-valued continuous functions on [0, 1]. With respect to the pointwise ordering (i. e., f :::; 9 in A if an only if f (x) :::; 9 (x) for all x E [0,1]), A is an Archimedean Riesz space. Define a multiplication • in A by
(f. g)(x)
=
{
f(x)g(x)
f (1/2) 9 (1/2)
(0 :::; x :::; 1/2) ; (1/2 :::; x :::; 1)
LATTICE ORDERED ALGEBRAS
103
lor all I, 9 E A. Then A is an almost I-algebm with respect to the multiplication •. For every natuml number n :::: 1, define the function In E A by (O :::; x :::; 1/2 - l/n) ; (1/2 - l/n :::; x :::; 1/2) ; (1/2:::; x :::; 1/2 + l/n) ; (1/2+1/n:::;x:::;1).
I f (:1") _ { n/2 - n:1: " ,-n/2 + nx 1
Then sup Un : n = 1,2, ... } exists in A and equals the function e defined bye (x) = 1 lor all x E [0,1]. On the other' hand,
(O :::; x :::; 1/2) ; (1/2:::;x:::;1)
lor all n E {I, 2, ... }, and clearly the set {e. In : n = 1,2, ... } does not have a supremum in A. We conclude that. is not order continuous. For more information about elementary theory of almost f-algebras, the reader is encouraged to consult [13], [23], [34], and [35]. At this point, we turn our attention to some properties of d-algebras. It follows directly from the definition of d-algebras that an i-algebra A is a d-algebra if and only if the multiplication map induced by any fixed element in A + is a Riesz (or lattice) homomorphism. It follows that a necessary and sufficient condition for an i-algebra A to be a d-algebra is that the identity If gl = 1/IIgi holds for all f, 9 in A. Contrary to (almost) I-algebras, d-algebras need not be commutative nor have positive squares. Next we give an example of a non-commutative d-algebra in which not all squares are positive. EXAMPLE
2.2. Let in this example A be the algebra of real {2 x 2)-matrices of
the form
(~
g)
with the usual addition, scalar multiplication, matrix product and partial or·dering. It is not hard to see that A is an Archimedean d-algebra. But A is not commutative and not all squares in A are positive. Indeed, if p=
(~ ~)
and q =
(~ ~)
then pq
=q
and qp = O.
Moreover, the square
is not positive. As for almost f-algebras, multiplication by a fixed element in a d-algebra is, in general, not order continuous. Point in case is the almost I-algebra that we considered in Example 2.1, which also is a d-algebra. Our main reference about d-algebras is [13]. In what follows, we look at some of the connections between the three kinds of i-algebras that we consider in this paper. It is immediate that any I-algebra is both, an almost f-algebra and a d-algebra. Almost f-algebras need not be d-algebras as we see in the next example.
KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
104
EXAMPLE 2.3. Take A as in Example 2.1, and let () E A be the function () (x) = { 1/2 - x x - 1/2
(0::; x ::; 1/2) ; (1/2::; x ::; 1).
For f, 9 E A, define (f
-9
)( ) _ {
x -
/,1-X f(s)g(s)ds
(0::; x ::; 1/2);
1/2
(1/2::; x ::; 1).
() (x) f (x) 9 (x)
Then A is an almost f -algebra under the multiplication -. However, A is not a d-algebra. Indeed, let e, f in A be defined by e(x)=l and
={
f (x)
forallxE[O,l]
-4x + 1 4x-3
(0::; x ::; 1/2) ; (1/2::;x::;1).
If - el (0) = 0 and (If I- e) (0) = If I- Igl fails in A.
Observe that
If - gl =
/,1
If(s)1 ds -; O. Thus the property
1/2
Since every almost f -algebra is commutative and has positive squares, Example 2.2 shows that d-algebras need not be almost f-algebras. However, if a d-algebra A is commutative or has positive squares then A is automatically an almost f-algebra [22, p. 60]. Summarizing part of the relations, we have the following diagram f-algebra::::} commutative d-algebra ::::} almost f-algebra. For more detail, see [22], [13] and [57]. The next lines deal with nilpotent element in i-algebras. The set of all nilpotent elements in the i-algebra A is denoted by N (A). In other words,
N (A)
= {f E A : r = 0 for some n =
1, 2, ... } .
Given a natural number p, we define Np (A)
= {f
E A : fP
= O} .
The i-algebra A is said to be semiprime if 0 is the only nilpotent element in A, that is, if N (A) = {O} . If A is an f-algebra then the following equalities hold N (A)
= N2 (A) = {f E A
: fg
= 0 for all 9 E A}.
(see [76, Proposition 10.2] or [92, Theorem 142.5]). If A is an almost f-algebra then N (A)
= N3 (A) = {J E A
: fg2
= 0 for all 9 E A}
= {f E A : fgh = 0 for all g, hE A}
(see [51, Theorem 3.11]) and, as for f-algebras, N2 (A)
= {f
E A : fg
= 0 for
all 9 E A}
(see [23, Lemma 5.3]). If A is a d-algebra then N (A)
= N3 (A) = {f E A
: gfh
= 0 for all 9
• A}
(see [13, Theorem 5.5] or [29, Theorem 5]). However, and co we documented for f-algebras and almost f-algebras, in d-algebl
ry to what he equality
LATTICE ORDERED ALGEBRAS
105
N2 (A) = {J E A: fg = 0 for all 9 E A} does not necessarily hold as is illustrated by the following example.
EXAMPLE 2.4. Take A as in Example 2.1 and define the multiplication. in A by (f. g)(x) = f (0) 9 (1) for all f, 9 E A. Let f E A be the function defined by f (x) = 1 - x for all x E [0,1]. Clearly, f. f = 0 but f. e i= 0 where e E A is defined bye (x) = 1 for all x E [0,1].
Finally, note that any f-algebra with multiplicative identity is semiprime and any semiprime almost f-algebra or semiprime d-algebra is automatically an falgebra (see Section 1 in [13]). As a final comment we remark that an l-algebra which has positive squares and has a multiplicative identity need not be an falgebra.
3. i-algebra multiplications in C (X) Let C (X) be the set of all real-valued continuous functions on a compact Hausdorff' topological space X. Under pointwise addition and scalar multiplication, C (X) is a real vector space. Moreover, C (X) is an Archimedean Riesz space with respect to the pointwise ordering (i.e., f ~ 9 in C (X) if and only if f (x) ~ 9 (x) for all x E X). By defining the multiplication in C (X) pointwise as well (Le., (fg) (x) = f(x)g(x) for all f,g E C(X) and all x E X), the space C(X) is easily seen to have the structure of an f-algebra with e as unit element, where e (x) = 1 for all x E X. Now consider another associative multiplication. in C (X). The main topic of this section is to produce necessary and sufficient conditions for C (X) to be an f-algebra (respectively, an almost f-algebra, a d-algebra) with respect to this new multiplication •. The first theorem in this direction goes back to Conrad (see [38, Theorem 2.2]), who obtained the following. THEOREM 3.1. Let. be an associative multiplication in C (X). Then C (X) is an f -algebra with respect to • if and only if there exists a positive function w E C (X) such that (f.g)(x) =w(x)f(x)g(x) for all f,g E C (X) and all x EX.
In fact, Conrad established the theorem above for any Archimedean f-ring with unit element. The representation formula given in Theorem 3.1 above was obtained in an alternative way by Scheff'old in [80, Korollar 1.4]. While Conrad's proof is purely algebraic and order theoretic, the proof presented by Scheff'old relies on analytic tools like the Riesz representation theorem. With the same analytic tools Scheff'old also obtained the following representation theorem for almost f-algebra multiplications in C (X) (see [80, Theorem 1.2]). THEOREM 3.2. Let. be an associative multiplication in C (X). Then C (X) is an almost f -algebra with respect to • if and only if there exists a family (/Jx : x E X) of positive measures such that (f. g)(x) for all f, 9 E C (X) and all x E
x.
=
Ix
f(s) 9 (s) d/Jx (s)
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KARIM BOULABIAR, GERARD BUSKES. AND ABDELMAJID TRIKI
Since every commutative d-algebra is an almost I-algebra, the previous theorem remains valid for commutative d-algebra multiplications in C (X) as well. Recently in [25, Corollary 3.2], Boulabiar proved the following representation formula for any (not necessarily commutative) d-algebra multiplication in C (X). THEOREM 3.3. Let. be an associative multiplication in C (X). Then C (X) is a d-algebra with respect to • if and only if there exist (i) a positive function wE C (X), and (ii) functions h,k: X -+ X (continuous on coz(w) = {x EX: w(x)"! O}) such that (J. g) (x) = w (x) I (h (x)) 9 (k (x)) for all I,g E C(X) and all x E X.
Notice that if C (X) is a d-algebra with respect to the multiplication. then • is commutative if and only if the functions hand k coincide on coz (w) (where 11" k and ware as in Theorem 3.3). The latter observation yields. in addition to the formula cited in Theorem 3.2 above, another representation for commutative d-algebra multiplications on C (X) . More abstract versions of the results above will be given in Section 8 below. 4. Multiplication by an element as an operator Let A be an (-algebra and recall that C b (A) denotes the ordered algebra of all order bounded operators on A. For every f E A, we define the map 7rf on A by 7rf (g) = f 9 for all 9 E A. Clearly, 7rf is an order bounded operator on A for all lEA. The map p : A -+ Cb (A) defined by p (J) = 7rf for all I E A is obviously an algebra homomorphism, that is, p (J g) = p (J) p (g) for all I, 9 E A. Hence the range p (A) of p is a subalgebra of Cb (A). In this section, we will see that if A is an almost f-algebra then p (A) can canonically be equipped with an ordering, under which p (A) is an Archimedean I-algebra. A corresponding result will also be given for commutative d-algebras and f-algebras. Let A be an almost I-algebra. Since
N2 (A) = {f E A : Ig = 0 for all 9 E A}, = 7r9 if and only if f by putting
7rf
-
9 E N 2 (A). This allows us to define an ordering on p (A)
(0) The ordering defined by (0) coincides with the ordering inherited from Cb (A), namely, 7rf is positive with respect to (0) if and only if 7rf is a positive operator on A. Under the usual addition and composition of operators, and with the ordering defined by (0), p (A) is an Archimedean ordered subalgebra of Cb (A). In fact, we have the following theorem (see Theorem 4.2 and Theorem 4.4 in [23]). THEOREM 4.1. Let A be an Archimedean almost I -algebra. Then p (A) is an Archimedean I -algebra with respect to the addition and composition of operators, and the ordering inherited from Cb (A). The lattice operations in p (A) are given by 7r f
V 7r9
= 7rfV 9'
7r f 1\ 7r9
= 7rf /1.9
for all f, 9 EA.
In particular, (7rf)+
= 7rf+,
(7rf)-
= 7rf-'
l7rfl
= 7rlfl
for all lEA.
LATTICE ORDERED ALGEBRAS
107
In other words, p defines a surjective Riesz homomorphism from A onto p (A).
Theorem 4.1 of course holds in commutative d-algebras. Moreover, let A be a commutative d-algebra. For f E A and 9 E A + the equalities 17I'fl (g)
= 71'tft (g) = Iflg = sup{lfllhl : Ihl
::; g}
= sup {Ifhl : Ihl ::;g} = sup {17I'f (h)1 : Ihl ::;g} imply that 71'f has an absolute value in Lb (A), which coincides with its absolute value in p (A). We collect the latter observations for commutative d-algebras. THEOREM 4.2. Let A be an Archimedean commutative d-algebra. Then p (A) is an Ar'chimedean f -algebra when equipped with the addition and composition of operators, and the ordering inherited from Lb (A). Moreover, the absolute value 71'tft of 71'f in p (A) coincides with the absolute value of 71' f in Lb (A) for all f E A. that is, 17I'fl (g) = 71'tft (g) = sup {17I'f (h)1 : Ihl ::; g} for all 9 E A+.
We obtain the f-algebra case as a corollary. COROLLARY 4.3. Let A be an Archimedean f-algebra. Then p (A) is an fsubalgebra of the Archimedean f-algebra Orth (A) of all orthomoTphisms of A.
The fact that the range of p in Corollary 4.3 is an f -algebra was first proved in [22, Corollary 3, p. 57] by Birkhoff and Pierce, while the fact that Orth (A) itself is an f-algebra has been proved in [18] by Bigard and Keimel and in [39] by Conrad and Diem. This topic was also discussed in great detail by de Pagter in his thesis [76, Proposition 12.1]. Note that if A is an f-algebra then A is semiprime if and only if p is one-to-one as a map from A into Orth (A). In this case, A and p (A) are isomorphic a.'! f-algebras. Also, if A is an f-algebra then A has a multiplicative identity if and only if the map p is one-to-one and onto as a map A ---> Orth (A), and consequently A and Orth (A) are isomorphic as f-algebra.'!.
5. Uniform completion and Dedekind completion Let A be an Archimedean i-algebra. The closure A1'U of A in its Dedekind completion A8 with respect to the uniform topology is a uniformly complete Riesz space. Using Quinn's Definition 2.12 in [79], A1'U is the uniform completion of A. The following theorem was obtained by n'iki in [91]. THEOREM 5.1. Let A be an Archimedean f-algebra (respectively, almost falgebra, d-algebra, f -algebra). Then the multiplication in A extends uniquely to a multiplication in A1'U such that A1'U is a uniformly complete i-algebra (respectively, almost f -algebra, d-algebra, f -algebra) with respect to this extended multiplication. Moreover, if A is semiprime (respectively, has a unit element e) then ATU is semiprime (respectively, has e as unit element).
We now turn our attention to the Dedekind completion of the i-algebra A. Johnson in his paper [64] proved that if A is an f-algebra (or even an Archimedean f-ring), then the multiplication in A extends uniquely to an f-algebra multiplication in A8. The uniqueness of such an extended multiplication in A8 of course arises from the order continuity of the multiplication in the f-algebra A. Alternative proofs of this extension can be found in [76, pp. 66-67] and [59, p. 166].
108
KARIM BOULABIAR. GERARD BUSKES. AND ABDELMAJID TRIKI
THEOREM 5.2. Let A be an Archimedean f -algebra. Then the multiplication in A extends uniquely to a multiplication in A" such that Ad 'is a Dedekind complete falgebra with respect to this extended multiplication. FUrthermore, if A is semiprime (respectively, A has a unit element e) then A" is semiprime (respectively. has e as unU element).
The corresponding results for (I-algebras in general, or almost f-algebras and d-algebras in particular, is much harder because of the absence of order continuity of the multiplication. Nonetheless, extensions of the multiplication to the Dedekind completion often exist, though such extensions are no longer necessarily unique. For almost f-algebras Buskes and van Rooij proved the following (see [35, Theorem 10]). THEOREM 5.3. Let A be an Archimedean almost f-algebra. Then the multiplication in A extends to a multiplication 'in A" such that A" is a Dedekind complete almost f -algebra with respect to that extended m'ultiplication.
Using the previous result as a starting point, Boulabiar and Chil in [28, Corollary 3] proved that from amongst the extensions provided, A" can be equipped with a commutative d-algebra multiplication whenever A is a commutative d-algebra. Then in [37, Theorem 7], Chil wa.c; able to drop the commutativity condition and prove the following theorem. THEOREM 5.4. Let A be an Archimedean d-algebra. Then the multiplication in A extends to a multiplication in Ad such that A" is a Dedekind complete d-algebra with respect to that extended multiplication.
In summary, all but one of the problems concerning Dedekind completions that Huijsmans raised in his survey paper [57] have now been solved. The remaining problem, though admittedly outside the scope of this survey, is the following. PROBLEM 5.5. Let A be an Ar'chimedean (I-algebra. Does the multiplication in A extend to a multiplication in A" so that A" is a Dedekind complete (I-algebra?
6. Powers in i-algebras Let A be a uniformly complete i-algebra and let P E lR+ [Xl, ... , Xn] be a homogeneous polynomial of degree a non zero natural number p. In their paper [16]' Beukers and Huijsmans considered the following problem: does there exist in A a 'p-th root' of P(iI, ... ,fn) for iI, ... ,fn in A+? They gave an affirmative answer in the case where A is a semiprime f-algebra. More precisely, they prowd the following theorem (see [16, Theorem 5]). THEOREM
6.1. Let A be a uniformly complete semiprime f-algebra and let
P E lR+ [Xl, ... , Xn] be a homogeneo1Ls polynomial of degree a non zero natural numbe1'p. Thenfo1' every fl, ... ,fn E A+ there exists a 'unique f E A+ such that fP = P (iI, ... , fn). As a consequence, one has the following corollary (see Corollary 6 in [16]). COROLLARY 6.2. Let A be a uniformly complete f -algebra with unit element and p E {I, 2, ... }. Then for each f E A +, there exists a unique 9 E A + such that
gl'
= f.
LATTICE ORDERED ALGEBRAS
109
Note that the previous result was first proved for p = 2 in [17, Theorem 4.2 and Cororllary 4.3] by Beukers, Huijsmans and de Pagter. Also, it should be noted that Theorem 6.1 above is proved alternatively by Buskes, de Pagter, and van Rooij in [31, Corollary 4.11], a paper that deals with a more general functional calculus on Riesz spaces and f-algebrar; to which we will return in the next section. The problem corresponding to Theorem 6.1 for almost f-algebras was considered by Boulabiar and follows next (see Theorem 3 in [24]). jR+
THEOREM 6.3. Let A be a uniformly complete almost f -algebra and let P E [Xl, ... , Xn] be a homogeneous polynomial of degree a natural number p. Then
for every II, ... , fn E A+ there exists a (not necessarily unique) f E A+ such that fP = P (II, ... , fn)·
Observe that, where roots are unique in semiprime f-algebras, this is no longer always the case for almost f-algebras. We illustrate this with an example. EXAMPLE 6.4. Let A = C ([-1,1]) be the uniformly complete Riesz space of all real-valued continuous functions on [-1, 1] and define w E A by
W(x)={ O-x
(-1::;x::;0); (0::;x::;1).
For every f, g E A, we put
(f. g) (x) =
{
w(x)f(x)g(x)
(-1::; x::; 0);
lO/(S)g(S)dS
(0::; x::; 1).
Then A is an almost f -algebra with respect to the multiplication.. h, g, Ct, {3 E A defined by g(x) =
Ixl
h (x)
Consider
= exp (x),
and Ct
(x)
= Jx 2 + exp (2x)
, and {3 (x)
for all x E [-1,1]' where X[O,I] (x) Then Ct • Ct
=1
= X.X[O,I] (x) + Jx 2 + exp (2x)
if x E [0,1] and X[O,I] (x)
=0
if x E [-1,0).
= {3 • {3 = 9 • g + h • h.
At this point, we define for each non zero natural number p, Ap =
{II ... fp : II, ... , fp
E A}.
In what follows, we will investigate the order structure as well as the algebra structure of Ap (since Al = A, we suppose that p ~ 2). The sets A2 and A3 were first considered in [35] by Buskes and van Rooij and then in great detail by Boulabiar in [24] from which we summarize the results in the following theorem (see Theorems 4, 5, and 6 in [24]). THEOREM 6.5. Let A be a uniformly complete almost f -algebra and let p ~ 3 be a natural number. Then Ap is a uniformly complete semiprime f -algebra under the ordering and multiplication inherited from A. The positive cone At of Ap is defined by
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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
The lattice operations I\p and V p in Ap are given by
fP I\p gP = (f 1\ g)P and the absolute value
1.l p
and
fP
Vp gP
= (f V g)P
for all 0 ~ f,g E A,
in Ap is defined by If Pip
=
Ifl P
for all f E A.
Contrary to Ap (p 2: 3), A2 need not be a Riesz space under the ordering inherited from A as is proved by the next example. EXAMPLE 6.6. Consider A = C ([0,1]) with the pO'intwise addition, scalar rrmltiplication and partial ordering. For f, 9 E A, define
l
(f _ g)(x) = {
(0 x - 1/2
~
x ~ 1/2);
0
f(s)g(s)ds (1/2 < x ~ 1). o Then A is a uniformly complete almost f -algebra under the multiplication - and h is an element of A2 if and only if h(x) = 0 for all x E [0,1/2] and the restriction of h to [1/2,1] belong to C 1 ([1/2,1]). Hence A2 is not a Riesz space under the order inherited from A.
The following example proves that though Ap (p 2: 3) is a Riesz space, in general it is not a Riesz subspace of A. EXAMPLE 6.7. Take A = C ([-1, 1]) with the pointwise addition, scalar multiplication and ordering, and define w EA· by
w (x)
=
{ -x 0'
(-l~x~O);
(0 ~ x ~ 1) .
For f, 9 E A, define
(f - g) (x) =
{
w(x)f(x)g(x)
(-l~x~O);
10/(S)g(S)ds
(O~x~l).
Clearly, A is a uniformly complete almost f -algebra under the multiplication _. Define 0 E A by o (x)
= 2x + 1
for all x E [-1, 1].
It follows that
10 - 0 - 01 (1) =
1/10
=110 - 0 - ob (1) = (101-101-101) (1) =
1/8.
If, however, A is a commutative d-algebra then some of the unpleasantness of the preceding example disappears. COROLLARY 6.8. Let A be a uniformly complete commutative d-algebra and p 2: 2 be a natural number. Then Ap is a uniformly complete f -subalgebra of A. If in addition p 2: 3 then Ap is semiprime.
In spite of the improvement in the conclusion of Corollary 6.8 over the conclusion for the more general situation of almost f-algebras, A2 still need not be semiprime. This is illustrated in the next example.
LATTICE ORDERED ALGEBRAS
111
EXAMPLE 6.9. Let A be the coordinatewise ordered vector space R3 with the multiplication defined by:
n; R}
Then A is a uniformly complete commutative d-algebm and
A,
~{(
X,Y E
Dbuiou,ly, A, is an f-""balg,bm of A. FUrth,""ore, (
~ ) ' ~ 0 and A, is not
semiprime. For f -algebras we have the following corollary. COROLLARY 6.10. Let A be a uniformly complete f -algebm and let p 2: 2 be a natuml number. Then Ap is a uniformly complete semiprime f -subalgebm of A.
There is a universal way in which A 2 , or more generally Ap for any p 2: 2 can be described. We provide the details of that description for A2 next (see [36]). Let E and F be Riesz spaces. A bilinear map q> : E x E ~ F is called orthosymmetric if whenever f I\g = 0 for f,g E E we have q>(f,g) = 0 (the notion of orthosymmetric bilinear map was introduced by Buskes and van Rooij in [34]). The bilinear map q> is a Riesz bimorphism if it is a Riesz homomorphism in each variable separately (more about Riesz bimorphisms can be found in [29]). Let E be a Riesz space. The pair (E8, 8) is called a square of E, if E8 is a Riesz space and if (1) 8: E x E ~ E8 is an orthosymmetric Riesz bimorphism, and (2) for every Riesz space F, whenever q> : E x E ~ F is an orthosymmetric Riesz bimorphism there exists a unique Riesz homomorphism q>8 : E8 ~ F such that q>8 08 = q>. The existence and uniqueness of squares for any Riesz space follows easily from the Riesz space tensor product as constructed by Fremlin in [43]. To understand the structure of the square of a Riesz space is best not done via this tensor product. The set A2 described above is often more helpful. The connection between semi prime f-algebras and squares of uniformly complete Riesz spaces is described in the next theorem. After reading that theorem the reader might feel like moving the lower index 2 in A2 to an upper index. THEOREM 6.11. Let E be a uniformly complete Riesz subspace of an Archimedean semiprime f -algebm G whose multiplication is indicated by a period e. Put E2 := {x e y : x,y E E} as before. Then E2 is a Riesz subspace of G and (E2,e) is a square of E.
7. Functional Calculus on f-algebras The theorem that we presented in Section 6 on the existence of p-th roots of homogeneous polynomials in f-algebras is a very special case of a rich functional calculus on uniformly complete f-algebras. The idea behind functional calculus
112
KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
for Riesz spaces in general is straightforward. For elementary functions on JRN one ought to be able to simply substitute elements of the Riesz space into these functions and get elements of the Riesz space as output. The idea of how to execute this substitution of elements in sufficiently simple functions essentially goes back to Yudin and in the form that we represent it to Lozanovsky [70]. The technical problem surmounts to what the class of sufficiently simple functions really looks like. Let n EN. We denote by 1i(JRN) the Riesz space of all continuous functions r.p : JRN ---+ JR for which
r.p(tx) = tr.p(x) for all x E JRN and all t ~ O. Let E be a Riesz space, r.p E 1i(JRN) and
iI, ... , fn
E E. We say that
r.p(iI, ... '/n) exists in E if there is an element 9 of E such that
w(g) = r.p(w(iI), ... ,w(fn)) for every real-valued Riesz homomorphism w on the Riesz subspace of E generated by iI, ... , f n, g. For any given E, r.p and iI, ... ,/n there exists at most one 9 with this property. This 9 is also indicated by
r.p(iI, ... , fn). In this situation we have the following theorem (see Lozanovsky [70]). THEOREM 7.1. Let E be a uniformly complete Riesz space and Then r.p(iI, ... , fn) exists fOT every r.p E 1i(JRN). The map
iI, ... , fn
E
E.
r.p(iI, ... , fn) (r.p E 1i(JRN)) is a Riesz homomorphism from 1i(JRN) into E. r.p
---+
Remark. In a way, r.p(iI, ... , fn) is independent of E. Indeed, if D is any Riesz subspace of E that is uniformly complete and contains iI, ... , fn then r.p(iI, ... , fn) relative to D means the same as r.p(iI, ... , fn) relative to E. In particular, every Riesz subspace of E that is uniformly complete and contains iI, ... , fn must also contain r.p(fl, ... , fn). By A(JR N) we denote the set of all continuous functions r.p : JRN ---+ JR that are of polynomial growth and for which limt!o rlr.p(tx) exists uniformly on bounded subsets of JRN (the latter condition is equivalent to the existence of a 't/J E 1i(JRN) such that r.p(x) = 't/J(x) + 0(11 x II) (x ---+ 0)). Observe that A(JRN) is an f-algebra. Let E be a semiprime f-algebra, r.p E A(JRN) and iI, ... , fn E E. We say that
r.p(iI, ... ,fn)exists in E if there is agE E with
w(g) = r.p(w(iI), ... ,w(fn)) for every real-valued multiplicative Riesz homomorphism w defined on the f-subalgebra of E generated by iI, ... , fn,g. There exists only one such g, which is then called r.p(iI, ... , fn). This definition is in accordance with the one we gave for 1i(JRN) if r.p E 1i(JRN). For 1i(JRN) we have the following theorem (see [31, Theorem 4.10]).
LATTICE ORDERED ALGEBRAS THEOREM
ft, ... , fn
113
7.2. Let E be a uniformly complete semiprime f-algebra and let
E E. Then cp(ft, ... , fn) exists for every cp E A(~JII). The map
cp-+cp(ft, ... ,fn) (cpEA(~JII)) is a multiplicative Riesz homomorphism from A(~JII) into E.
8. Relationships between i-algebra multiplications
Let A be an i-algebra with multiplication denoted by juxtaposition, and assume that A is equipped with another associative multiplication e. In the first theorem of this section, we present a relationship between the two multiplications in A, under the conditions that A is a unital f-algebra with respect to the initial multiplication and an (almost) f-algebra with respect to the other multiplication e. For proofs, see [38, Theorem 2.2] and [23, Theorem 5.2]. THEOREM 8.1. Let A be an Archimedean f -algebra with identity element e and assume that A is furnished with another associative multiplication e. Then (i) A is an f-algebra under e if and only if
feg=(eee)fg
for allf,g E A, and
(ii) A is an almost f-algebra under e if and only 'if f e9
= e e (I g)
for all
J, 9 E A.
The corresponding problem in the case where A is d-algebra with respect to e is rather more difficult. Indeed, since then e need not be commutative, one cannot write the product f e 9 as a function of the product fg (I, 9 E A). However, there exists another way (involving f, 9 and the initial multiplication in A) to express the product f e g. This is the subject of the next result. First recall that the maximal ring of quotients Q (A) of the Archimedean f -algebra A with unit element e is again an Archimedean J-algebra with the same e as multiplicative identity. Moreover, A is an f-subalgebra of Q (A), a fact proved by Anderson in [4] (see also the recent paper [71, Cororllary 2.7.1] by Martinez). For the definition of the maximal ring of quotients of a ring, the reader can consult e.g. [66]. The proof of the following theorem can be found in [25, Theorem 4.3]. THEOREM 8.2. Let A be an Archimedean f-algebra with 'identity element e and let e be another associat'ive multiplication in A. Then A is a d-algebra with respect to e if and only if there exist two algebra and Riesz homomorphisms cp and 'l/J from A into its maximal ring of quotients Q (A) such that
f e 9 = (e e e) cp (I) 'l/J (g)
for all f, 9 E A.
As mentioned at the end of Section 3, the two preceding theorems are abstract versions of the corresponding results, given in that section, for the C (X)-case. In the second part of this section, we are interested in A being a commutative dalgebra with respect to the initial multiplication rather than an f-algebra with unit element. However, we will impose the additional assumption that A is uniformly complete. Uniform completeness is not needed for all our results but we will use the set A2 and remind the reader of the special nature of that set under the extra condition of uniform completeness (see Corollary 6.8). If there exists a positive operator T OIl A2 such that
(T)
f e9 =
T (lg)
for all
f, 9 E
A
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KARIM BOULABIAR, GERARD BUSKES. AND ABDELMA.HD TRIKI
then e is an almost f-algebra multiplication and N2 (A) c N; (A), where
N; (A)
= {f
E A :
f e f = O} .
In what follows, we show in detail what happens if we assume that A with the initial multiplication is a (uniformly complete) commutative d-algebra, and the inclusion N2 (A) c N; (A) holds. Under those circumstances, we then relate a necessary and sufficient condition for the new multiplication to be an almost f-algebra, a d-algebra or an f-algebra to the existence of some posit.ive operator T satisfying the relation (T). The details follow in the next theorem, the proof of which can be found in [23, Theorems 5.4 and 5.5]. THEOREM 8.3. Let A be a uniformly complete commutative d-algebra and assume that A is an I!-algebra with respect to another associative multiplication e such that p = 0 implies f e f = O. Then the following statements hold.
(i) A is an almost f -algebra under
e if and only if ther-e exists a positive operator T from A2 into A such that
feg=T(fg)
forallf,gEA.
(ii) A is a commutative d-algebra under
e if and only if there exists a Riesz homomorphism T from A2 into A such that
f e 9 = T (fg)
for all f, 9 E A.
(iii) A is an f-algebra under e if and only if there exists an operator T from A2 into A such that To 7rf E Ort.h (A) for all f E A+, where 7rf (g) = fg for all 9 E A, and f e9
=T
(f g)
for all f, 9 E A.
We remark that A2 in the previous theorem is a Riesz space (see Section 6). Next, we produce an example which shows that in Theorem 8.3 above, the hypothesis 'A is a commutative d-algebra' cannot be replaced by 'A is an almost f-algebra'. EXAMPLE 8.4. Take A = C ([-1,1]) with the usual operations and order and define a, (3 E A by
-(4x+l) a(x)= { 0 4x -1
(-I~x~-1/4); ~ x ~ 1/4); (1/4 ~ x ~ 1)
(-1/4
and (3 (x) = {
~4x + 1
(-1 ~x~ 1/4); (1/4 ~ x ~ 1).
For f, 9 E A, define a(x)f(x)g(x) (f x g)(x) =
{
(-1~x~1/4); ~ 3/4);
(1/4 ~ x °J(X-3/4)
f(s)g(s)ds
(3/4~x~1)
-(x-3/4)
and (feg) (x) =f3(x)f(x)g(x)
LATTICE ORDERED ALGEBRAS
115
for all x E [-1,1]. Then A is an almost f-algebra (respectively, an f-algebra) with respect to the multiplication x (respectively, .). It follows that N; (A)
= N 2x
(A)
= {f
E A : f (x)
= 0 for
all x E [-1, I/4]}.
Consider B which is simultaneously an algebra isomorphism, such that
T (f) = wS (f)
for all lEA.
Very recently in [30J, Boulabiar, Buskes, and Henriksen extended the latter result to all order bounded linear biseparating maps on arbitrary (not necessarily closed under inversion) unital I-algebras over the reals as well as over the complex numbers (for the theory of complex I-algebras, we refer to [17]). The theorem they obtained is the following. THEOREM 15.1. Let A and B be (real or complex) I-algebras with unit elements, and let T : A ---> B be an order bounded linear biseparating map. Then T is a weighted isomorph'ism.
In the previously mentioned memoir by Abramovich and Kitover, we find the following theorem. A d-isomorphism between two uniformly complete Riesz spaces A and Jyf is automatically order bounded as soon as every universally a-complete projection band in A is essentially one-dimensional (see [1, Corollary 15.3]). This theorem is used in [30], under the same condit.ions on A, to show that every linear biseparating map between two uniformly complete I-algebras A and B is a weighted isomorphism. We point out that a complex I-algebra is by definition uniformly complete. Thus the phrase 'uniformly complete unital I-algebra' is understood to mean either a uniformly complete unital I -algebra over the reals, or simply a unital I-algebra over the complex numbers. For t.he proof of the next theorem, see Proposition 5.1 and Theorem 5.2 in [30J. THEOREM 15.2. Let A and B be unilormly complete unital I -algebras and assume that every universally a-complete projection band in A is essentially onedimensional. Then every linear biseparating map from A onto B is order bounded and then a weighted 'isomorphism.
To obtain the result by Araujo, Beckenstein and Narici cited above as a consequence of the preceding theorem, Boulabiar, Buskes and Henriksen proved t.hat
LATTICE ORDERED ALGEBRAS
129
every algebra of all scalar-valued continuous functions on a completely regular topological space has the property that every universally a-complete projection band is essentially one-dimensional (see [30, Theorem 5.5]). THEOREM 15.3. Let X be a completely regular topological space X. Then every universally a-complete projection band in the Riesz space C (X) is essentially onedimensional.
Combining Theorems 15.2 and 15.3, we arrive at the next result. COROLLARY 15.4. Let X and Y be completely regular topological spaces. Then every biseparating linear map T : C (X) -> C (Y) is a weighted isomorphism. In particular, C (X) and C (Y) are isomorphic as f -algebras if and only if there exists a linear biseparating map from C (X) onto C (Y).
It is well known that if X and Yare completely regular topological spaces and S is an isomorphism from C (X) into C (Y) then there exists an homeomorphism h from vY into vX such that S (f) = f 0 h, where vX and vY denote the realcompactifications of X and Y, respectively (see Section 10 in [47]). The latter fact, together with Corollary 15.4 above, directly leads to the next corollary, which was proved earlier in an alternative way by Araujo, Beckenstein and Narici in [5, Proposition 3]. COROLLARY 15.5. Let X and Y be completely regular topological. Then for every l'inear biseparating map T : C (X) -> C (Y) there exist a non-vanishing function wE C (Y) and an homeomorphism h : vY -> vX such that
T (f) (y) = w (y)
f
(h (y))
for all f E C(X) and y E Y.
It is shown in [47, Theorem 8.3] that two realcompact X and Yare homeomorphic if and only if C (X) and C (Y) are isomorphic as f-algebras. Another classical result of rings of continuous functions theory is that if X is a completely regular topological space then C (X) and C (vX) are isomorphic as f-algebras (see Remark 8 (a) in [47]). It follows immediately that if X and Y are two completely regular topological spaces, and C (X) and C (Y) are isomorphic as f-algebras, then vX and v Yare homeomorphic. The latter implies that, without further assumptions, the conclusion that vX is homeomorphic to vY in Corollary 15.5 is best possible. Under additional assumptions, however, X and Y may themselves be homeomorphic as is shown in the next Corollary. COROLLARY 15.6. Let X and Y be completely regular topological spaces and assume either (i) or (ii) below. (i) X and Yare realcompact. (ii) The points of X, as well as those ofY, are Go-points. If there exists a linear biseparating map from C (X) onto C (Y) then X and Yare homeomorphic.
Under the condition (i), the result in Corollary 15.6 above follows straightforwardly from Corollary 15.5 while under the condition (ii), it stems directly from [73] by Misra. Finally, it should be noted that the result of Theorem 15.2 above is false if the universally a-complete projection bands in A fail to be essentially one-dimensional. Indeed, Example 7.13 in [1] produces a linear biseparating map T on the Dedekind
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KARIM BOULABIAR, GERARD BUSKES, AND ABDELMAJID TRIKI
complete (and then uniformly complete) unital i-algebra Lo ([0, 1]) of all equivalence classes of measurable functions on [0,1)' which is not order bounded and thus cannot be a weighted isomorphism.
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[61] Huijsmans, C. B. and B. de Pagter, Averaging operators and positive contractive projections, J. Math. Ana. Appl., 113 (1986), 163-184. [62] Ionesco Tulcea, A. and C. Ionesco Tulcea, On the lifting property I, J. Math. Anal. Appl., 3 (1961), 537-546. [63] Jarosz, K., Automatic continuity of separating linear isomorphisms, Bull. Canadian Math. Soc., 33 (1990), 139-144. [64] Johnson, D. G., The completion of an archimedean I-ring, J. London Math. Soc, 40 (1965), 493-493. [65] Kudhicek, V., On some types of R-rings. Sb. Vysoke. Ucen{ Tech. Bmo 1-2 (1962), 179-181. [66] Lambek, J. Lectures on Rings and Modules, Blaisdell, 1966. [67] Luxemburg, W. A. J., Some Aspects 01 the Theory 01 Riesz Spaces, Univ. Arkansas Lecture Notes Math. 4, Fayetteville, 1979 [68] Luxemburg, W. A. J., A remark on a paper by D. Feldman and M. Henriksen concerning the definition of I-rings, Nederl. Akad. Wetensch. Indag. Math., 50 (1998), 127-130. [69] Luxemburg, W. A. J. and A. C. Zaanen, Riesz spaces I, North-Holland, Amsterdam, 1971. [70] Lozanovsky, G., The functions of elements of vector lattices, Izv. Vyssh. Uchebn. Zaved. Mat., 4 (1973), 45-54. [71] Martinez, J. The maximal ring of quotient I-ring. Algebra Univ .• 33 (1995), 355-369. [72] Meyer-Nieberg, P., Banach Lattices, Springer Verlag, Berlin Heidelberg New York, 1991. [73] Misra, P. R., On isomorphism theorems for C(X), Acta. Math. Acad. Sci. Hungar., 39 (1982), 179-180. [74] Nagasawa, M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Semin. Rep., 11 (1959), 182-188. [75] Nakano, H., Modern Spectral Theory, Tokyo Math. Book Series II, Maruzen, Tokyo, 1950. [76] de Pagter, B., I-algebras and orthomorphisms (Thesis, Leiden, 1981). [77] Phelps, R. R., Extremal operators and homomorphisms, Trans. Amer. Math. Soc., 108 (1963), 265-274. [78] van Putten, B., Disjunctive linear operators and partial multiplication in Riesz spaces, (Thesis, Wageningen, 1980). [79] Quinn, J., Intermediate Riesz spaces, Pacific J. Math., 56 (1975), 225-263. [80] Scheffold, E., FF-Banachverbandsalgebren, Math. Z., 177 (1981),193-205. [81] Scheffold, E., Der Bidual von F-Banachverbandsalgebren, Acta Sci. Math., 55 (1991), 167179. [82] Scheffold, E., Uber Bimorphismen und das Arens-Produkt bei kommutativen DBanachverbandsalgebren, Preprint. [83] Scheffold, E., Uber den ordnungsstetigen Bidual von FF-Banachverbandsalgebren, Arch. Math., 60 (1993), 473-477. [84] Steinberg, S. A., Quotient rings of a class of lattice-ordered rings, Can. J. Math., 25 (1973), 627-645. [85] Steinberg, Stuart A. On the unitability of a class of partially ordered rings that have squares positive. J. Algebra 100 (1986), no. 2, 325-343. [86] Steinberg, Stuart A. On lattice-ordered algebras that satisfy polynomial identities. Ordered algebraic structures (Cincinnati, Ohio, 1982), 179-187, Lecture Notes in Pure and Appl. Math., 99, Dekker, New York, 1985. [87] Steinberg, Stuart A. Unital $l$-prime lattice-ordered rings with polynomial constraints are domains. Trans. Amer. Math. Soc. 276 (1983), no. 1, 145-164. [88] Steinberg, Stuart A. On lattice-ordered rings in which the square of every element is positive. J. Austral. Math. Soc. Ser. A 22 (1976), no. 3, 362-370. [89] Toumi, M. A., On some I-subalgebras of d-algebras, To appear in Math. Reports [90] Triki, A., Stable I-algebras, Algebra Univ., 44 (2000) 65-86. [91] Triki, A., On algebra homomorphisms in complex almost I-algebras, Comment. Math. Univ. Carolinae, 43 (2002), 23-31. [92] Zaanen, A. C., Riesz Spaces II, North Holland, Amsterdam-New York-Oxford, 1983. [93] Zaanen, A. C., Introduction to Operator Theory in Riesz Spaces, Springer Verlag, Berlin, 1997. [94] Zelazko, W., Banach algebras, Elsevier, Amsterdam-London-New York, 1973.
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DEPARTEMENT DU CYCLE AGREGATIF, INSTITUT PREPARATOIRE AUX ETUDES SCIENTIFIQUES ET TECHNIQUES, UNIVERSITE DU 7 NOVEMBRE
A CARTHAGE,
BP 51, 2070-LA MARSA, TUNISIA
E-mail address:karim.boulabiar«lipest.rnu.tn DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MISSISSIPPI, UNIVERSITY, MS-38677, USA
E-mail address: mmbuskes«lsunset. olemiss. edu DEPARTEMENT DE MATHEMATIQUES, FACULTE DES SCIENCES DE TUNIS, UNIVERSITE TUNIS EL MANAR, 1060-TUNIS, TUNISIA
E-mail address:abdelmajid.triki«lfst.rnu.tn
Contemporary Mathematics Volume 328, 2003
An extension of a theorem of Wermer, Bernard, Sidney and Hatori to algebras of functions on locally compact spaces Eggert Briem
ABSTRACT. It follows from a theorem of J. Wermer that if A is a uniformly closed algebra of continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space X, the property that b2 E Re A for all bE Re A implies A = Co(X). In other words, if the function h(t) = t 2 operates by composition on ReA then A = Co(X). This result was generalized by O. Hatori. He proved that in place of the function h(t) = t 2 one can put any real-valued function defined in a neighbourhood of 0, thus extending a similar result for the compact case. Here a simple alternative proof is given for the locally compact case.
1. Introduction
Let X be a compact Hausdorff space and B a uniformly closed subspace of CIR(X), the space of all continuous real-valued functions on X, which separates the points of X and contains the constant functions. A version of the Stone-Weierstrass theorem says that if b2 E B for all b E B then B = CIR(X). Clearly this result does not hold if, instead of assuming that B is uniformly closed, one assumes that B is a Banach space in some norm which dominates the sup-norm, as the example of any non-trivial real Banach function algebra shows. However, if B is the real part of a uniform algebra, a theorem of J. Wermer says that if b2 E B for all b E B then
B = CIR(X). Let us say that a real-valued function h, defined on an interval I of the real line, E B whenever b E Band b maps X into I. Thus b2 E B for all bE B means that h(t) = t 2 operates on B. The Stone-Weierstrass theorem W8.', generalized by K. de Leeuw and Y. Katznelson (see [4], theorem 4.21), they showed that h(t) = t 2 can be replaced by any continuous non-affine function (Le. a function not ofthe type h(t) = o:t+!3) defined on an interval, and the theorem of J. Wermer was similarily generalized by A. Bernard, S. Sidney and O. Hatori [1], [5] and [8]. Here one can even do without the continuity assumption, a function operating on the real part of a uniform algebra of infinite dimension is automatically continuous.
operates on B if hob
1991 Mathematics Subject Classification. Primary 46JlOj Secondary 46E15. Key words and phmses. uniform algebra, locally compact space, functional calculus. © 2003 American Mathematical Society
135
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EGGERT BRIEM
In the case where X is locally compact, the functional calculus for a uniformly closed subspace of Co(X, JR), the space of all continuous real-valued functions vanishing at infinity on X, may be non-trivial (cf. [2]). What then about the real part of a uniformly closed subalgebra of Co(X), the space of all continuous complex-valued functions vanishing at infinity on X? Wermer's theorem, [10], clearly applies to this situation. It turns out that the functional calculus for the real part of a uniformly closed subalgebra, which is not a C* -algebra, of Co(X) is trivial. This result, which is due to O. Hatori, is to be found in [7]. To prove this one has to do more than just adapt the proofs for the compact case to the locally compact situation. The purpose of this paper is to give a simple alternative proof using antisymmetric decomposition of uniform algebras. 2. Proofs and results Let X be a locally compact Hausdorff space and let Xl denote the one-point compactification of X, with Xoo denoting the point at infinity. The functions in Co(X) have a natural extension to functions in C(Xl)' the space of all continuous complex-valued functions on Xl. If A is a uniformly closed subalgebra of Co(X), that separates the points of X and does not vanish identically at any point of X, then Al = AEI1C is a uniformly closed subalgebra of C(Xd that separates the points of Xl and contains the contant functions and A = {a E All a(x oo ) = O}. Thus, we can assume that there is a uniformly closed subalgebra Al of C(Xd containing the constant functions, where Xl is compact, and a point Xoo in Xl, such that A is obtained by restricting the functions in the set {a E Al Ia(x co ) = O} to the set X = Xl \ {xoo}. We need some results from the theory of uniform algebras. A subset E of Xl is a set of antisymmetry for Al if the only functions in Al that are real-valued on E are constant functions. Maximal sets of antisymmetry are closed generalized peak sets (intersections of peak sets), they form a partition of Xl and a continuous complexvalued function f on Xl is in Al if and only if its restriction to each maximal set of antisymmetry E is in the space AIlE of restrictions of functions in Al to E. Thus Al(Xd = C(Xd if and only if each maximal set of antisymmetry is a singleton. Also, AIlE is uniformly closed for each maximal set of antisymmetry, there is actually for each a in AIlE an element a' in Al with a' = a on E and II a' 1100=11 a llco,(E). (The latter norm is the supnorm w.r.t E). This material can f. ex. be found in [3]. If Al is the disc algebra on the closed unit disc in the complex plane and A is the algebra of functions vanishing at the origin, then any non-zero function in A takes real values of opposite signs in any neighbourhood of O. Thus any real-valued function, defined on an interval which is not a neighbourhood of 0, operates trivially on Re A. We shall therefore always assume that the interval on which an operating function is defined is a neighbourhood of O. We can now state the main result of this note. THEOREM 1. Let A be a uniformly closed subalgebra of Co(X) which separates the points of X and does not vanish identically at any point in X and let Re A be the space of real parts of functions in A. If Re A has a non-affine operating function h: I ........ JR where I, an interval, is a neighbourhood of 0, then Re A = Co(X, JR) and thus also A = Co(X).
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137
The first step in the proof of Theorem 1 is to show that operating functions are continuous. To prove this we need the following lemma. LEMMA 1. Let E be a maximal set of antisymmetry for A1 and let b E Re A. Then b(E) is either an interval or a singleton.
Proof. Suppose b(E) is not connected, and that b = Rea for some a in A 1. Then we can find two disjoint closed rectangles Ro and R1 in the complex plane such that a(E) is a subset of Ro u R1 but not a subset of anyone of the two rectangles. By Runge's theorem there is sequence of polynomials (Pn) converging uniformly to o on Ro and to 1 on R 1. Since AdE is uniformly closed, the sequence (Pn 0 a) converges on E to an element of A1IE. But the sequence converges on E to a function which takes only the values 0 and 1, contradicting the fact that E is a set of antisymmetry. 0 LEMMA 2. Let E be a maximal set of antisymmetry for A 1 such that E\ {xoo } =I- 0. Then there is an Xo E E such that for every to E I (resp. to in the interior of I), there exists bo E ReA with bo(xo) = to and bo(X) C I (resp. bo(X) is a subset of the interior of I).
Proof. If to = 0, the existence of the desired function is easily proved. We consider the case where to > O. (The proof for the case to < 0 is similar.) Let Xo be a generalized peak point in E, other than x oo , for A 1 • Simple calculations show that we can take 1 E A with I(xo) = 11/1100 = 1. Let c: be a positive number such that (-2c:, to] C I (resp. (-2c:, to] is a subset of the interior of I), and R the closed rectangle with corners -c: ± i and to ± i. Let
' E JR+ s.t. lui::::; >'Iel}. If b ERe A and b + ie E A then be and be(b2 - e2 ) belong to B(e). Let also
M(e) = {g
E
Co(X, JR) Ig. cl(B(e))
~
cl(B(e))},
a subalgebra of Co(X, JR). We note that since h is continuous h also operates on
cl(ReA).
ALGEBRAS ON LOCALLY COMPACT SPACES
139
For bl ERe A, for u E B(e), r, t E lR and
.. = A. By assumption there is an element {b A } in lOO(ReA) such that we have the inequality
I{fA} - {bA}1 < 1/2 on x and hence also on some open subset U of {3(A x Xt} containing X. By a simple calculation we have
Xo = {p E (3(A x Xl) I I{f}(p) - f(xo)1
:=; ~
Vf E CIR(Xl )}.
Thus there is a function fo in CIR(Xt} such that the set
{p
E
{3(A x Xt}
I I{fo}(p) -
fo(x)1
:=; 1/2}
is a subset of U. Put
Kx = {y E Xd Ifo(Y) - fo(x)1 :=; 1/2}, a compact neighbourhood of x. Then A x Kx is contained in U so that IfA - bA I :=; 1/2 on Kx for all A in A. Let M = SUPA II bA II, a finite quantity because (b A ) is in lOO(Re A). Take any f in Co(X, JR.) with II f lloo:=; 1. By induction we construct a sequence (b n ) of elements from ReA, with II bn II:=; M for all n, such that n-l
12n( f - L
2- i
bi ) - bnl < ~
i=O
on Kx. The function b =
E:o 2- b is in Band b = f on Kx. 0 i i
PROPOSITION 3. Let Xo E X and suppose that there is an open neighbourhood Xo of Xo such that every function in Co(X,JR.), which vanishes outside Xo can be unifromly approximated on X by elements from ReA. Then lOO(ReA) sepamtes the points of xo.
Proof. Suppose p,q are in xo. There is an element {fA} in lOO(Co(X,JR.», where each 1>.. vanishes outside X o, such that {fA}(P) = 0 and {fA}(q) = 1. By assumption we can for each A E A find a function bA in ReA with II fA - bA lloo:=; ~. Thus we have {bA}(p) :=; ~ and {bA}(q) ~ ~ since II{fA} - {bA}lIoo,{3(Axxt} :=; ~. Let a A = bA + iC A be in A for each A. Now, the A-net {e a ... - 1} is in lOO(A) and it separates p from q since {e an } does. Since
I{e a ... }(p)1 =
{e b... }(p)
:=; e l / 4
and I{e a ... }(q)1 = {e b... }(q) ~ e3 / 4
,
we see that lOO(A) separates p from q and thus lOO(ReA) does so as well. 0
EGGERT BRIEM
142
We also need the aforementioned extension of the Stone-Weierstrass theorem due to de Leeuw and Katznelson (see [4], Theorem 4.21). We can prove a version of this result, Proposition 4, in the same way as the original one, the proof is omitted. PROPOSITION 4. Let Y be a compact Hausdorff space and B a subspace ofCR(Y) which separates the points of Y and contains the constant functions. Suppose that B is also a normed space with the norm II· liB which dominates the uniform norm. Let h be a continuous function defined on an interval I. Suppose that h is nonaffine in every neighbourhood of an interior point to in 1. Suppose that there exists a positive real number 8 > 0 with (to - 8, to + 8) ~ I such that h 0 (to + u) E B for every u E B with Ilull < 8. Then B is uniformly dense in CIR(Y).
Proof of Theorem 1. We are going to use the local version of Bernard's Lemma. We thus have to show that the conditions stated there are satisfied. For this we use the result of de Leeuw and Katznelson above. The main obstacle is that we can not conclude that if {b.>.} is in [00 (Re A) then h e {b.>.} is in [00 (Re A) although the composite function is defined, i.e. we can not conclude that composition with h maps bounded nets to bounded nets. To overcome this difficulty we use a method of Sidney, [8], to obtain local boundedness for composition with h. Suppose A =I- Co(X) and thus also Al =I- C(X 1 ). Let E be a maximal set of antisymmetry for Al containing more than one point. Let further to be an interior point of I such that h is not affine in any neighbourhood of to and let bo be a function in Re A which maps X into the interior I for which to is an interior point of bo(E). By Lemma 2 such a function exists. We now choose E > 0 such that if b is in the E-ball, (ReA)., of ReA then ho(bo+b) is defined and to is an interior point of (b o + b)(E). We write (ReA).
= U{b E (ReA). I I h 0 (bo + b) II::; n}. n
The Baire Category Theorem shows that the closure of one of the sets on the right hand side has an interior point and thus there is a function b1 E Re A and positive numbers 8, lvI, with I b1 I +8 < E, such that
I he (bo +b 1 +c) II::; lvI for c in a dense subset of the 8-ball of Re A. Let b = bo + b1 , and let [00 (Re A) be as in the local version of Bernard's Lemma. We then have
h 0 {b + c.>.} E cl([OO(Re A)), the uniform closure of [OO(ReA), if {c.>.} E [OO(ReA) and such that b(xo) = to, an interior point of b(E). We restrict to Xo and deduce that
h 0 {to
I
{c.>.}
11< 8.
Take Xo E E
+ c.>.} E cl([OO(ReA)lxo),
for every {c.>.} E [OO(ReA)lxo whose quotient norm satisfies II {c.>.} 11< 8. Since {c} is constant on xo for any c E ReA, the space [OO(ReA)lxo contains the constant functions, it also separates the points of Xo by Proposition 2 and 3. Then Proposition 4 shows that [OO(ReA)lxo is dense in CIR(xo) and thus, by the local version of Bernard's Lemma, ReAIK = CIR(K) for some compact neighbourhood K of Xo. The theorem of Sidney and Stout, [9], then shows that AIK = C(K). By Proposition 2, there exists an open neighbourhood Xo of Xo such that every f E Co(X, 1R) which vanishes outside Xo can be approximated uniformly on X by
ALGEBRAS ON LOCALLY COMPACT SPACES
143
elements of ReA. We may assume K c Xo. Let Kl be a compact neighbourhood of Xo such that K 1 is in the interior of K. Since E is antisymmetric and contains more than one point, EnK l contains more than one point. On the other hand, we will show that {f E Co(X, IR) II = 0 outside Kd CAl. It will follow that En Kl = {xo}, which will be a contradiction proving A = Co(X, IR). Let 1 E Co(X, IR) with 1 = 0 outside K 1 . A function u E Co(X, IR) with lui::; 1 on X, u = 1 on K l , and u = 0 outside K can be uniformly approximated by functions in ReA. Thus, for every positive integer n, there exists bn E ReA such that 1 - ,& ::; bn ::; 1 on K 1 and bn ::; ,& outside K. Without loss of generality we may assume that Ibnl ::; 1 on X. Take Cn E A with Recn = bn and put an = (eCn-l)n. Then an E AI, e-1. ::; lanl ::; Ion Kl and lanl ::; e- n+1. outside K. Since AllK = C(K) and thus AllKl = C(Kd, there exists a positive real number M such that for every positive integer n there exists gn E Al such that gnan = 1 on Kl with IIgnlloo ::; M. Since AllK = C(K), there is a function af E Al with af = 1 on K. Then afgnan E Al and by a simple calculation lIafgnan - 11100 --+ 0 as n --+ 00, that is, 1 E AI' 0 References [lJ A. Bernard, Espaces des parties relies des Iments d'une algebre de Banach de fonctions, J. Funct. Anal. 10 (1972), 387-409. [2J E. Briem, Approximations from Subspaces of Co(X), J. Approx. Theory 112 (2001), 279-294. [3J A. Browder, Introduction to Function Algebras, W. A. Benjamin, Inc. !969. [4J R.B. Burckel, Characterizations of C(X) among its subalgebras (Lecture Notes in Pure and Appl. Math. 6). Marcel Dekker, New York 1972. [5J O. Hatori, Functions which operate on the real part of a uniform algebra, Proc. Amer. Math. Soc. 83 (1981), 565-568. [6J O. Hatori, Separation properties and operating functions on a space of continuous functions, lntemat. J. Math. 4 (1993), 551-600. [7J O. hatori, Range transformations on a Banach function algebra. IV, Proc. Amer. Math. Soc. 116 (1992), 149-156. [8J S. J. Sidney, Functions which operate on the real part of a uniform algebra, Pac. J. Math. 80 (1979), 265-272. [9J S. J. Sidney and E. L. Stout, A note on interpolation, Proc. Amer. Math. Soc. 19 (1968), 380-382. [10J J. Wermer, The space of real parts of a function algebra, Pac. J. Math. 13 (1963), 1423-1426. SCIENCE INSTITUTE, UNIVERSITY OF ICELAND. REYKJAVIK, ICELAND
E-mail address:briemillhi.is
Contemporary Mathematics Volume 328, 2003
Some mapping properties of p-summing operators with Hilbertian domain Qingying Bu
ABSTRACT. We prove that if H is a Hilbert space, Y is a Banach space and u : H ---+ Y is absolutely p-summing for some p :::: 1, then for any 1 < q < 00, u takes absolutely q-summable sequences in H into members of iq®Y, the projective tensor product of lq and Y.
Given a real or complex Banach space X and 1 ::; p < 00, we denote by and f;eak(X) the Banach spaces of sequences in X with norms II(xn)nlle;trong(X) = 11(llxnll)nllep and II(xn)nlle;eak(X) = sUP",*eB x * II(x*xn)nllep, respectively (cf. [4, pp. 32-36]). For 1 < p < 00, let fp(X) denote the space of all (strongly p-summable) sequences in X such that E~=l Ix~(xn)1 < 00 for each (X~)n E f't},eak(x*), normed by f~trong(x)
II(xn)nlltp(X) = sup
{I ~ x~(xn)1
:
II(x~)nlle~~eak(X.) ::; 1} ,
where pI is the conjugate of p, i.e., lip + 11pl = 1. With this norm fp(X) is a Banach space (cf. [1, 3]). Note: In [2] it was shown that fp(X) is exactly fp®X, the projective tensor product of fp and X. In this note we use this identification of fp(X) with fp®X to deduce a surprising mapping property of absolutely p-summing operators that have a Hilbert space domain. While the main result of this note can be derived from some by-now famous results of Kwapien, it was discovered because of the identification of fp(X) with fp®X, moreover, this identification leads itself to a proof that is a clean and clear application of Khinchin's inequality, Kahane's inequality, and Pietsch's Domination theorem - all fundamental aspects of the theory of p-summing operators. From the definitions, we have for 1 < p < 00, fp(X) ~ f;trong(x) ~ f;eak(x), and 11·lIe~..ak(X) ::; 1I'lIl~trong(X) ::; 11·llep(x)· Moreover, in case dimX = 00, all the containments are proper. For Banach spaces X and Y and a continuous linear operator 'U : X ---+ Y, define ft. : XN ---+ yN by (xn)n 1---+ (uxn)n. Then ft. is a linear operator. Thanks 2000 Mathematics Subject Classification. 46B28. © 145
2003 American Mathematical Society
146
QINGYING BU
to the Closed Graph Theorem, each of
it: f;eak(x)
---->
f;eak(y);
it:
f~trong(x) ----> f~trong(y);
it: fp(X)
---->
fp(Y)
is a continuous operator with
Ilitll£;:,eak(x)_e;:,eak(Y) = Ilitll£~trong(x)_e~trong(y) =
Ilitllep(x)-£p(Y) = Ilull·
We should mention here Khinchin's inequality (cf. [4, p. 10]) and Kahane's inequality (cf. [4, p. 211]) each of which plays a critical role in this paper. Let rn(t) denote the Rademacher functions (cf. [4, p. 10]), namely, rn : [O,IJ ----> ~, n E Pi! defined by rn(t) := sign(sin2n7rt).
< p < 00, there are positive constants Ap, Bp such that for any scalars a1, a2, ... ,an, we have
Khinchin's Inequality. For any 0
Kahane's Inequality. If 0 < p, q < which
([II t, r,('lx, I ,d')
1/, ,;
00,
then there is a constant Kp,q > 0 for
K",' ([II
t,
r,('lx,lI' dt )
1/,
regardless of the choice of a Banach space X and of finitely m.any vectors Xl, X2, ... , Xn from.X.
In 1967, A. Pietsch [5J introduced p-summing operators between Banach spaces, namely, a Banach space operator u : X ----> Y is called p-summing operator, 1 :::; p < 00, if it takes f~eak(x) into f~trong(y). Let IIp(X, Y) denote the space of all psumming operators from a Banach space X to a Banach space Y. If u : X ----> Y is a p-summing operator then, thanks to the Closed Graph Theorem, it : f~eak(x) ----> f~trong(y) is continuous. So we define the p-summing norm 7rp(u) on IIp (X, Y) to be 7rp (u) = Ilitll£;:,eak(x)_£~trong(y). With this norm IIp (X, Y) is a Banach space. There is an equivalent definition of p-summing operators, namely, a Banach space operator u : X ----> Y is p-summing if and only if there is a constant c > 0 such that for any XI,X2,'" ,Xn E X, (1)
In this case, 7rp (u)
= inf{c
> 0: for all possible c in (I)}.
In 1973, J. S. Cohen [3J introduced strongly p-summing operators between Banach spaces, namely, a Banach space operator u : X ----> Y is called strongly p-summing operator, 1 < p < 00, if it takes f~trong(x) into fp(Y). Let Dp(X, Y) denote the space of all strongly p-summing operators from a Banach space X to a Banach space Y. If u : X ----> Y is a strongly p-summing operator then, thanks to the Closed Graph Theorem, it : f~trong(x) ----> fp(Y) is continuous. So we define a
PROPERTIES OF p-SUMMING OPERATORS
147
strongly p-summing norm Dp(u) on Dp(X, Y) to be Dp(u) = lIulll~trong(X)-+lp(Y)' With this norm Dp(X, Y) is a Banach space. There is an equivalent definition of strongly p-summing operators, namely, a Banach space operator U ': X ~ Y is strongly p-summing if and only if there is a constant c > 0 such that for any Xll X2,'" , Xn E X, and any yi, Y2,'" , y~ E Y*,
(2)
In this case, Dp(u) = inf{c > 0: for all possible c in (2)}.
Note: Actually, U E Dp(X, Y) means that u takes absolutely p-summable sequences in X into members of ip®Y.
Main Theorem. Let 1 < p, q < 00; and let H be a Hilbert space and Y be a Banach space. Then IIp(H, Y) ~ Dq(H, Y), i.e., if u : H ~ Y is absolutely p-summing, then u takes absolutely q-summable sequences in H into members of
lq®Y.
PROOF. First consider H = i2 for n E N. Let u E IIp(l2' Y). By Pietsch's Domination Theorem [4, p. 44], there is a regular probability measure J.L on Bl'.J: such that for any x E i 2, lip (
Iluxll ::; 1Tp (U)' Lt'.J: I(x, z)IP dJ.L(z) Now for
Xl,X2,'"
,Xm
Ei
)
(3)
2, and Yi'Y2"" ,Y;" E Y*, we have n
Xk =
~Xk ·e· L.-J ,'&"',
k = 1,2,··· ,m.
i=l
Then m
m
L
I(UXk,
yZ)1
n
L I(Lxk,iuei, yZ)1 k=l
k=l
<
.. is unique, for each x E K \ (FI U F 2 ) but independent of y and z, then FI and F2 are said to be parallel faces; parallel faces are automatically norm closed. If in addition y and z are unique then FI and F2 are called split faces. See, for example, [1, 2] for the general theory of compact convex sets and related topics. Let K be a compact convex set. Recall that the facial topology on 8K is given by defining
{F n 8K : F is a closed split face of K} to be the family of all closed sets. The facial topology is weaker than the relative topology on 8K. The centre Z(A(K)) of A(K) is the set of all those functions in A(K) whose restriction to 8K is facially continuous. The central functions h E Z(A(K)) are characterised by the following property (see, for example, [1, Corollary 11.7.4] or [2, Theorem 3.1.4]): for all f E A(K), there exists 9 E A(K) such that g(x) = h(x)f(x) for all x in 8K. The uniqueness of the continuous affine function 9 is clear, since a continuous affine function on a compact convex set is completely determined by its values on the extreme boundary, and consequently we may write 9 = h . f. In this way it is useful to think of the central functions as the multipliers of A(K). A compact convex set K is a Choquet simplex whenever for all bounded (real) linear functionals cp in A(K)* and 0 > 0 the set K n (cp + oK) is either empty or of the form 'IjJ + (3K for some '¢ in A(K)* and {3 ~ 0 (see, for example, [12]); note that {3 = 0 allows K n (cp + oK) to be a singleton. In a Choquet simplex every closed face is split (see [2, Theorem 2.7.2]).
THE UNIQUE DECOMPOSITION PROPERTY AND BANACH-STONE THEOREM
153
In [6, 7, 8], Ellis defined the unique decomposition property of K by the condition that for every
. - (1- >.) = 2>' - 1. This establishes the uniqueness of>. = ((Tl)(s) + 1)/2 and the result follows. Tl(s)
0
We now specialise to the case when K and 8 have the unique decomposition property. LEMMA 3.2. Let 8 1 and 8 2 be as in (3.1). 8uppose that K has the unique decomposition property. Then 8 1 and 8 2 are complementary split faces of 8. PROOF. By Lemma 3.1, for each s E 8\(81 U 8 2) we may write s = >.x + (1 - >.)y where x E 8 1 , Y E 8 2 and 0 < >. < 1, and>' is unique. We consider the decomposition T* s = >'T*x - (>. -1)T*y. Since x E 8 1 we have T*x E K and hence >'T*x is positive. Similarly, T*y E -K and hence (>. - I)T*y is positive. Also
IIT*sll = 1 = >. + (1- >.) = Ii>'T*xli + 11(>' -
I)T*yli· Thus, by the unique decomposition property, >'T*x and (>. -1)T*y are unique. By the uniqueness of >., we have T*x and T*y are unique. Since T is surjective, T* is injective and the result follows. 0
THE UNIQUE DECOMPOSITION PROPERTY AND BANACH-STONE THEOREM
155
By replacing T by T- 1 in (3.1) we may 'decompose' K by defining (3.2) K1 = {k E K : T- 11(k) = I} and K2 = {k E K : T- 11(k) = -I}. Applying Lemmas 3.1 and 3.2 to T- 1 we see that K1 and K2 are complementary split faces of K whenever S has the unique decomposition property. We say that T is a weighted composition operator whenever there exists a central function h in A(S) and a continuous affine mapping (1: S -+ K such that Tf = h· f 0 (1 for all f E A(K); that is, Tf(s) = h(s)f((1(s)) for all sEaS. The following proposition asserts that the linear isometry T is a weighted composition operator, with Tl(s) = ±1 on as, if and only if Tl is central. PROPOSITION 3.3. Let T: A(K) -+ A(S) be a linear mapping. Then the following are equivalent: a) T is an isometry and Tl is central; b) T is a weighted composition operator of the form T f = h . f 0 (1 for all f E A(K) where (1 is an affine homeomorphism and h(s) = ±1 for all sEas. PROOF. See [4, Theorem 3.3] or [13].
o
We now apply the above decompositions of K and S to prove our main theorem. THEOREM 3.4. Suppose that K and S have the unique decomposition property. Then the real affine function spaces A(K) and A(S) are linearly isometric if and only if K and S are affinely homeomorphic. Moreover, every linear isometry from A(K) onto A(S) may be written as a weighted composition operator. PROOF. It suffices to show necessity. Suppose that T is a linear isometry from A(K) onto A(S). We 'decompose'S into the complementary split faces Sl and S2 of (3.1) and, similarly, K into the complementary split faces K1 and K2 of (3.2). Since (T-1)* = (T*)-l, we have T*(Sd = K1 and T*(S2) = -K2' and hence we may define (1: S -+ K by (1(,xx + (1 - ,x)y) = ,xT*(x) - (1 - ,x)T*(y)
whenever x E Sl, Y E S2 and 0 :5 ,x :5 1. We see that (1 is an affine homeomorphism from S = co (Sl U S2) onto K = co (K1 U K 2). Moreover, to show that T is a weighted composition operator it suffices, by Proposition 3.3, to show that h = Tl is central. Let f E A(S) then, since (1: S -+ K is an affine homeomorphism, we may write f = gO(1 for some g E A(K). Note that for x E Sl we have Tg(x) = T*x(g) = g((1(x)) = h(x)f(x). Similarly for x E S2 we have Tg(x) = T*x(g) = -g((1(x)) = h(x)f(x). Therefore, for all x E as £;; Sl U S2 we have Tg(x) = h(x)f(x), and the result follows. 0 References [1] E.M. Alfsen, Compact convex sets and boundary Integmls, Ergebnisse der Mathematik, 57, (Springer-Verlag, Berlin-Heidelberg-New York, 1971). [21 L. Asimow and A.J. Ellis, Convexity theory and its applications in functional analysis, London Math. Soc. Monograph, 16 (Academic Press, London, 1980). [3] E. Behrends, M -Structure and the Banach-Stone Theorem, Lecture Notes in Mathematics 736, (Springer-Verlag, Berlin-Heidelberg-New York, 1979). [4] A. Curnock, J. Howroyd, and N.-C. Wong, Isometries of affine function spaces, preprint. [5] J. Dixmier, C*-Algebms (North-Holland Publishing Co., Amsterdem-New York-Oxford, 1982).
AUDREY CURNOCK, JOHN HOWROYD, AND NGAI-CHING WONG
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[6] A.J. Ellis, An intersection property for state spaces, J. London Math. Soc., 43 (1968), 173176. [7] A.J. Ellis; Minimal decompositions in partially ordered normed spaces, Proc. Camb. Phil. Soc., 64 (1968),989-1000. [8] A.J. Ellis, On partial orderings of normed spaces, Math. Scand., 23 (1968), 123-132. [9] A.J. Ellis and W.S. So, Isometries and the complex state spaces of uniform algebras, Math. Z., 195 (1987), 119-125. [10] A. Grothendieck, Un result at sur Ie dual d'une C*-algebre, J. Math. Pures Appl., 36 (1957), 97-108. [11] A.J. Lazar, Affine products of simplexes, Math. Scand., 22 (1968), 165-175. [12] R.R. Phelps, Lectures on Choquet's Theorem, Second Edition, Lecture notes in Mathematics 1757 (Springer-Verlag, Berlin, 2001). [13] T.S.R.K. Rao, Isometries of Ac(K), Proc. Amer. Math. Soc., 85 (1982), 544-546. SCHOOL OF COMPUTING, INFORMATION SYSTEMS AND MATHEMATICS, SOUTH BANK UNIVERSITY, LONDON SE1 OAA, ENGLAND. E-mail address:curnocaOsbu.ac.uk DEPARTMENT OF MATHEMATICAL SCIENCES, GOLDSMITHS COLLEGE, UNIVERSITY OF LONDON, LONDON SE14 6NW, ENGLAND. E-mail address:masOljdhOgold.ac.uk DEPARTMENT OF ApPLIED MATHEMATICS, NATIONAL SUN YAT-SEN UNIVERSITY, KAOHSJUNG 80424, TAIWAN, R.O.C.
E-mail
address:wong~ath.nsysu.edu.tw
Contemporary Mathematics Volume 328, 2003
A Survey of Algebraic Extensions of Commutative, Unital Normed Algebras Thomas Dawson ABSTRACT. We describe the role of algebraic extensions in the theory of commutative, unital normed algebras, with special attention to uniform algebras. We shall also compare these constructions and show how they are related to each other.
Introduction Algebraic extensions have had striking applications in the theory of uniform algebras ever since Cole used them (in [5]) to construct a counterexample to the peak-point conjecture. Apart from this, their main use has been in (a) the construction of examples of general, normed algebras with special properties and (b) the Galois theory of Banach algebras. We shall not discuss (b) here; a summary of some of this work is included in [29]. In the first section of this article we shall introduce the types of extensions and relate their applications. The section ends by giving the exact relationship between the types of extensions. Section 2 contains a table summarising what is known about the extensions' properties. A theme lying behind all the work to be discussed is the following question:
(Q) Suppose the normed algebra B is related to a subalgebra A by some specific property or construction. (For example, B might be integral over A: every element b E B satisfies ao + ... +an_1bn-1 +bn = 0 for some ao, ... ,an -1 EA.) What properties of A (for example, completeness or semisimplicity) must be shared by B? This is a natural question, and interesting in its own right. Many special cases of it have been studied in the literature. We shall review the related body of work in which B is constructed from A by adjoining roots of monic polynomial equations. Throughout this article, A denotes a commutative, unital normed algebra, and A its completion. The fundamental construction of [1] applies to this class of 1991 Mathematics Subject Classification. Primary 46J05, 46J10. This research was supported by the EPSRC. © 2003 American Mathematical Society 157
THOMAS DAWSON
158
algebras. Algebraic extensions of more general types of topological algebras have received limited attention in the literature (see [19], [21]). If E is a subset of a ring then (E) will stand for the the ideal generated by E. 1. Types of Algebraic Extensions and their Applications 1.1. Arens-Hoffman Extensions. Let a(x) = ao + ... + an_lx n- l + xn be a monic polynomial over the algebra A. The basic construction arising from A and a(x) is the Arens-Hoffman extension, Ao. This was introduced in [1]. Most of the obvious questions of the type (Q) for Arens-Hoffman extensions were dealt with in this paper and in the subsequent work of Lindberg ([18]' [20], [13]). See columns two and three of Table 2.2. All the constructions we shall meet are built out of Arens-Hoffman extensions. DEFINITION 1.1.1. A mapping 0: A - B between algebras A and B is called unital if it sends the identity of A to the identity of B. An extension of A is a commutative, unital normed algebra, B, together with a unital, isometric monomorphism
O:A-B.
The Arens-Hoffman extension of A with respect to a(x) is the algebra Ao := A[x]/(a(x)) under a certain norm; the embedding is given by the map v: a t-+ (a(x)) + a.
To simplify notation, we shall let x denote the coset of x and often omit the indeterminate when using a polynomial as an index. It is a purely algebraic fact that each element of Ao has a unique representative of degree less than n, the degree of a(x). Arens and Hoffman proved that, provided the positive number t satisfies the inequality t n ~ ~~:~ lIakll tk, then
I~
bkX
k
=
~ IIbkll t
k
(bo, ... , bn -
l
E
A)
defines an algebra norm on Ao. The first proposition shows that Arens-Hoffman extensions satisfy a certain universal property which is very useful when investigating algebraic extensions. It is not specially stated anywhere in the literature; it seems to be taken as obvious. 1.1.2. Let A(l) be a normed algebra and let 0: A(l) _ B(2) be a unital homomorphism of normed algebras. Let al (x) = ao + ... + an_lXn - l + xn E A(l) [x] and B(1) = A~l}. Let y E B(2) be a root of the polynomial a2(x) := O(ad(x) := O(ao) + ... +O(an_l)X n - l +xn. Then there is a unique homomorphism 1>: B(l) _ B(2) such that PROPOSITION
B(1)
~
II
/9
r
B(2)
is commutative and 1>(x) = y.
A(l)
The map 1> is continuous if and only if 0 is continuous. PROOF.
This is elementary; see [7]
o
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS
159
1.2. Incomplete Normed Algebras. A minor source of applications of Arens-Hoffman extensions fits in nicely with our thematic question (Q): these extensions are useful in constructing examples to show that taking the completion of A need not preserve certain properties of A. The method uses the fact that the actions of forming completions and ArensHoffman extensions commute in a natural sense. A special case of this is stated in [17]; the general case is proved in [7), Theorem 3.13, and follows easily from Proposition 1.1.2. It is convenient to introduce some more notation and terminology here. Let O(A) denote the space of continuous epimorphisms A --+ Cj when n appears on its own it will refer to A. As discussed in [1], this space, with the weak *-topology relative to the topological dual of A, generalises the notion of the maximal ideal space of a Banach algebra. In fact, it is easy to check that 0 is homeomorphic to 0(..4), the maximal ideal space of the completion of A. The Gelfand transform of an element a E A is defined by
a: n --+ C; W
1--+
w(a)
and the map sending a to a is a homomorphism, r, of A into the algebra, C(O), of all continuous, complex-valued functions on the compact, Hausdorff space O. We denote the image of r by A. A good reference for Gelfand theory is Chapter three of [24]. DEFINITION
1. 2.1 ([1)). The algebra A is called topologically semisimple if
r
is injective. If A is a Banach algebra then this condition is equivalent to the usual notion of semisimplicity. The precise conditions under which Aa is topologically semisimple if A is are determined in [1]. In [17] Lindberg shows that the completion of a topologically semisimple algebra need not be semisimple. In order to illustrate Lindberg's strategy we recall two standard properties of normed algebras.
1.2.2. The normed algebra A is called regular if for each closed subset E ~ 0 and wE O-E there exists a E A such that a(E) ~ {O} and a(w) = 1. The algebra is called local if A contains every complex function, f, on 0 such that every wE 0 has a neighbourhood, V, and an element a E A such that flv = alv. DEFINITION
It is a standard fact that regularity is stronger than localness; see Lemma 7.2.8 of [24). EXAMPLE 1.2.3. Let A be the algebra of all continuous, piecewise polynomial functions on the unit interval, I, and a(x) = x 2 - id/ E A[x]. Let A have the supremum norm. By the Stone-Weierstrass theorem, A = C(I) and hence n is identifiable with I. Clearly A is regular. We leave it as an exercise for the reader to find examples to show that Aa is not local. This is not hardj it may be helpful to know that in this example the space O(Aa) is homeomorphic to {(s, oX) E I xC: oX 2 = s}. This follows from facts in [1]. In the present example, neither localness nor regularity is preserved by (incomplete) Arens-Hoffman extensions.
THOMAS DAWSON
160
Finally we can explain the method for showing that some properties of normed algebras are not shared by their completions because, in the above, 'non-regularity' is not preserved by completion of Ao (nor is 'non-localness'). To see this, note that Ais clearly regular if A is and so by a theorem of Lindberg (see Table 2.2) the ArensHoffman extension (A)o is regular. But, by a result of [17] referred to above, this algebra is isometrically isomorphic to the completion of Aa. Of course Lindberg's original application was much more significant; there are simpler examples of the present result: for example the algebra of polynomials on I.
1.3. Uniform Algebras. It is curious that the application of Arens-Hoffman extensions to the construction of integrally closed extensions of normed algebras did not appear in the literature for some time after [1]. It was seventeen years later until a construction was given in [22]. Even then the author acknowledges that the constuction was prompted by the work of Cole, [5], in the theory of uniform algebras. Cole invented a method of adjoining square roots of elements to uniform algebras. He used it to extend uniform algebras to ones which contain square roots for all of their elements. Apart from feeding back into the general theory of commutative Banach algebras (mainly accomplished in [22] and [23]) his construction provided important examples in the theory of uniform algebras. We shall describe these after recalling some basic definitions. DEFINITION 1.3.1. A uniform algebra, A, is a subalgebra of C(X) for some compact, Hausdorff space X such that A is closed with respect to the supremum norm, separates the points of X, and contains the constant functions. We speak of 'the uniform algebra (A, X)'. The uniform algebra is natural if all of its homomorphisms wEn are given by evaluation at points of X, and it is called trivial if A = C(X).
Introductions to uniform algebras can be found in [4], [11], [26], and [16]. An important question in this area is which properties of (A, X) force A to be trivial. For example it is sufficient that A be self-adjoint, by the Stone-Weierstrass theorem. In [5] an example is given of a non-trivial uniform algebra, (B,X), which is natural and such that every point of X is a 'peak-point'. It had previously been conjectured that no such algebra existed. We shall describe the use of Cole's construction in the next section, but now we reveal some of the detail. PROPOSITION 1.3.2 ([5],[7]). Let U be a set of monic polynomials over the uniform algebra (A, X). There exists a uniform algebra (AU, XU) and a continuous, open surjection 7r: XU ---> X such that (i) the adjoint map 7r*: C(X) ---> C(XU) induces an isometric, unital monomorphism A ---> AU, and (ii) for every a E U the polynomial 7r*(a)(x) E AU[x] has a root Po E AU. PROOF. We let XU be the subset of X x such that for all a E U f(a)(K)
JO
cU consisting of the elements (K, >.)
+ ... + /(0) (K)>.n(0)-1 + >.n(o) n(a)-1 a 0
= 0
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORM ED ALGEBRAS
161
where o:(x) = f~Ol) + ... + f~(l)_lXn(Ol)-l + xn(Ol) E U. The reader can easily check that XU is a compact, Hausdorff space in the relative product topology and so the following functions are continuous: 11":
XU -+ X; (~, >.) ...... ~
POI: XU -+ C; (~, >.) ...... >'01
(0:
E U).
The extension AU is defined to be the closed subalgebra of C(XU) generated by 1I"*(A) U {POI: 0: E U} where 11"* is the adjoint map C(X) -+ C(XU) ; g ...... go 11". It is not hard to check that AU is a uniform algebra on XU with the required properties. 0 We shall call AU the Cole extension of A by U. Cole gave the construction for the case in which every element of U is of the form x 2 - f for some f E A. It is remarked in [22] that similar methods can be used for the general case; these were independently, explicitly given in [7]. By repeating this construction, using transfinite induction, one can generate uniform algebras which are integrally closed extensions of A. Full details of this, including references and the required facts on ordinal numbers and direct limits of normed algebras, can be found in [7]. Again this closely follows [5]. Informally the construction is as follows. Let v be a non-zero ordinal number. Set (Ao,Xo) = (A,X). For ordinal numbers T with 0 < T ~ V we define (A~" , X!:" )
(A.,-, X T )
if T = a
+1
and
= { l~ . (A u,Xu)u
2.2. Table. Cole extensions have only been defined for uniform algebra'l; the algebra is therefore assumed to be a uniform algebra throughout column one. Colulllns two and three, a'l mentioned above, refer to Arens-Hoffman extensions of a normed algebra, A, by a monic polynomial a(x); in column three it is given that a(:1:) is separable. Type of Extension: Property:
Cole
A.-H.
Ac.
A.-H. standard Narmania a sep.
for normed algebras
complete topologically semisimple non-local local regular
1.
2. 3. 4. 5.
• • • ?
• 0
•
• • •
•
0
? '?
?
'?
'!
• • • • •
• •
• • • •
0
0
•
0
0
•
•
0
•
'?
?
for Banach algebras
6. 7. S. 9. 10.
local regular dense invertible group sup-norm closed symmetric 11. amenable 12. weakly amenable
?
0 0 0 0
for uniform algebms
13. non-trivial 14. trivial 15. natural
• • •
Key • property is always preserved o property is sometimes, but not always preserved
? ?
? ?
?
•
• •
0
0
0
0
0
0
0
0
168
THOMAS DAWSON
? not yet determined - it doesn't always make sense to consider this property here References for the Entries. If we do not mention an entry here, it can be taken that the result is an immediate consequence of the definition or was proved in the same paper in which the relevant extension was introduced (that is in [5], [1], [22], or [23]). The results of row three are not hard to obtain, using appropriate versions of Proposition 1.5.1. Localness and regularity were discussed in Section 1.2. The main result about this is due to Lindberg in [18]; the same section of his paper also deals with the results on the symmetry of Arens-Hoffman extensions. That regularity passes to direct limits of such extensions has been widely noted by many authors, for example in [15]. Results of row eight follow from [8]; the case of Cole extensions was partially covered in [15], but the reasoning is not clear. The property of being sup-norm closed was investigated in [13]; this work was generalised in [28]. Finally, examples of amenable Banach algebras which do not have even weakly amenable Arens-Hoffman extensions have been known for a long time. For example, the algebra C E9 C under the multiplication (a, b) (e, d) = (ae, be + ad) is realisable as an Arens-Hoffman extension of C. Examples with both A and Ao semisimple have been found by the author. However the entries marked "?' in rows eleven and twelve represent intriguing open problems. 3. Conclusion The table in Section 2.2 still has gaps, and there are many more rows which could be added. For example it would be interesting to be able to estimate various types of 'stable ranks' (see [2]) of the extensions in terms of the stable ranks of the original algebras. (The condition G(A) = A is equivalent to the 'topological stable rank' of A not exceeding 1.) Remember too that there are many more questions which can be asked, of the form: 'if n has the topological property P, does n(Ao) have property P?' By way of a conclusion we repeat that algebraic extensions have proved immensely useful in the construction of examples of uniform algebras. There is therefore great scope for and potential usefulness in augmenting Table 2.2. It might also be valuable to reexamine the techniques used to obtain the entries to produce more general results (of the kind in [28] for example) in the context of question (Q).
References 1. Arens, R. and Hoffman, K., Algebraic Extension of Normed Algebras., Proc. Am. Math. Soc. 7 (1956), 203-210. 2. Badea, C., The Stable Rank of Topological Algebras and a Problem of R. G. Swan., J. Funet. Anal. 160 (1998), 42-78. 3. Batikyan, B. T., Point Derivations on Algebraic Extension of Banach Algebra., Lobachevskii J. Math. 6 (2000), 3-37. 4. Browder, A., Introduction to Function Algebras., W. A. Benjamin, Inc., New York, 1969.
ALGEBRAIC EXTENSIONS OF COMMUTATIVE NORMED ALGEBRAS
169
5. Cole, B. J., One-Point Parts and the Peak-Point Conjecture., Ph.D. Thesis, Yale University, 1968. 6. Dales, H. G., Banach Algebms and Automatic Continuity., Oxford University Press Inc., New York,2000. 7. Dawson, T. W .. Algebmic Extensions of Normed Algebms., M.Math. Dissertation, accessible from the web at: http://xxx.lanl.gov/abs/math.FA/0102131, University of Nottingham, 2000. 8. Dawson, T. W., and Feinstein, J. F., On the Denseness of the Invertible Group in Banach Algebms., Proc. Am. Math. Soc. (to appear). 9. Feinstein, J. F., A Non-Trivial, Strongly Regular Unif01m Algebm., J. Lond. Math. Soc. 45 (1992), 288-300. 10. Feinstein, J. F., Trivial Jensen Measures Without Regularity., Studia Math. 148 (2001).6774. 11. Gamelin, T. W., Uniform Algebms., Prentice-Hall Inc., Engelwood Cliffs, N. J., 1969. 12. Gorin, E. A., and Lin, V ..J., Algebmic Equations with Cont'inuous Coefficients and Some Problems of the Algebmic Theory of Bmids., Math. USSR Sb. 7 (1969), 569-596. 13. Heuer, G. A., and Lindberg, J. A., Algebmic Extensions of Continuous Function Algebms., Proc. Am. Math. Soc. 14 (1963),337-342. 14. Johnson, B. E., Norming C(O) and Related Algebms., Trans. Am. Math. Soc. 220 (1976), 37-58. 15. Karahanjan, M. I., Some Algebmic Chamcterizations of the Algebm of All Continuous Functions on a Locally Connected Compactum., Math. USSR Sb. 35 (1979),681-696. 16. Leibowitz, G. M., Lectures on Complex Function Algebm,~., Scott, Foresman and Company, Glenview, Illinois, 1970. 17. Lindberg, J. A .• On the Completion of Tractable Normed Algebms., Proc. Am. Math. Soc. 14 (1963),319-321. 18. Lindberg, J. A., Algebmic Extensions of Commutative Banach Algebms., Pacif. J. Math. 14 (1964), 559-583. 19. Lindberg, J. A., On Singly Genemted Topological Algebras, Function Algebras (ed. Birtel, F. T.), Scott-Foresman, Chicago, 1966, pp. 334-340. 20. Lindberg, J. A., A Class of Commutative Banach Algebms with Unique Complete Norm Topology and Continuous Derivations., Proc. Am. Math. Soc. 29 (1971), 516-520. 21. Lindberg, J. A .. Polynomials over Complete l.m.-c. Algebms and Simple Integml Extensions., Rev. Roumaine Math. Pures Appl. 17 (1972), 47-63. 22. Lindberg, J. A., lntegml Extensions of Commutative Banach Algebms., Can. ,/. Math. 25 (1973), 673-686. 23. Narmaniya, V. G., The Construction of Algebmically Closed Exten,~ions of Commutative Banach Algebms., Trudy Tbiliss. Mat. Inst. Razmadze Akad. 69 (1982), 154-162. 24. Palmer, T. W., Banach Algebras and the Geneml Theory of *-Algebms. Vol. 1, Cambridge University Press, Cambridge, 1994. 25. Read, C. J., Commutative, Radical Amenable Banach Algebms., Studia Math. 140 (2000), 199-212. 26. Stout, E. L., The Theory of Uniform Algebms., Bogden and Quigley Inc., Tarrytown-onHudson, New York, 1973. 27. Taylor, J. L., Banach Algebms and Topology, Algebras in Analysis. (ed. Williamson, J. H.), Academic Press Inc. (London) Ltd., Norwich:, 1975, pp. 118-186. 28. Verdera, J., On Finitely Genemted and Projective Extensions of Banach Algebms., Proc. Am. Math. Soc. 80 (1980), 614-620. 29. Zame, W. R., Covering Spaces and the Galois Theory of Commutative Banach Algebms., J. Funet. Anal. 27 (1984), 151-171.
Acknowledgements The author would like to thank the Division of Pure Mathematics and the Graduate School at the University of Nottingham for paying for his expenses in
170
THOMAS DAWSON
order to attend the 4th Conference on FUnction Spaces (2002) at the Southern Illinois University at Edwardsville. The author is grateful to Mr. Brian Lockett who provided him with a translation of the paper [23]. Special thanks are due to Dr. J. F. Feinstein who offered much valuable advice and encouragement and also proofread the article. DIVISION OF PURE MATHEMATICS, SCHOOL OF MATHEMATICAL SCIENCES, UNIVERSITY OF NOTTINGHAM, UNIVERSITY PARK, NOTTINGHAM, NC7 2RD, UK.
E-mail address:pmxtwdlDnottingham.ac.uk
Contemporary Mat.hematics Volume 328, 200a
Some more examples of subsets of Co and Ll[O, 1] failing the fixed point property P.N. Dowling, C.J. Lennard, and B. Thrett We give examples of closed, bounded. convex, non-weakly compact subsets of Co on which the right shift is expansive, and we construct two nonexpansive self-mappings (one affine and one non-affine) on these sets which fail to have a fixed point. We also prove that every closed, bounded, convex subset of L1 [0,1] with a non-empty interior fails the fixed point property for nonexpansive mappings. Finally, we extend this result by showing that every closed, bounded, convex subset of L1 [0,1] that contains a non-trivial order interval must fail the fixed point property. ABSTRACT.
1. Introduction
In [5], Llorens-Fuster and Sims construct examples of closed, bounded, convex subsets of Co that are not weakly compact but are compact in a topology that is slightly coarser than the weak topology, and they nonetheless fail the fixed point property for nonexpansive mappings. These examples led Llorens-Fuster and Sims to conjecture that closed, bounded, convex non-empty subsets of Co have the fixed point property if and only if they are weakly compact. This conjecture has been recently settled in the affirmative [2, 3]. All the examples constructed in [5] had a common feature - they all support a nonexpansive right shift. In the first part of this short note we produce a collection of sets of the type considered by Llorens-Fuster and Sims, but which do not support a nonexpansive right shift and yet they fail the fixed point property for nonexpansive mappings. In fact, we will produce two nonexpansive fixed point free mappings: one affine and the other non-affine. Variations on the themes of these examples are important in the papers [2, 3]. Llorens-Fuster and Sims [5] also proved that a closed, bounded, convex subset of Co with non-empty interior fails the fixed point property for nonexpansive mappings. In the second part of tllis note we prove an analogous statement in the setting of L1 [0, 1]. We also generalize this result to show that every closed, bounded, convex 2000 Mathematics Subject Classification. Primary 47HI0, 47H09, 46E30. The authors wish to thank Professor Kaz Goebel for his helpful suggestions concerning the proof of Theorem 3.2. The second author thanks the Department of Mathematics and Statistics at Miami University for their hospitality during part of the preparation of this paper. He also acknowledges the financial support of Miami University.
© 171
2003 American l\1athematical Society
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
172
subset of L1 [0, 1J that contains a non-trivial order interval must fail the fixed point property. We refer the reader to the text of Goebel and Kirk [4J for any unexplained terminology. 2. The fixed point property in Co We begin this section with the Llorens-F\lster and Sims examples. We have slightly modified their examples to simplify the computations. Let r denote the set of sequences 'Y = ("(n)n in the interval (0,1). For all 'Y E r, define K., by
K...,
:=
{x
=
("(ntn)n E Co 112: h 2: t2 2: t3 2: ... 2: O} .
We define the right shift Ton K..., by
T( ("(1 t 1 , 'Y2t2, 'Y3 t3, 'Y4t4, ... ))
:=
("(1, 'Y2t 1, 'Y3 t2, 'Y4t3, ... ).
Note that if x = ("(ntn)n and y = ("(nsn)n are elements of K..." then Ilx - yll sUPn 'Ynlt" - 8 n l and IIT(x) - T(y)11 = SUPn 'Yn+1Itn - 8 n l· Clearly, if the sequence ("(n)n is decreasing, then T is a nonexpansive mapping on K..., - this is the case considered by Llorens-Fuster and Sims [5J. However, it is equally obvious that if the sequence ("(n)n is strictly increasing, then T is an expansive mapping on K...,; that is, IIT(x) - T(y)11 > IIx - yll whenever x i= y. We will show that even though some of these sets do not support a nonexpansive right shift, they do support nonexpansive fixed point free mappings. To simplify our computations we will only consider the sets K..., where the sequence 'Y = ("(n)n is in (0,1), is strictly increasing and satisfies 1 - 'Yn < 4- n for all n E N. EXAMPLE 2.1. Let I be the identity mapping on K..." let T be the right shift defined above, T2 = ToT, T3 = ToT 0 T, and so on. Define R: K..., --+ K..., by
R
:= ~I
+ -b T + -1a T2 + 21
4
T3
+... .
A simple calculation shows that if x = ("(1 t1, 'Y2t2, 'Y3t3, ... ) E K..." then
R(x) = ('Y1(~t1
+ ~),'Y2(~t2 + it1 + i),'Y3(~t3 + it2 + kt1 + ~), ... ). that if R(x) = x, then tn = 1 for all n E N, and thus
It is easily seen x is not an element of K...,; that is, R is fixed point free on K...,. To see that R is nonexpansive on K..." let x = ("(ntn)n and y = ("(nsn)n be elements of K...,. Then, 12 + - 4 1 + ... + 2,,1...,1 ) < 1 for each n E N. since 1 - 4- n < 'Yn < 1, we have 'Yn( -...,,, "",,-1 Consequently,
IIR(x) - R(y)11
sup hnl~(tn - sn) n
+ i(tn-1
- sn-d
+ ... + 2~ (t1
-
< sup hn(~ltn - snl + iltn-1 - 8 n -11 + ... + 2~' It 1 n
< sup {'Yn(_l+ _1_ + ... + +) max 'Yilti - Sil} n 2...,,, 4"",,-1 2""1 l::;'i::;n
< sUP'Ynltn n
8n
l
Ilx-yll· Thus R is a nonexpansive mapping on K...,.
st)l} Sl
J)}
173
EXAMPLES OF FIXED POINT FREE MAPS
The mapping R, given in example 2.1, is an affine mapping on K"I' Our next example is non-affine on K"I' EXAMPLE 2.2. For an element x = (xn)n in K"I' we denote by bl,X) the sequence bl,Xl,X2,X3, ... ). We define a mapping S: K"I -+ Co by S(x):= X, for each x E K"I' where x = (Xl, X2, X2, ... ) is the decreasing rearrangement of the sequence bl' x); that is, X = bl, x)*. Note that
X= (Xl,X2,X3, ... ) =
(11
(~~) ,12 (~:) ,13 (~:) , ... ).
Also Xl 2: X2 2: X3 2: ... 2: 0 and 0 < 11 < 12 < 13 < ... < 1. Therefore ~ > ~ > ~ > ... > O. Since ~ = max("("x) > 1 S(x) does not necessarily belong ~-n-n~ ~ -, to K"I' However, the mapping S is nonexpansive on K"I because the operation of decreasing rearrangement is nonexpansive on Co, so for all x and y in K"I' we have IIS(X) - S(Y)II = IIx -
yll
IIbl,X)* - bl,Y)*11 :::; Ilb1.x) - b1.y)11
=
=
Ilx - YII·
We now introduce a modification U of S that will be nonexpansive and fixed point free on K"I' Define U : K"I -+ Co by
Since
11j AXj - 1j AYj I :::; IXj - Yj I for all j E N, it follows that IIU(x) - U(Y)II :::;
Ilx - yll = IIS(x) - S(Y)II :::; IIx - YII·
Thus U is a nonexpansive mapping on K"I' Furthermore, since 1j AXj =
1j (1 A ~) for all j E Nand 1 2: 1 A ~ 2: 1 A ~ 2:
1 A ~ 2: ... 2: 0, U maps K"I into K"I' To finish, we will show is fixed point free on K"I' Suppose, to get a contradiction, that there exists x E K"I such that x = U(x). Thus, for all j EN, Xj = 1j AXj. A well-known fact about decreasing rearrangements that we will use is that for for each mEN, all W E
ct,
WI
+ ... + Wm
:::;
wi + ... + w;;'.
Since 1 = (1n)n is strictly increasing with limit 1, while X is decreasing with 1 > Xl 2: 11, there exists a unique kEN such that Xk+1 < 1k+l and Xk 2: 1k. Thus, for all mEN with m > k, we have, k + Xl
+ ... + Xm+l > (Xl + ... + Xk) + (Xl + ... + xm+d (Xl
+ ... + Xk) +
bl A Xl + ... + 1k A Xk + 1k+l A Xk+1 + ... + 1m+1 A Xm+1 = (Xl + ... + Xk) + bl + ... + 1k + Xk+l + ... + xm+d bl + ... + 1k) + (Xl + ... + Xk + Xk+1 + ... + Xm+1) bl + ... + 1k) + (Xl + ... + Xm+l) > bl+"'+1k)+bl+Xl+"'+Xm),
174
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
It follows that for all
Tn
E N with Tn
> k, k
Xm
+l
>
bl+"'+'Yk)-k+'Yl='YI-L(1-'Yj) j=1
> 'Y1 -
=
L (1 -
'Yj) ~ 'Yl -
j=1
=.
L 4-
J
1
= 'Yl -
:3
1
>
:3'
j=1
This contradicts the fact that x E Co and so completes the proof that U is fixed point free on K-y.
3. The fixed point property in £1 [0, 1] One of the most notable works in metric fixed point theory is the construction of Alspach [1] of a non-empty weakly compact convex subset of Ll[O, 1] which fails the fixed point property. We begin this section by recalling some of the details of Alspach's construction. Let C:= {f E Ll[O, 1] : 0::; f(t) ::; 1, for all t E [0, I]}. Now define T: C -+ C by
Tf(t) := {min{2f (2t), I} max{2f(2t - 1) - 1, O}
for 0 ::; t ::; ~ for ~ < t ::; 1.
for all f E C. Alspach showed that the mapping T is an isometry on C which has two fixed points; namely 0 and X[O,lj' Alspach also showed that T is an isometric self map of the closed convex subset Co := {f E C : J~ f dm = 1/2} of C, such that T is fixed point free on Co. Here, m denotes Lebesgue measure. We now follow a modification of Alspach's example due to Sine [7]. Define S : C -+ C by S(f) := X[O,lj - f, for all f E C. The mapping S is clearly an isometry of C onto C. Thus the mapping ST is a nonexpansive mapping on C. Sine proved that ST is fixed point free on C. In [5], Llorens-FUster and Sims prove that a closed bounded convex subset of Co with non-empty interior fails the fixed point property. We will use the above construction of Alspach, and modification by Sine, to prove a result analogous to the Llorens-FUster and Sims result in the setting of Ll [0, 1]. Specifically we prove the following result. THEOREM 3.1. Let K be a closed, bounded, convex subset of Ll [0, 1] with nonempty interior. Then K fails the fixed point property for non expansive mappings. PROOF. By translating and scaling, we ca.n assume that K contains the unit ball of £1 [0, 1]. Consequently, the set C, constructed above, is a subset of K. Define the mapping R : K -+ K by
Rf(t) := min{lf(t)l, I}, for 0::;
f
E
t::; 1, for all f
E
K.
It is easily seen that R is a. nonexpansive mapping on K and R(f) E C for all K. Now define U: K -+ K by
U(f) := ST(R(f)), for all
f
E
K.
The mapping U is nonexpansive since all of the mappings, R, S, and Tare nonexpansive.
EXAMPLES OF FIXED POINT FREE MAPS
175
We now show that U is fixed point free. Suppose that f E K is a fixed point of U, that is, U(f) = f. Since f E K, R(f) E C, and since ST maps C into C, f = U(f) = ST(R(f)) E C. Note that the mapping R restricted to C is the identity on C. Therefore, f = ST(R(f)) = ST(f) and so f is a fixed point of ST in C. This contradicts Sine's result that ST has no fixed point in C [7], and thus the proof is complete. D THEOREM 3.2. Let K be a closed, bOllnded, convex sllbset of Ll[O, 1] that contains an order interval [h,g] := {f E L1[0, 1] : h ::; f ::; 9 a.e.}, for some h,g E Ll[O, 1] with h ::; 9 a.nd h ::f g. Then K fa'ils the fixed point property for nonexpansive mappings,
a a
a
PROOF. By translating by -h, we may assume that h = and 9 ~ a.e. with 9 non-trivial. Next, note that there exists a real number c > and a measurable set E with Lebesgue measure rn(E) > 0, such that 9 ~ CXE. By rescaling K by lie, we may assume without loss that c = 1. Now, define the mapping R : K ~ [0, xel ~ K by R(f) := ifi /\ XE. Note that R is nonexpansive and R equals the identity on [0, xel. At this stage, consider E. There exists to in the interval [0,1] such that rn(En [0, to]) = ~ rn(E), Let E1.1 := En [0, to] and E1.2 := En (to, 1]. Clearly E is the disjoint union of E1.1 and E 1,2 and rn(E 1,1) = rn(El,2) = ~ rn(E). Proceed iteratively from here. Similarly to above, there exist pairwise disjoint measurable subsets E2,1, E 2,2, E2,3 and E2,4 of [0, 1] such that El,l = E 2,1 U E2,2, E 1 ,2 = E 2,3 U E 2.4 , and each rn(E2,k) = ~ rn(E). Repeating this construction inductively, we produce a family of measurable subsets (EO,1 := E, En,k : n E N, k E {I, ... , 2n}) of [0,1] such that (XE",k)n.k is a dyadic tree in Ll [0,1]. Moreover, letting the measure v be defined on the measurable subsets of E by v = (1/rn(E))rn, it follows that the Banach space L 1 ( E, v) is isometrically isomorphic to L 1 ( [0, 1], m) = L 1 [0, 1] via the mapping Z defined as follows: Z(XEn, k):= X[k-l k), 2l'r'2"'t"
for each XE",k' Then Z is extended to L := the linear span of the functions XE",k in the usual way. Of course, Z is an isometry on L. Finally, since L is dense in Ll(E, v), with dense range in Ll [0,1], Z extends to a linear isometry from L1(E, v) onto Ll[O, 1]. Let W := ST be Sine's variation on Alspach's example, as described above, and note that W maps the order interval C := [0, X[O,I]] into C. Let's use W to define E : [0, xel ~ [0, XE], by E := Z-1 W Z. We have that E is a fixed point free Ll [0, 1]-isometry on [0, xel. Finally, we define U : K ~ [0, XE] ~ K via U := E R. In a manner analogous to the argument in the proof of Theorem 3.1 above, we see that U is a fixed point free Ll[O, 1]-nonexpansive mapping on K. D REMARK 3.3. In [6], MatlI'ey proved that closed, bounded, convex, non-empty subsets of reflexive subspaces of Ll [0, 1] have the fixed point property for nonexpansive mappings. Consequently, Maurey's result, in tandem with Theorem 3.2, shows that reflexive subspaces of U [0,1] cannot contain a non-trivial order interval. In fact, as pointed out by the referee, the argument in the proof of Theorem 3.2 shows
176
P.N. DOWLING, C.J. LENNARD, AND B. TURETT
that infinite-dimensional subspaces of £1[0,1] which contain non-trivial order intervals actually contain isometric copies of £1 [0, 1] and thus are nonreflexive. The authors thank the referee for his/her comments. References 1. D. Alspach, A fixed point free nonexpansive mapping, Proc. Amer. Math. Soc., 82 (1981), 423-424. 2. P.N. Dowling, C.J. Lennard and B. Thrett, Characterizations of weakly compact sets and new fixed point free maps in co, to appear in Studia Math. 3. P.N. Dowling, C.J. Lennard and B. Thrett, Weak compactness is equivalent to the fixed point property in co, preprint 4. Kazimierz Goebel and W.A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, 1990 5. Enrique Llorens-Fuster and Brailey Sims, The fixed point property in co, Canad. Math. Bull. 41 (1998), no. 2, 413-422. 6. B. Maurey, Points fixes des contractions de certains faiblement compacts de Ll, Seminaire d'Analyse Fonctionelle, 1980-1981, Centre de Mathematiques, Ecole Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19. 7. R. Sine, Remarks on an example of Alspach, Nonlinear Anal. and Appl., Marcel Dekker, (1981), 237-241. DEPARTMENT OF MATHEMATICS AND STATISTICS, MIAMI UNIVERSITY, OXFORD, OH
45056
E-mail address: dowlinpnGmuohio. edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF PITTSBURGH, PITTSBURGH, PA
15260
E-mail address: lennard+lDpi tt. edu DEPARTMENT OF MATHEMATICS AND STATISTICS, OAKLAND UNIVERSITY, ROCHESTER,
48309 E-mail address: turettlDoakland. edu
MI
Contemporary Mathelnatics Volume 328. 2003
Homotopic composition operators on Hoo (Bn) Pamela Gorkin, Raymond Mortini, and Daniel Suarez We characterize the path components of composition operators on Hoo(B n ), where Bn is the unit ball of en. We give a geometrical equivalence for the compactness of the difference of two of such operators. For n = 1, we give a characterization of the path components of the algebra endomorphisms. ABSTRACT.
1. Introduction
Consider the Hardy space H2 on the unit disk D. Littlewood's subordination principle tells us that for an analytic self-map ¢ of D and a function f in H2, the function f 0 ¢ is once again in H2. Thus one defines the composition operator C'" on H2 by C",(f) = f 0 ¢. The interplay of operator theory and function theory leads to several interesting results. One of these results is Berkson's theorem on isolation of composition operators (see [1] and [14]): THEOREM 1 (Berkson). Let ¢ be an analytic self-map of D. If ¢ has mdial limits of modulus one on a set E of positive measure, then for every other analytic self-map 'l/J of D, the following estimate holds:
IIC", - C",II2:
Jmea;(E),
where C'" and C'" are the corresponding composition opemtors on H2.
Thus, Berkson's theorem tells us that every such operator is isolated in the set of composition operators in the operator norm topology. For example, the identity operator, C z , is at least a distance of ..[f72 from every other composition operator on H2 (as is C"', where ¢ is any inner function). However, not every composition operator is isolated. If ¢ is analytic and ¢ : D ~ sD for some s with 0 < s < 1, then it is easy to check that
Thus, C'" is not isolated. Inner functions induce highly noncompact operators, as well as isolated operators. The operators C'" for which ¢(D) is contained in sD for some s with 0 < s < 1 2000 Mathematics Subject Classification. Primary 47B33; Secondary 47B38. Key words and phrases. composition operator, path components, compact differences.
© 177
2003 American lvlathematical Society
178
PAMELA GORKIN. RAYMOND MORTINI. AND DANIEL SUAREZ
are compact. As Shapiro and Sundberg [14J indicate in their paper, "compact composition operators are dramatically nonisolated." They show that the set of compact composition operators is path connected, and therefore these operators are never isolated. It is interesting to ask which composition operators are, in fact, isolated. Shapiro and Sundberg studied this problem, and showed (among other things) that if ¢ is au analytic self-map of D that is not an extreme point of the algebra HOC(D), then C is not isolated; in their words, "isolated composition operators can only be induced by extreme points." This allowed them to exhibit an example of a non-compact non-isolated operator. They also raised several questions at the end of their paper: (1) Characterize the components in Comp(H2), the space of all composition operators on H2. (2) Which composition operators are isolated? (3) Characterize composition operators whose difference is compact. Before stating the final question, we remind the reader that the essential norm of an operator T defined on a Banach space H is the distance to the compact operators; that is, liT lie = inf{IIT - KII : K compact on H}. It is clear that IITII ~ IITlle, and therefore every essentially isolated operator is isolated. In fact, because of the abundance of weakly null sequences in H2, all the results on isolation appearing in Shapiro and Sundberg's paper hold true if we replace the norm with the essential norm (see [14], p. 148). Thus they raised the following question. (4) Is every isolated operator essentially isolated? Other papers of interest on this subject include [10J. Of course, one is not limited to the space H2, and the study of composition operators on various spaces has lead to a large body of literature. In this paper, we are interested in the same problems for composition operators on HOC (Bn), where Bn denotes the open unit ball in en. While the problem 011 H2 seems to be difficult, MacCluer, Ohno and Zhao [l1J were able to obtain partial results about operators on the algebra HOO(D). They showed that for two analytic self-maps of the disk, C and C.p are in the same path component in the space of composition operators, Comp(HOC(D)), if and only if IIC", - c.pll < 2. In particular, an operator C is isolated in Comp(HOC(D)) if and only if IIC", - C",II = 2 for any other analytic self-map 'IjJ of the disk. The authors show that this result can be rephra.. - Cvlli < 2 and IICob C. These maps are all endomorphisms of the algebra HOC (Bn). For the special case of n = 1 we will say more in the final section of the paper. We are interested here in estimates on the essential norm of the difference of two composition operators. If T is a bounded operator, we denote its essential norm by IITlle. THEOREM
6. Let ¢> and '!/J be holomorphic self-maps of Bn such that
max{II¢>II, II'!/JII} =
1.
HOMOTOPIC COMPOSITION OPERATORS ON H"""(Bn)
183
Let
e=
max {lim sup p(¢(Z), 'lj!(Z)) , lim sup p(¢(Z), '¢(Z))} . 1'I/J(z)l-l
1",(z)l-l
Then (3.1) PROOF. By hypothesis there is a sequence of points {Zj} in Bn, such that p(¢(Zj),-¢(Zj)) ~ e, and one of them, say {¢(Zj)}, converges to a point ( on the boundary of Bn. Without loss of generality we may assume that p( ¢( Zj ), 'lj!( Zj)) > e-1fj. Let kj E Hoo(Bn) be a function of norm one, whose existence is guaranteed by the pseudohyperbolic distance definition, satisfying kj(¢(zj)) > e - 1fj and kj('lj!(zj)) = O. Consider the functions
f(z) = (1 + (z, () )/2 and g(z) = (1 - (z, () )/2. Then f,g E HOO(B n ), f(() = 1, If(7])1 < 1 for all 7] E aBn satisfying 7] -I- (, and g(() = O. We will now produce a sequence of functions, {hj}, tending to zero weakly for which Ihj(¢(zj)) - hj (1/I(Zj)) I ~ e. We proceed as follows. Let j E N. Since f(¢(zj)) ~ 1, we may choose Zmj so that If(¢(zmj))lj > 1 - 1 fj. Now since 9 -I- 0 on Bn, there exists an integer Ij such that
J
Ig(¢(zmj))1 1/ 1j 2: J1- 1fj. Consider the functions hj = (gl/1 j )(J3)kmr Then Ilhjll ~ 1, hj (1/I(zmj)) = 0, and Ihj(¢(zmJ)1 > (1 - 1fj)(e - 1/mj). We note that hj ~ 0 on the Shilov boundary and, by Lemma 4, hj ~ 0 weakly. Thus for any compact operator K we have IIC", - C,p
+ KII > 2:
IIC",hj - C,phj + Khjll Ihj(¢(zmj)) - hj('lj!(zmj))
+ (Khj)(zmJI ~ e·
= Ihj(¢(zmJ) + (Khj)(zmj)1
This proves the lower inequality in (3.1). For the upper inequality, let I: > 0 and choose 8 with 0 to 1 so that
p(¢(Z), 'lj!(z)) ~
e + I:
on the set {1¢(z)1
< 8 < 1 close enough
> 8} U {1'lj!(z)1 > 8}.
Now choose a = a(I:,8) E (0,1) close enough to one so that p(¢(z),a¢(z)) < I: and p('lj!(Z) , ml'{z)) < I: on the set {1¢(z)1 ~ 8}n{I1/I(z)1 ~ 8}. Since max{lla¢lI, Ila'lj!ll} ~ a < 1, the operator K ~f Co", - Co,p is compact. If Z and ware any two points of B'\ we may view Z and w as elements of the dual space of Hoo(Bn) in the obvious way. Therefore, for any function f in the unit ball of Hoo(Bn) we may apply (2.2) to conclude that If(z) - f(w)1 ~ 2p(z,w). Henceforth, applying (2.2) to the functions f and fo(z) = f(az), we have
I(C",f)(z) - (C,pf)(z) - (Kf)(z)1
~
If(¢(z)) - f(a¢(z))1 + If('lj!(z)) - f(a'¢(z))1 < 2p(¢(z), a¢(z)) + 2p('lj!(z), a'lj!(z)) < 4c
when z E {I¢I ~ 8} n Hl/)I ~ 8}, while
I(C",f)(z) - (C,pf)(z) - (Kf)(z)1
< If(¢(z)) - f('lj!(z)) I + If(a¢(z)) - f(a1/l(z))1 < 2p(¢(z), 'lj!(z)) + 2p(¢(z), 'lj!(z)) < 4e + 41:
184 when
PAMELA GORKIN. RAYMOND MORTINI, AND DANIEL SUAREZ Z E
{1cf>1
> 8} U {I'!/JI > 8}. Since the function f is arbitrary, IIG - G", - KII
4e + 4c, and since c is arbitrary we obtain (3.1).
~
D
COROLLARY 7. Let cf> and '!/J be two holomorphic self-maps of the unit ball. Then G - G", is compact if and only if either max {11(z)I~1
lim sup p(cf>(z), '!/J(z))
= O.
1"'(z)I~1
PROOF. It is clear that G and G", are compact, if max {11cf>11, 11'!/J11l < 1. On the other hand, if max {11cf>11, II'!/JII} = 1 and e is the parameter of Theorem 6, then (3.1) says that Gel> - G", is compact if and only if e = o. D Our next goal is to characterize the path components of composition operators on Hoo(Bn). We write G '" G", to indicate that there is a norm-continuous homotopy of composition operators joining Gel> with G",. Also, if K denotes the ideal of compact operators, we write G "'e G", to indicate that there is an essential norm-continuous homotopy of classes {G", + K: cp: Bn -4 B n holomorphic} joining Gel> + K with G", + K. Let cf> be a holomorphic self-map of Bn. For x E M (HOO (Bn)) we can define
cf>(x) E M(Hoo(Bn)) by the rule : B n -4Bn to a self-map of M(Hoo(Bn)), which we also denote by cf>. The continuity of this extension is immediate. We now have everything we need to prove the main theorem of this paper. As indicated in the introduction, this theorem unifies and extends many of the results appearing in [11], as well as [8]. THEOREM 8. Let cf> and '!/J be holomorphic self-maps of the unit ball in en. Then the following are equivalent.
(a) (b) (c) (d)
G '" G",. G "'e G",. IIGeI> - G",II < 2. SUPzEBn p(cf>(z), '!/J(z)) < 1.
PROOF. (a) => (b) is obvious. (c) ¢:} (d). A boundary for HOCJ (Bn) is a closed set F c M (HOCJ (Bn)) such that Ilfll = SUPxEF If(x)1 for all f E HOCJ(Bn). It is clear that the closure B n of B n in M(HOCJ(Bn)) is a boundary for HOCJ(B n ), and since oHOCJ(Bn) is the intersection of all the boundaries [4, p. 10], then oHOCJ(Bn) c F. The equivalence then follows from (2.1). (b) => (c). By hypothesis there is a family {cf>t}, with t E [0, 1], of holomorphic self-maps of Bn such that cf>o = cf>, cf>1 = '!/J and for every c > 0 there is some 8 > 0 satisfying
< c: if It - sl < 8. Then we can take finitely many points ty = 0 < ... < tm = 1 in [0,1] such that IIGt. - GeI>tHl lie < 1/2 for every i = 1, . .. , m - 1. We claim that IIGt - Gel>. lie
(3.2)
sup p(cf>t. (z), cf>t'+l (z)) < 1
zEBn
for every i. In fact, if r = max{llcf>d,lIcf>tHlll} < 1, then both functions map B n into the closure of r Bn, and since the pseudohyperbolic diameter of this ball
HOMOTOPIC COMPOSITION OPERATORS ON
Hoo(Bn)
185
is smaller than 1, we are done. If some of the maps have norm 1, then the first inequality of (3.1) tells us that there is some 0 < 8 < 1 close enough to 1 such that sup p(¢t;{Z), ¢tHl (Z)) < 3/4, {I., 1~t5}U{I"+11~6} while the set {I¢t, I < 8} n {I¢t'+ll < 8} is mapped by both functions into the ball 8Bn, whose pseudohyperbolic diameter is smaller than 1. Our claim follows. Since the closure of Bn in M(Hoo(Bn)) contains the Shilov boundary, (3.2) and (2.1) imply that IIC" - C' and '1/), such that C'" '" C'" but C - C,,) is not compact. Let n = {UJ ED: ~1-=-!:1 > ~} be a nontangential region in D at the point Z = 1. We want to estimate p(w, (w + 1)/2) for wEn. We recall that p(z, w) = Iz - wi/II - zwl for z, wED. By straightforward calculation,
( ( + 1)/2) -1
p w, w
=
11 -lwl 2+ 1- wi Il-w I
(1 -lwl)(1 + Iwl) 1 3 Il-w I + ~
~
and
- Iwl21 11+ 1l-w
p(w, (w + 1)/2)-1
> ~ (1 + 1- Iw12) 1- w
1-lwl 2
1+ 11_wI2(1-~w)
1+ 1+ Iwl > ~
>
2
- 2
when wEn. That is,
1
2
3 ~ p(w, (w + 1)/2) ~ 3
(4.1)
for all wEn. Let c.p : D---+n be a one-to-one and onto holomorphic function and define 4>, 'I/J : Bn---+Bn by
= (c.p(Z1), 0, ... ,0) and that 114>11 = 11'1/)11 = 1. For z
4>(Z1,"" zn) It is clear that
'I/J(Z1, ... , zn) E
= ((c.p(zt) + 1)/2,0, ... ,0).
Bn, a straightforward calculation shows
p(¢(z), 'I/J(z)) = p(c.p(zd, (c.p(zd
+ 1)/2).
Since c.p(zd E n, the inequalities in (4.1) show that p(4)(z), 'I/J(z)) E [1/3,2/3]. Therefore Theorem 8 says that C'" '" C"', while Corollary 7 says that C - C'" is not compact..
5. Endomorphisms of HOO(D) In this section we investigate the path components of elldomorphisms of H OO (D). For x E M(HOO(D)), the Gleason part of x is P(x) = {y E M(HOO(D)) : p(x,y) < I}. Since the condition p(:r, y) < 1 is an equivalence relation, the Gleason parts form a partition of M(HOO(D)). In [7] Hoffman produced a continuous and onto map Lx: D---+P(x) such that Lx(O) = x and foLx E HOO(D) for every x E M(HOO(D)) and f E HOO(D). There are two possibilities: either Lx(z) = x for all zED (so P(x) = {x}) or Lx is one-to-one. We write G = {x E M(HOO) : Lx is one-to-one} and
r
= {x E M(HOO) : Lx = {x}}.
It is well-known that every endomorphism T of HOO(D) can be factored as T = C",CL", , where 4> is a holomorphic self-map of D and x E M(HOO(D)). Although it is clear that this factorization is not unique, two different factorizations of the same endomorphism are related in the following way (see [2]): if p(x, y) < 1, then there is a biholomorphic map r of D (depending on x and y) such that Ly(z) = Lx(r(z))
HOMOTOPIC COMPOSITION OPERATORS ON
Hoo(B n )
for every zED. This means that every endomorphism of the form T also be factored as
187
= Cq,CLy can
T = Cq,CLy = Cq,CTCL", = CToq,CLx ' Of course, if x E rand p(x, y) < 1, then x = y and T = C Lr . LEMMA 10. Let x E G and A = Ej=1 )..,jCq,j' where composition operators on H'XJ(D). Then IIACLJI = IIAII.
)..,j
E C and Cq,j are
PROOF. Since IIACLxll ::; IICLxllliAIl ::; IIAII, one direction is easy. For the other direction, if 0 < f < 1, there exists a function f in the ball of HOO(D) such that IIA(f)11 > (1 - f)IIAIi. By the definition of the norm, there exists r with o < r < 1 such that n
L )..,jf(cPj(z)) ZETD j=l
sup IA(f)(z)1 = sup
ZETD
> (1 - f)2I1AII·
By a result of Hoffman [7, p. 91]' there exist Blaschke products bk such that (bk 0 Lx)(z) ~ z uniformly on compact subsets of D. But rD is a precompact subset of D, and therefore cPj (r D) is precompact for each j. That is, there is 0 < 0: < 1 such that Uj=l cP j(rD) C o:D. Fix {J with 0: < {J < 1. Since f is analytic, there is 8> 0 such that for z, wE (JD with Iz - wi < 8 we have If(z) - f(w)1 < f. Clearly we can also require 8 < (J - 0:. Therefore we may choose k sufficiently large so that I(b k 0 Lx)(cPj(z)) - cPj(z)1 < 8 for all z E rD. Thus, for k that large, z E rD and f as above, (b k o Lx)(cPj(z)) E (3D and consequently If(bk(Lx(cPj(z))) - f(cPj(z))1 < f. Therefore there exists a constant At depending only on nand )..,l"",)..,n such that II ACLx II
~
sup I(ACLx(f
0
bk))(z)1
zErD
n
sup
IL
zErD j=l
)..,j(f 0 bk 0 Lx)(cPj(z))1
n
>
sup
IL
zErD
> Letting
f ~
)..,jf(cPj(z))I- Mf
j=l
(1 - f)2I1AII- Mf.
0 yields the desired result.
o
THEOREM 11. Let T 1, T2 E End(HOO(D)). Then the following ar'e equivalent.
(a) T1 rv T2 in End(HOO(D)). (b) IIT1 - T211 < 2. (c) There exist x E M(HOO(D)) and holomorphic self-maps cP. 'ljJ of D such that T1 = Cq,C Lx • T2 = CtfJCLx and IICq, - CtfJlI < 2. PROOF. Suppose that (a) holds. Then there is a homotopy
G: [0, 1] ~End(HOO(D)) with G(O) = T1 and G(I) = T 2. We can find finitely many points 0 = it < ... < tn = 1 such that IIG(tj) - G(tj+d II < 2 for j = 1, ... , n - 1. Lemma 3 then says that IIG(O) - G(I)1I < 2. Suppose that (b) holds and write T1 = CLxoq, and T2 = CLyo"" where x,y E M(HOO(D)) and L' by (g, 8' J) = (8g, J) for all f E L' and gEL (see [5], Section 97). Then 8' E Cb(L'). If there is no reason for confusion, we will denote (n, E, p,) by (E) only. Q
In
DEFINITION 1. Let E and F be vector lattices and let T : E ---> F be a linear operator. Then (i) T is positive (denoted by T ~ 0) whenever Tx ~ 0 for all x ~ 0; T is called strictly positive (denoted by T »0) if Tx > 0 for all x > O. (ii) T is order continuous whenever Txc< -+ 0 in order for every net (x satisfying X -+ 0 in order. (iii) T is order bounded if it maps order bounded subsets into order bounded subsets. Q )
Q
For a Banach function space (E, II . liE) defined on some finite measure space ~ E ~ Ll (E, I.L), we define the following (see [2), p69).
(n, E, p,) for which Loo(E, p,)
DEFINITION 2. (i) The linear map T : E -+ E is called averaging if for all f E Loo(E) and all gEE we have that T(JTg) = Tf· Tg. (ii) T: E -+ E is called contractive if II T II ~ 1. DEFINITION 3. Let (n, E, p,) be a probability space (i.e. 1.L(n) = 1) and let Eo be a sub-a-algebra of E. For fELl (E), we denote by lFP(J I Eo) the IL-a.e. unique Eo-measurable function with the property that
i
lFf'(J I Eo)dl.L
=
i
fdp,
for all A E Eo. The function lFP(J I Eo) is called the conditional expectation of f with respect to Eo. If there is no reason for confusion, we will denote the p,-a.e. Eo-measurable function lElL('IE o) by lE('IE o) only. The existence of lE(J I Eo) is a consequence of
CHARACTERIZATION OF CONDITIONAL EXPECTATION.
191
the Radon-Nikodym theorem. The conditional expectationlE('IEo) can be extended from a mapping from Ll (E) into itself, to a mapping from M+(E) into itself. If f E M+(E,), then 1E(f I Eo) E .l\J+(E) is defined by 1E(f I Eo) = suplE(fn I Eo), where 0 :::; fn E Ll (E) (71. = 1,2, ... ) satisfy 0 :::; fn i f J,L-a.e. The conditional expectation operator has the following properties. For a proof of properties (i) to (vi) we refer to [4], p7; for property (vii) we refer to [3], p7. 1. (i) lE(o:f + /1g I Eo) = 0:1E(f I Eo) + /11E(g I Eo) for all f,g E M+(E) and for all 0:::; 0:,/1 E R (ii) 0 :::; f :::; 9 in M+(E) implies that 0 :::; 1E(f I Eo) :::; lE(g I Eo) and if 1E(lfll Eo) = 0, then it follows that f = O. By virtue of positivity we have 11E(f I Eo)1 :::; 1E(lfll Eo). (iii) 0:::; fn i f IJ,-a.e. implies that 0:::; lE(fn I Eo) i 1E(f I Eo) J,L-a.e. (iv) lE(gf I Eo) = glE(f I Eo) for all f E M+(E) and all 9 E M+(Eo). (v) If 9 E M+(Eo) and f E M+(E), then fA gdJ,L = J~ fdJ,L for all A E Eo if and only if 9 = 1E(f I Eo) IJ,-a.e. (vi) If Eo c Ao are sub-a-algebras of E, then 1E(f I Eo) = 1E(1E(f I Ao) I Eo) for all 0:::; f E M+(E). (vii) If f E M+(E) is such that 1E(f I Eo) E Lo(E), then we also have that f E Lo(E). PROPOSITION
DEFINITION
4. The domain domlE('IE o) of 1E('IEo) is defined by
domlE('IE o) := {f E Lo(E) : 1E(lfll Eo) E Lo(Eo)}. Clearly, domlE(·IE o) is an ideal in Lo(E) which contains L 1 (E). For f E dom 1E(·1 Eo), we define: 1E(f I Eo) := 1E(f+ I Eo) -1E(r I Eo). This defines a positive linear operator 1E('IEo) : domlE('IE o)
--+
Lo(Eo) C Lo(E).
Let (n, E, l.l) be a probability space and let L carrier n. Set
c
M(L) = {m E Lo(E) : 1E(lmfll Eo) E L
Lo(n, E, J,L) be an ideal with
V f E L}.
Since L C Lo(E), we have that mf E domlE('IE o) for all m E M(L) and f E L. For m E M(L) we define Smf : L --+ L by
Smf := lE(mf I Eo)
V f E L.
Sm is order bounded and ISml :::; Simi' Sm is also order continuous. The following proposition will be applied in the sequel. A proof can be found in [3], (p8). PROPOSITION
2. Let (n, E, J,L) be a probability space and Eo C E a sub-a-
algebra. (i) If f E domlE('IE o) and 9 E Lo(Eo), then it follows that gf E domlE('IE o) and lE(gf I Eo) = glE(f I Eo). (ii) If f E Lo(E), then f E domlE('IE o) if and only if there exists a sequence {A n }:'=1 in Eo such that An in and .
{
JAn
IfldJ,L
", < yx, z >",=< y, zx* >", for any y, z E E and 0: E A. x* is not necessarily unique. In case, E is proper (viz. Ex = (0), implies x = 0), then x* is unique and * : E -+ E : x f-+ x* is an involution (see [4: p. 451, Definition 1.1 and p. 452, Theorem 1.3]). Throughout of this work the considered algebras are over the field of complexes. To fix notation we recall the following. Let (E, (P"')"'EA) be a complete locally m-convex algebra and
(1.1)
P'" : E
-+
E/ker(p",) == E", : x
f-+
p"'(x) :=x + ker(p",)
the respective quotient maps. Then Ilx",ll", := p"'(x), x E E, 0: E A defines on E", an algebra norm, so that E", is a normed algebra and the morphisms P"" 0: E A are continuous. E"" 0: E A denotes the completion of E", (with respect to II . II",). A is endowed with a partial order by putting 0: :::; /3 if and only if p"'(x) :::; P(3(x) for every x E E. Thus, ker(p(3) 1): C*-property implies Pa(e)(l-po(e)) = 0 for every Q E A. If po(e) = 0 for some Q E A, then Ileoli o = 0, where eo = e + ker(Pa) is the respective unit
THE KRULL NATURE OF LOCALLY C'-ALGEBRAS
199
element in the factor algebra EOl == EOl (see aslo [1: p. 32, Theorem 2.4] and [11: p. 91, Theorem 4.1]). Thus eOl = 0, which is a contradiction. Thus, pOl(e) =I- 0 for every Q E A, hence POl (e) = 1 for every Q E A. 0 As a consequence of Corollary 2.2 and Proposition 2.5 we have the next. COROLLARY 2.6. Every proper complete locally m-convex H* -algebm with continuous involution and a normal unit is a Krull algebm. Our next aim is to provide another proof to the fact that a F'rechet locally C*-algebra is Krull (see Corollary 2.2). To do this, we use the notion involved in the next. DEFINITION 2.7. A projective system {(EOl , fOl,8)}OlEA of topological algebras is called perfect, if the restrictions to the projective limit algebra
= ~EOl = {(x Ol ) E
II
EOl: fOl,8(X,8) = XOl , if Q:S; (3 in A} OlEA of the canonical projections 7r0l : I10lEA EOl -+ E Ol , Q E A, namely, the (continuous algebra) morphisms
(2.4)
(2.5)
E
fOl = 7rOl IE =limE", : E
-+
EOl ,
Q
E A,
t--
are onto maps. The resulted projective limit algebra E = lim EOl is called perfect (topological) algebm. ~
LEMMA 2.8. Every Frechet locally m-convex algebm (E, (Pn)nEN) gives a perfect projective system of normed algebms. PROOF. For any n
:s; m in N, the connecting maps
(2.6)
with
fnm(X + ker(Pm)) = x + ker(Pn) are onto algebra morphisms (see, for instance, [11: p. 86, (3.6) and (3.7)]). So, since {(En, fnm)}nEN is a denumerable projective system of normed algebras, it follows that fn, n E N (see (2.5)) are onto, as well (see [10: p. 229, Theorem 8]). 0 The proof in the next result is an adaptation of that in Proposition 2.1. PROPOSITION 2.9. Any perfect projective limit of Krull algebms is a Krull algebm. PROOF. Let {(EOl,fOl,B)}OlEA be a perfect projective system of Krull algebras. Consider the projective limit algebra E = limEOl (see (2.4)), which is a closed subalgebra of the cartesian product topological algebra I10l EA EOl (see, for instance, [11: p. 84, Lemma 2.1]). For a proper closed left ideal I in E, fOl(I) is a (closed) left ideal in E Ol , Q E A. If f Ol (I) = EOl for every Q E A, then ~
1= limfOl(I) ~
= limfOl(I) = limEOl = E, ~
~
(see also [ibid. p. 87, Lemma 3.2]), which is a contradiction. Thus, fOl(I) =I- EOl for some Q E A. Since E Ol , Q E A is a Krull algebra, there exists a closed maximal regular left ideal, say M, with fOl(I) ~ M and hence I ~ f;;l(1Ol(1)) ~ f;;l(M),
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MARINA HARALAMPIDOU
where J;;l(M) is a closed maximal regular left ideal in E and this terminates the proof for closed left ideals. Similarly, for closed right ideals. D THEOREM
2.10. Any Prichet locally C*-algebm is a Krull algebm.
PROOF. Let (E, (Pn)nEN) be an algebra as in the statement. By [1: p. 32, Theorem 2.4], the respective normed algebras En, n E N in the Arens-Michael decomposition of E, are C* -algebras and hence Krull (see, for instance, [2: p. 56, Theorem 2.4.5]. In particular, {(En' Jnm)}nEN is a perfect system of normed algebras (see Lemma 2.8 and relation (2.6)). Proposition 2.9 assures that the projective limit algebra lim En is a Krull algebra and hence E is a Krull algebra, as it fol+-lows from Corollary 1.3 and the fact that E ~ lim En within a topological algebra +-isomorphism (see (1.3)). D References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C. Apostol, b*-algebms and their representation, J. London Math. Soc. 3(1971), 30-38. MR 44:2040. J. Dixmier, C*-Algebms, North-Holland, Amsterdam, 1977. MR 56:16388. R.S. Doran and V.A. Belfi, Chamcterizations of CO-Algebras. The Gel'fand-Na'tmark Theorems, Marcel-Dekker, 1986. MR 87k:46115. M. Haralampidou, On locally convex H*-algebms, Math. Japon. 38(1993), 451-460. MR 94h:46088. M. Haralampidou, Annihilator topological algebras, Portug. Math. 51(1994), 147-162. MR 95f:46076. M. Haralampidou, On complementing topological algebms, J. Math. Sci. 96(1999), 3722-3734. MR 2000j:46085. M. Haralampidou, On the Krull property in topological algebms (to appear). A. Inoue, Locally C* -algebras, Mem. Faculty Sci. Kyushu Univ. (SerA) 25(1971), 197-235. MR 46:4219. A. EI Kinani, On locally pre-C*-algebm structures in locally m-convex H*-algebms, Thrk. J. Math. 26(2002), 263-271. G. Kothe, Topological Vector Spaces, I, Springer-Verlag, Berlin, 1969. MR 40:1750. A. Mallios, Topological Algebms. Selected Topics, North-Holland, Amsterdam, 1986. MR 87m:46099. E.A. Michael, Locally multiplicatively-convex topological algebms, Mem. Amer. Math. Soc. 11(1952). (Reprinted 1968). MR 14,482a. Z. Sebestyen, Every C*-seminorm is automatically 8ubmultiplicative, Period. Math. Hung. 10(1979), 1-8. MR 80c:46065.
DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ATHENS, PANEPISTIMIOPOLlS, ATHENS 15784, GREECE E-mail address:mharalamOcc.uoa.gr
Contemporary Mathematics Volume 328, 2003
Characterizations and automatic linearity for ring homomorphisms on algebras of functions Osamu Hatori, Takashi Ishii, Takeshi Miura, and Sin-Ei Takahasi ABSTRACT. Automatic linearity results for certain ring homomorphisms between two algebras, in particular, semi-simple commutative Banach algebras with units are proved. For this purpose a representation by using the induced continuous mapping between the maximal ideal spaces and ring homomorphisms on the field of complex numbers is given. Ring homomorphisms on certain non-complete metrizable algebras into the algebras of analytic functions are also considered. A characterization of the kernel of complex-valued ring homomorphism on a commutative algebra is given. As a corollary of the results a complete description of ring homomorphisms on the disk algebra into itself is given in terms of prime ideals.
Introduction A ring homomorphism between two algebras is a mapping which preserves addition and multiplication. If we assume that the mapping is linear, then it is an ordinary homomorphism. In the case where the two algebras are just the field C of complex numbers, the assumption cannot be avoided; there are ring homomorphisms of C into C which are not linear nor conjugate linear (cf. [9]). The history of ring homomorphisms on C probably dates back to the investigation of Segre [19] in the nineteenth century and that of Lebesgue [12]. A similar remark applies to finite-dimensional Banach algebras. But this is not the case for several infinitedimensional ones; for instance, Arnold [1] proved that a ring isomorphism between the two Banach algebras of all bounded operators on two infinite-dimensional Banach spaces is linear or conjugate linear (cf. [5]). Kaplansky [8] proved that if p is a ring isomorphism from one semi-simple Banach algebra A onto another, then A is a direct sum Al EB A2 EB A3 with A3 finite-dimensional, p linear on All and p conjugate linear on A 2 . It follows that a ring isomorphism from a semi-simple commutative Banach algebra onto another with infinite and connected maximal ideal space is linear or conjugate linear. 2000 Mathematics Subject Classification. Primary 46JlO, 46E25; Secondary 46J40. The first, the second, and the fourth author were partialy supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science, Sports and Culture, Japan.
© 201
2003 AJnerican Mathematical Society
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O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
It is interesting to study ring homomorphisms on Banach algebras which are not necessarily injective or surjective. We may expect that a number of ring homomorphisms on infinite-dimensional Banach algebras are automatically linear or conjugate linear. By a routine work we see that a a ring homomorphism is real-linear if it is continuous. On the other hand, we can also arrive at automatic linearity for several ring homomorphisms by results in [14, 15, 20, 21, 13]; they studied and characterized *-ring homomorphisms between commutative Banach algebras with involutions and ring homomorphisms on regular commutative Banach algebras with additional assumptions. The heart of this paper is automatic linearity results for certain ring homomorphisms of a much more general nature. Throughout the paper A and B denote semi-simple commutative Banach algebras with units eA and eB respectively. The maximal ideal space for A is denoted by MA. In this paper, we denote the Gelfand transform of a E A also by a. For a ring homomorphism of C into C, we simply say a ring homomorphism on C. Let T be a ring homomorphism on C and x E MA. Then the complex-valued mapping p on A defined by
p(a) = T(a(x)),
aEA
is a typical example of a ring homomorphism. Semrl [20, Example 5.4] showed that there exists a complex-valued ring homomorphism other than this type. In section 2 we show that if a ring homomorphism of A into B satisfies a certain condition, say (m), then it is represented by a modified version of the above. Many ring homomorphisms satisfy this weak and rather natural condition (m): *-ring homomorphisms on involutive algebras; p{A)(y) = C for every y E M B ; p(A) contains a subalgebra of B. Thus our result generalizes the previous ones in [14, 20, 21, 13]. In section 3, by using results in section 2, we deduce some automatic linearity results for ring homomorphisms: p with (m) is real-linear on a closed ideal of finite co dimension in A; if p(CeA) = CeB and p(A) contains an element with an infinite spectrum, then p is linear or conjugate linear. It is a natural question: under the two hypotheses (1) p(CeA) c CeB and (2) p(A) contains an element with an infinite spectrum, does it follow that p is linear or conjugate linear? We give an affirmative answer under stronger hypotheses: (1) and (2)' p(A) contains an element whose spectrum contains a non-empty open subset. Problems in the same vein are also considered not only for Banach algebras but also for algebras of analytic functions. Bers [3] proved that if U and V are plane domains and H(U), H(V) are the rings of analytic functions on U, V respectively, then any ring isomorphism of H(U) onto H(V) is induced by a conformal (or anti-conformal) equivalence of V with U, thus the ring isomorphism is linear (or conjugate linear). Nakai [17] and Rudin [18] have shown this also holds for open Riemann surfaces. Ring homomorphisms which are not necessarily injective or surjective are also considered by many mathematicians (cf. [7, 10]). Among them, Becker and Zame [2] have proved automatic continuity and linearity for ring homomorphisms from certain complete metrizable topological algebras into the algebra of analytic functions on connected, reduced analytic spaces. In section 4 we also consider ring homomorphisms into the algebras of analytic functions. In particular, we consider the case of a ring homomorphism p from the algebra Rs of rational functions on C with poles off a subset SeC into an algebra of analytic funtions. Here Rs is a metrizable topological algebra, but it cannot be
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a Banach algebra by the Baire category theorem. We show that, for certain subsets S, p is automatically linear or conjugate linear, if the range of p contains a 11011constant function. We also give an example of S such that a ring hommomorphism that is neither linear nor conjugate linear, and whose range contains non-constant functions, is possible. In the final section we study ring homomorphisms into C: ring homomorphisms whose ranges contain only constant functions. We characterize the kernels of ring homomorphisms from a unital commutative algebra into C, which is compared with the one to one correspondence between maximal ideals and complex (linear) homomorphisms on commutative Banach algebras. As a corollary we show that there exists an injective ring homomorphism from an algebra which consists of analytic functions into C. We also give a complete description of the ring homomorphisms on the disk algebra in terms of prime ideals. We say that a ring homomorphism 7 on C is trivial if 7 = 0 or 7(Z) = Z (resp. z) for every Z E C. Other ring homomorphisms on C are said to be non-trivial. We note some properties of non-trivial ring homomorphisms on C, which are used later in this paper. For a proof of the existence of non-trivial ring homomorphisms, historical comments, and further properties, see [9]. It is easy to see that every non-zero ring homomorphism 7 on C fixes rational real numbers and 7(i) = i or -i. If 7 is non-trivial, then 7 does not preserve complex conjugation. (This is a standard fact. Here is a proof. Suppose 7 does preserve complex conjugation: 7(Z) = 7(Z) for every Z E C. Then 7(JR) C JR, that is, 7 is a ring homomorphism on the set of all real numbers R If x> 0, then 7(X) = (7( JX))2 > O. It follows that 7 is order preserving on R Since 7(r) = r for every rational real number r, we have 7(X) = x for every real number x. Thus 7(Z) = Z (resp. 7(Z) = z) for every Z E C if 7(i) = i (resp. 7(i) = -i), which is a contradiction.) It is easy to see that T is non-trivial if and only if 7 is discontinuous at every (resp. one) point in C. Thus, if T is non-trivial, then it is unbounded on every neighborhood of zero. It follows that there exists a sequence {w n } of complex numbers which converges to 0 such that IT(Wn)1 tends to infinity as n -> 00 if 7 is non-trivial. If the ring homomorphism on C is onto, then it is said to be a ring automorphism on C. Note that there is a non-zero ring homomorphism on C which is not a ring automorphism. We also note that there is a non-trivial ring automorphism on C (cf. [9, 11]).
1. Partial representation If ¢ is a non-zero complex homomorphism on A, then there exists a unique x E MA such that ¢(a) = a(x) for every a E A. By this fact a well-known representation of a (linear) homomorphism VJ from A into B follows: There exists a continuous mapping defined on {y E MB : VJ(a)(y) :I 0 for some a E A} into MA such that
VJ(a)(y)
= a((y)),
a E A,
y E {y E MB : VJ(a)(y)
:I 0 for
some a E A}.
On the other hand, if is a continuous mapping of MB into AfA and Ty is a ring homomorphism on C for every y E AfB , then
p(a)(y) = Ty(a((y)),
a E A,
y E MB
defines a ring homomorphism from A into the algebra of all complex-valued functions on M B . Thus it defines a ring homomorphism from A into B under the condition that T. (a ( (. )) is in B for every a E A, and this is the case when M B
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
204
is finite. A problem is the converse: Is every ring homomorphism represented as above? A negative answer is known even in the case where B = C by the example due to Semd [20, Example 5.4]. Nevertheless, we show that a partial representation is still possible in this section. DEFINITION 1.1. Let p be a ring homomorphism of A into Band y a point in M B . The induced ring homomorphism py of A into C is defined by
py(a) = p(a)(y), Let Ie : C
-+
A be defined by IdA)
= AeA
aE
A.
for every A E C. We denote Ty
= pyole.
For every y E M B , the induced mapping Ty is a ring homomorphism on C. DEFINITION 1.2. Let p be a ring homomorphism of A into B. We denote: Mo = {y E MB : Ty = a}; !vlt = {y E MB : Ty(Z) = Z for every Z E C}; ALl = {y E MB : Ty(Z) = Z for every Z E C}; Md,l = {y E MB : Ty is non-trivial and Ty(i) = i}; Md,-l = {y E !vIB : Ty is non-trivial and Ty(i) = -i}. LEMMA 1.3. Let p be a ring homomorphism of A into B. Then M o, Ml U Md,l and M-1 UMd,-l are clopen (closed and open) subsets of MB. The subsets M1 and M-l are closed in M B . PROOF. By the definitions it is easy to see that Mo = {y E MB : p(ieA)(y) = a}, M1 U Md,l = {y E MB : p(ieA)(y) = i}, and ALl U Md,-l = {y E MB : p(ieA)(y) = -i}, so they are clopen since p(ieA) is continuous on M B . Next we show that Jl,ft is a closed subset of MB. Let y E Md,l' Since Ty is non-trivial, there exists a complex number A such that Ty(A) =I- A. Put
Then G is an open neighborhood of y. We also see that G n M1 = 0. It follows that ]\,{1 is a closed subset of MB since M1 U M d,l is clopen. In the same way, we see that M-1 is a closed subset of M B . 0 Suppose that p is a ring homomorphism of A into B. If y E M 1 , then it is easy to see that Py is a non-zero complex homomorphism on A. Thus there exists a unique cp(y) in MA with
p(a)(y) = a(CP(y)),
a E
A.
In a way similar to the above we arrive at a partial representation as follows:
p(a)(y) =
a, { a(CP(y)), a(CP(y)),
yEMo, yE M 1 , y E M_ 1 .
If y E Md,l U Md.-1, then the situation is complicated, in particular, ring homomorphisms with large Md,l U Md,-l are possible (cf. [20, Examples 5.3 and 5.4]).
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
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2. Ring homomorphisms which satisfy the condition (m) In general the kernel of a non-zero ring homomorphism of A into C is a prime ideal and need not be a maximal ideal. (See section 5 in this paper.) In this section we consider ring homomorphisms P of A into B which satisfy the condition that the kernel of the induced ring homomorphism Py for each y E MB defined by
Py(f) = p(f)(y),
f EA
is a maximal ideal. DEFINITION 2.1. Let P be a ring homomorphism of A into B. We say that P satisfies the condition (m) if Py is zero or ker Py is a maximal ideal of A for every yEMB. By the following Lemma 2.2, if py(A) = C for every y E M B , in particular, if p(A) :J CeB, then (m) is satisfied. A *-ring homomorphism also satisfies the condition (m). (See the proof of Corollary 2.5.) LEMMA 2.2. Let Po be a non-zero ring homomorphism of A into C. Then the following are equivalent. (1) The kernel ker Po of Po is a maximal ideal of A. (2) The equation po(A) = PO(CeA) holds. (3) There exist a non-zero ring homomorphism 7 on C and an x E MA such that the equation po(a) = 7(a(x)) holds for every a E A. In this case 7 = Po 0 Ie. Such a 7 and x are unique. (4) The mnge Po(A) is a subfield ofC which contains a non-zero complex number. PROOF. First we show that (1) implies (2). Suppose that ker Po is a maximal ideal. Then there exists a non-zero complex homomorphism
2n + ITYn (al(xn) + ... + an-l(xn))l· Then a2 = onbn is a desired function for n.) Then E::'=l an converges in A, say to a. Then a(xn) = al(x n ) + ... + an(xn) since the Banach norm on A dominates the uniform norm on AlA. On the other hand
so that p(a) is unbounded, which is a contradiction proving that q>(Md) is a finite set. Let q>(Md) = {Xl, ... , xn} and Yj = q>-l(Xj) n Md for each j = 1,2, ... , n. Choose an a E A such that a(xl) = 1, a(x2) = ... = a(xn) = O. Then p(a)(y) = 1 if y E Y l while p(a)(y) = 0 if y E Md \ Y l . Because p(a) is continuous, Y1 is clopen in Md; but Md is open in M B , so Y l is open in MB. In the same way we see that Yj is an open subset of MB for each j = 2,3, ... n. Finally we prove that q> is continuous. Since Yj is open and q>(Yj) = Xj, we only need to prove that q> is continuous at each point in Ml U M_ l . Let y E Ml and {y>.hEA be a net which converges to y. Without loss of generality we may assume that {y>.} c Ml U Md,l since Ml U Md,l is clopen. Suppose that {q>(y>.)} does not converge to q>(y), that is, there is an open neighborhood G of q>(y) such that for every A E A there exists a A' 2:: A with q>(YN) rf. G. There exist a finite number of points aI, ... , am in A and a positive real number e such that
{x
E
MA : laj(x) - aj(q>(y))1 < e,j = 1,2, ... , m}
C
G.
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
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Since {p(aj)(y,x)} converges to p(aj)(Y) for each j, there exists a Ao E A such that
JTy>. (aj(iP(y,x))) - aj(iP(y))J < e holds for every A ~ AO and j = 1,2, ... , m. Suppose that A ~ Ao, then there exists a A' ~ A such that iP(y,x/) f/. G, so that Jajl(iP(y,x/)) - ajl(iP(y))J ~ e for some j'. lt follows that YN E Md,l. We also see that iP(y,x/) E {Xl, ... ,Xn } \ {iP(y)}. There exists an a E A such that a(iP(y)) = 1 and a = 0 on {Xl, ... ,Xn } \ {iP(y)}. We conclude that for every A with A ~ Ao there exists a A' ~ A such that p(a)(y,x/) = 0 and p(a)(y) = 1, which is a contradiction since p(a) is continuous on M B . Thus we have that {iP(y,x)} converges to iP(y), so iP is continuous at y. In the same way we see that iP is continuous at each point in M_ I . We have proved that iP is continuous on MB \Mo. 0 Note that the set Md,l U Md,-l need not be a finite set or even a closed subset of AlB (cf. [20, Example 5.3]). In [21] the authors proved the following corollary in the case where A is regular and satisfies a certain additional condition. Now we can remove these conditions. COROLLARY 2.4. Let p be a ring homomorphism from A into B. Suppose that py(A) = C for every y E M B . Then there exists a continuous mapping iP of MB into MA and a non-trivial ring automorphism Ty on C for every y E Md,l U Md,-l s1Lch that Y E MI , a(iP(Y))' { p(a)(y) = a(iP(y)), Y EM_I, Ty(a(iP(y))), Y E Md,l U Md,-l.
Moreover iP(Md,1 U Md,-d is a finite subset of MA. PROOF. By Lemma 2.2 we see that ker py is a maximal ideal, so the condition (m) is satisfied. The conclusion follows by Theorem 2.3. In particular, Ty = Py 0 Ie is onto, thus it is a non-trivial automorphism on C for y E Md,l U Md,-l. 0
Theorem 2.1 in [13] for the case of unital and semi-simple commutative Banach algebras is also deduced from Theorem 2.3 COROLLARY 2.5. Suppose that A is involutive and B is symmetrically involutive. Let p be a *-ring homomorphism. Then MB = Mo U MI U M-I and there exists a continuous function iP from MB \ Mo into AlA such that
a(iP(Y))' p(a)(y) = { 0, --:a('-=-iP..,...( y77")) ,
yE
MI ,
yE Mo,
y EM_I·
PROOF. Since p is a *-ring homomorphism, it it easy to see that Ty(Z) = Ty(Z) for every Z E C and for every y E MB \ Mo. It follows that Ty is 0 or linear or conjugate linear. Thus the conclusion follows. 0
3. Automatic linearity One of the reasons for ring homomorphisms between infinite-dimensional Banach algebras to be linear or conjugate linear is that the range contains an element with large spectrum. In this section we show evidence of this.
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O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKA HAS I
COROLLARY 3.1. Let p be a ring homomorphism of A onto B. If MB contains no isolated point, then p is real-linear. If MB is infinite and connected, then p is linear or conjugate linear. PROOF. Since p is a surjection, py(A) = C for every y E M B , so the condition (m) is satisfied by Lemma 2.2. We also have that the induced mapping is injective. Thus Md,l U Md,-l is a (possibly empty) finite set. Because Md,l U Md,-l is open (by Lemma 1.3), each point of Md,l U Md,-l is isolated in M B . If MB contains no isolated point, then Md,l U Md,-l = 0. Thus p is real-linear. If lvIB is infinite and connected, then MB contains no isolated point, so Md,l U Md,-l = 0. It follows by Lemma 1.3 that MB = lvh or MB = M_ l . Thus P is linear or conjugate linear. 0 COROLLARY 3.2. Let p be a ring homomorphism of A into B. Suppose that p satisfies the condition (m). Then there exists a (possibly empty) finite subset {Xl, ... , xn} of M A such that p is real-linear on the finite-codimensional closed ideal {a E A: a(xj) = O,j = 1,2, ... ,n} of A. PROOF. Put {Xl,""X n } = (Md,l UMd,-d. (The set is finite by Theorem 2.3.) Then for every a E {a E A: a(xj) = O,j = 1,2, ... ,n} p(a)(y)
=
{Ty(a((Y))), 0,
y E Ml U M_l' Y E Mo U Md,l U Md,-l.
Since Ty is real-linear for every y E Ml U M_l' the conclusion follows.
o
COROLLARY 3.3. Let p be a ring homomorphism from A into B such that p(CeA) = CeB. Then we have that Mo = 0, and there exists a continuous mapping from MB into MA such that one of the following three occurs. (1) P is linear: p(a)(y)
= a((y)),
a E A,
y E MB .
a E A,
y E MB .
(2) p is conjugate linear: p(a)(y)
= a((y)),
(3) There exists a non-trivial ring automorphism T on C such that p(a)(y)
= T(a((y))),
a E A,
y E M B.
In particular, if there exists an a E A such that the spectrum of p( a) is an infinite set, then p is linear or conjugate linear.
PROOF. For every y E MB, we have py(CeA) = C, so py(A) = py(CeA) = c. Thus ker Py is a maximal ideal of A by Lemma 2.2, so that the condition (m) is satisfied. Since p(ieA) E CeB, MB = Ml U Md,l or MB = M_l U Md,-l. Suppose that Md,l U Md,-l = 0. Then (1) or (2) occurs. Suppose that there exists some Yd E Md,l and some Yl E M l · Then there is a complex number>' with Tyd (>') =I=- >., so that p(>.eA) is not a constant function, which contradicts our hypothesis. Thus Md,l =I=- 0 implies that MB = Md,l. It is also easy to see that Ty is identical for every y E Md,l since p(CeA) consists of constant functions. Thus (3) follows. In the same way we see that (3) follows if Md,-l =I=- 0. Suppose that there exists an a E A such that p(a)(MB) is infinite, then (3) does not occur since ( M B) is a finite set in this case. It follows that p is linear or conjugate linear. 0
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
209
Corresponding results for ring homomorphisms on rings of analytic functions are proved by Kra [10, Theorem I]. Suppose that p is a ring homomorphism from A into B which satisfies two conditions: p(CeA) C CeB; there exists an a E A such that p(a)(MB) is infinite. Does it follow that p is linear or conjugate linear? Although the authors do not know the answer, we can provide a positive answer under a stronger condition. THEOREM 3.4. Let p be a ring homomorphism from A into B. Suppose that the following two conditions are satisfied: (i) p(CeA) C CeB; (ii) there exists an a E A such that p(a)(MB) contains a non-empty open subset ofC. Then p is linear or conjugate linear. PROOF. Since p(CeA) C CeB we may suppose that pole is a non-zero ring homomorphism on C. We have two possibilities: po Ie(i) = i; po Ie(i) = -i. We show that, in the first case, po Ie(z) = z for every complex number z, so it will follow that p is linear on A. (In the same way we see that p is conjugate linear if pole( i) = -i.) Suppose that pole( i) = i. We show that pole is continuous on C. For this it is enough to show that pole is continuous at O. Suppose not. Then there is a sequence {w n } of non-zero complex numbers which converges to 0 such that {poIe(w n )} does not converge to O. Without loss of generality we may assume that Ip 0 Ie(w n ) I ~ 00 as n ~ 00. Let a be in A such that p(a)(MB) contains a non-empty open subset G of the complex plane. Let s be a complex number in G such that the real part and the imaginary part of s are both rational numbers. Put Zn = S + 1/ po Ie(w n ). Then there is a positive integer mo such that Zm E G for every m ~ mo since Ip 0 Ie(w n )I ~ 00 as n ~ 00, so ZmeB - p(a) ~ B- 1 . Thus we have (8 + l/wm)eA - a ~ A-I. Then 8 + l/wm is in the spectrum of a for every m ~ mo, which is a contradiction since Is + l/wnl ~ 00 as n ~ 00. It follows that pole is continuous at 0, thus on C, so pole( w) = w for every complex number w since p(ieA) = i. Then we see that p is linear on A. D Note that either of the two conditions (i) and (ii) in the above theorem itself does not suffice for p to be linear or conjugate linear. Let T be a non-trivial ring automorphism on C. Suppose that x E MA and tp from A into C is defined by tp(a) = a(x) for every a E A. Put P = TO tp. Then p is a ring fomomorphism with (i) since p(CeA) = C, but P is neither linear nor conjugate linear; p is not even real-linear. Let D be the closed unit disk in the complex plane. Let D + 3 = {z E C: Iz - 31::; I} and X = D U (D + 3). Define p(f)(z) = {f(Z)' f(z - 3),
zED zED+3
for every f E C(D). Then p is a ring homomorphism from C(D) into C(X) with the condition (ii). But p is neither linear nor conjugate linear. Even more is true. There is a ring homomorphism with the condition (ii) which is not real-linear. Recall that the disk algebra A(D) is the algebra of all complex-valued continuous functions on D which are analytic on the interior D of D. Suppose that
K = {O} U {l/n : n is a positive integer}, X
= {2} U D
and Y
= K U {z E C : Iz - 31::; I}.
210
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
Let
A = {f
E C(X) :
flD
E
A(D)}
and B = C(Y), where CO denotes the algebra of all complex-valued continuous functions on '. Let ¢ be the ring homomorphism from C into C(K) defined in [20, Example 5.3]. For f E A, put
f(2), y=O, p(f)(y) = { ¢(f(2))(1/n), y = lin, f(y - 3), Iy - 31 :::; 1. Then p is a ring homomorphism and satisfies the condition (ii). But (i) is not satisfied and p is not real-linear on A.
4. Ring homomorphisms into algebras of analytic functions Suppose that A is a completely metrizable topological algebra with an identity and r(X, Ox) is the algebra of global sections of a connected reduced complexanalytic space (X, Ox). Becker and Zame [2] proved among other things that if p is a ring homomorphism from A into r(X, Ox) such that the range of p contains a non-constant section, then p is linear or conjugate linear. This is not the case for ring homomorphisms on non-complete algebras. (Suppose that P is the algebra of polynomials on C and H(CC) is the algebra of entire functions. Let T be a nontrivial ring homomorphism on C and define p on P by p(Eanz n ) = ET(an)Zn for every polynomial E anz n . Then p is a ring homomorphism, but it is neither linear nor conjugate linear. By Theorem 5.1 there also exists an injective ring homomorphism from Pinto C since {O} is a prime ideal in P.) Nevertheless we show automatic linearity results for ring homomorphisms on certain non-complete metrizable algebras. THEOREM 4.1. Suppose that A is a complex (commutative or non-commutative) algebra with unit e. Suppose that Y is a non-empty set and B is a complex algebra of complex-valued functions on Y which contains the constant functions. Suppose that for every non-constant function b E B the range of b contains a non-empty open subset of c. Let p be a ring homomorphism from A into B. If there exists an element a in A such that the resolvent set of a contains a non-empty open subset G of C and p( a) is non-constant, then p is linear or conjugate linear. PROOF. It is easy to see that p( e) = 0 or 1 since the range of a non-constant function in B contains a non-empty open set. If p( e) = 0, then p is 0 on A and so p is linear. Suppose that p(e) = 1. In the same way as above, we see that p(ie) = i or -i. We show that p is linear if p(ie) = i. (If p(ie) = -i, then 15 defined by 15(a) = p(a) will be linear by what we will show, so p will be conjugate linear.) We will show that p(,Xe) = ,X for every complex number 'x. First we show that p(Ce) C cc. Suppose not. Then there is a complex number ,x such that p(,Xe) is a non-constant function. Note that Re'x or Im'x is irrational since p(xe) = x for every rational real number x by a simple calculation. Since p('xe)(Y) contains a non-empty open set, there exists r E p('xe)(Y) with Rer and Imr both rational. Then p(,Xe) - r is not an invertible element in B. Therefore (,x - r)e is not invertible in A, that is, ,x = r, which is a contradiction. Thus we have proved that p(Ce) C C, or p induces a ring homomorphism on cc. We denote the induced ring homomorphism also by p.
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
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Suppose that p is non-trivial on C. Since p(a) is not a constant, the interior of p( a) (Y) contains a complex number s whose real and imaginary parts are both rational, by our assumption for B. Since p is non-trivial, there exists a sequence {w n } of non-zero complex number such that Ip(wn)1 tends to infinity as n --+ 00 and s+ l/w n E G for every n. Since p(i) = i we see that p(s+ l/wn ) = s+ 1/ p(w n ) and we may assume that
s + l/p(w n ) E p(a)(Y) for every n. Thus p(a) - (s + 1/ p(wn )) is not invertible in B. It follows that a - (s + l/w n )e is not invertible in A, which is a contradiction, proving that p is trivial. Since p(i) = i, we have that p(A) = A for every complex number A. We conclude that p is linear on A. 0 The spectrum of each element in a Banach algebra is compact, so the conditions for A in Theorem 4.1 are satisfied by every Banach algebra with unit. Since the range of non-constant analytic function is a non-empty open subset of C, algebras of global sections on connected, reduced complex-analytic spaces satisfy the condition for B in Theorem 4.1. Thus we have the following, which is a version of a more general result of Becker and Zame [2, Theorem 3.1]. But our proof is considerably simpler. COROLLARY 4.2. Let Ao be a Banach algebra with unit. Suppose that p is a ring homomorphism from Ao into r(X, Ox), the algebra of global sections on a connected, reduced complex-analytic space (X , Ox). If p( Ao) contains a nonconstant section, then p is linear or conjugate linear.
Let S be a subset of C. We denote by Rs the algebra of all rational functions on C with poles off S. Although Rs is a wlital algebra, it cannot be a Banach algebra by the Baire category theorem. If S = C, then Rs = P, and so there is a ring homomorphism p on Rc into r( X , Ox) for (X, Ox) = C such that p(Rc) contains a non-constant function, while p is neither linear nor conjugate linear. In the case where C \ S contains an interior point, the situation is different; in this case we prove an automatic linearity result. COROLLARY 4.3. Let S be a subset of C whose complement contains interior points. Suppose that (X, Ox) is a connected, reduced complex-analytic space and r(X, Ox) is the algebra of global sections. Suppose that p is a ring homomorphism from Rs into r (X , Ox). If the range of p contains a non-constant section, then p is linear or conjugate linear. PROOF. In the same way as in the proof of Theorem 4.1 we see that p(C) C C. Suppose that z denotes the identity function: z(w) = w for every complex number w. Then we have that p(z) is non-constant. (Suppose not. Then p(f) is a constant section for every f E Rs.) On the other hand z - A is invertible for every A E C \ S, which contains a non-empty open set. Thus the conditions in Theorem 4.1 are satisfied. It follows by Theorem 4.1 that p is linear or conjugate linear. 0
Note that every ring homomorphism p of R0 into r(X, Ox) is constant-valued for the empty set 0. (We see that p(C) C C as before. Suppose that p(f) is not a constant section for some non-constant rational function f. Then there is a complex number r in p(f)(X) with rational real and imaginary parts. It follows that f - r is not invertible in R0, which is a contradiction.)
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O. HATORI, T. ISHII, T. MIURA. AND S.-E. TAKAHASI
Let A be one of the algebras P, njj or the disk algebra A(D), where jj denotes the closed unit disk in C, and H(D) the algebra of analytic functions on the open unit disk. Although both P and no are dense in the disk algebra, automatic linearity results for ring homomorphisms on these algebras are different from each other. Suppose that p is a ring homomorphism from A into H(D) such that the range of p contains a non-constant function. If A = njj (resp. A(D), then p is linear or conjugate linear by Corollary 4.3 (resp. Corollary 4.2). But that is not the case for A = P. The ring homomorphisms defined by p(2:a n z n ) = 2:1'(a n )zn for polynomials 2: anz n are neither linear nor conjugate linear for non-trivial ring homomorphisms l' on C.
5. Complex-valued ring homomorphisms In this section we consider ring homomorphisms into the complex number field C. Suppose that A is a complex algebra and p is a non-zero ring homomorphism from A into C. Then the kernel ker p of p is a prime ideal. Recall that a proper ideal I of A is said to be a prime ideal of A if fg E I implies that f E I or gEl. By using well-known results of algebra, we see the converse is also valid; for every prime ideal such that the cardinal number of the quotient algebra of the algebra by the ideal is equal to that of the continuum, there exists a ring homomorphism into C whose kernel coincides with the ideal. Let K be an extension field of a field k. (Here and after a field means a commutative field.) We recall a subset S of K is said to be algebraically independent over k if the set of all finite products of elements in S is linearly independent over k. A subset T of K which is algebraically independent over k and is maximal with respect to the inclusion ordering is said to be a transcendence base of Kover k. By definition, for every transcendence base T of Kover k, K is algebraic over the quotient field k(T) of the polynomial ring of T over k. There exists a transcendence base of Kover k (cf. [11, Theorem X.l.I]). Using the same argument as in [9] we can prove the following (cf. [11, 20]). (This might be a standard fact. But we present here with a proof for the convenient of the readers.) THEOREM 5.l. Let A be a commutative complex algebra with unit e. Suppose that I is a prime ideal of A such that the cardinal number of AI I is that of the continuum c. Then there exists a ring homomorphism p from A into C such that kerp=I. PROOF. The quotient algebra AI I has no non-zero divisor of zero, for I is a prime ideal. We denote by K the field of fractions over AI I. Let Q be the field of complex numbers whose real and imaginary parts are both rational. Let TK be a transcendence base for Kover Q and T a transcendence base for Cover Q. Then the cardinal number of TK (resp. T) is c since that of AI I (resp. q is c. There exists au injection a defined from TK onto T. Since TK is algebraically independent, there is a unique extension from Q(TK ) onto Q(T), which is also denoted by a, and a is a ring homomorphism. Since C is algebraically closed and K is an algebraic extension of Q(TK), there exists an extension of a which defines a ring homomorphism of K into C by Theorem VII.2.8 in [11]. We also denote it bya. Let h be the natural homomorphism of A onto AI I. Put p = a 0 h. Then p is the desired ring homomorphism. D
AUTOMATIC LINEARITY FOR RING HOMOMORPHISMS
213
Note that the corresponding ring homomorphism p is not unique. Let T be any non-zero ring homomorphism on C. Then TOp is a ring homomorphism on A with ker p = ker TOp. As a corollary of Theorem 5.1 we display a pathological feature of ring homomorphisms on algebras of analytic functions into C; even injection can be possible. COROLLARY 5.2. Let A be a unital algebra which consists of holomorphic functions on a domain in Then there exists an injective ring homomorphism of A into Co
cn.
PROOF. Since the ideal containing only zero is a prime ideal and the cardinality of A is the same as that of the continuum, there exists a ring homomorphism p of A into C whose kernel consists only of zero, by Theorem 5.1. Then p is an injective ring homomorphism. 0 Note that the injective ring homomorphism in Corollary 5.2 can never be surjective if A contains non-constant functions since A is not a field. Note also that every ring homomorphism from a unital commutative C* -algebra into C cannot be injective if the dimension of the algebra is greater than one since {O} is not a prime ideal in this case. Together with the results in the previous sections we give a complete description of ring homomorphisms on the disk algebra A(D). COROLLARY 5.3. Let p be a non-zero ring homomorphism on the disk algebra into itself. Then ker p is a prime ideal. If the range of p contains a non-constant function, then p is linear or conjugate linear; there exists 'P E A(D) with 'P(D) c fJ such that zED, f E A(D) p(f)(z) = f 0 'P(z), or
p(f)(z) = f
0
z E fJ,
'P(z),
f E A(fJ).
On the other hand, suppose that'P E A(D) with 'P(D) a(f)(z) = f
0
'P(z),
c
D. Then
ZED,
f
E A(fJ)
zED,
f
E A(D)
defines a linear ring homomorphism and a(f)(z) = f
0
'P(z),
defines a conjugate linear ring homomorphism.
PROOF. A(D) has no non-zero divisors of zero, so the kernel of any ring homomorphism from complex algebra with unit element into A(D) must be a prime (algebra) ideal. If p(A(D) contains a non-constant function, then by Theorem 4.1 we see that p is linear or conjugate linear. Suppose that p is linear. Then it is well known and easy to prove, since the maximal ideal space of A(D) is the closed unit disk D, that there exists 'P E A(fJ) with 'P(D) c D such that p(f)(z) =
f
0
'P(z)
holds for every f E A(D) and zED. Suppose that p is conjugate linear. Let h : A(fJ) -+ A(fJ) be defined as h(f)(z) = f(2),
f
E A(D),
zED.
214
O. HATORI, T. ISHII, T. MIURA, AND S.-E. TAKAHASI
Then hop is a linear ring homomorphism on the disk algebra. It follows that there exists r.p E A(D) with r.p(D) c D such that
h 0 p(f)(z)
=
1 0 r.p(z) ,
zED,
I
E
A(D).
Thus we see that
p(f)(z) = 1 0 r.p(z), holds for every I E A(D) and zED. Conversely, suppose that r.p E A(D) with r.p(D) c D. Then it is easy to see that u(f)(z)
=
1 0 r.p(z),
zED,
IE A(D)
zED,
IE A(D)
defines a linear ring homomorphism and
u(f)(z) = 1 0 r.p(z),
defines a conjugate linear ring homomorphism.
o
Let n be a positive integer and An(D) the subalgebra of those I in A(D) whose n-th derivative I(n) on D is continuously extended up to D. An(D) is a unital commutative Banach algebra with the norm IIIlIn = L~=o III(k)lIoo/k! for I E An(D), where II . 1100 is the supremum norm on D. Then Corollary 5.3 is also valid for An(D). Prime ideals in A(D) and An(D) are studied in [16]. (See also [4] for the case of A(D).) Mortini proved that every non-zero prime ideal is contained in a unique maximal ideal. He in fact showed that a non-zero and nonmaximal prime ideal in An(D) (resp. A(D)) is dense in exactly one of the ideals {J E An(D) : 1(>") = 1'(>..) = ... = I(j)(>..) = O} for some 0 ~ j ~ n (resp. {I E A(D) : 1(>") = O}), >.. E aD. We also see by a theorem of Dietrich [4] that the cardinal number of the set of all prime ideals of A(D) which is contained in a maximal ideal {J E A(D) : 1(>") = O}, >.. E aD is 2', the cardinal number of the set of all the subsets of the continuum. Thus we see that there are 2' ring homomorphisms on the disk algebra. Acknowlegement. The authers would like to thank Professor Ken-Ichiroh Kawasaki for his valuable comments. They also would like to thank the referees for their careful reading of the paper and their valuable comments.
References [1] B. H. Arnold, Rings of opemtors on vector spaces, Ann. of Math. 45(1944), 24-49 [2] J. A. Becker and W. R. Zame, Homomorphisms into analytic rings, Amer. Jour. Math. 101(1979), 1103-1122 [3] L. Bers, On rings of analytic junctions, Bull. Amer. Math. Soc. 54(1948), 311-315 [4] W. E. Dietrich, Jr., Prime ideals in uniform algebms, Proc. Amer. Math. Soc. 42(1974), 171-174 [5] M. Eidelheit, On isomorphisms of rings of linear opemtors, Studia Math. 9(1940),97-105 [6] J. B. Garnett, Bounded analytic functions, Academic Press, New York, 1981 [7] H. Iss'sa, On the meromorphic junction field of a Stein variety, Ann. Math. 83(1966), 34-46 [8] 1. Kaplansky, Ring isomorphisms of Banach algebms, Canadian J. Math. 6(1954),374-381 [9] H. Kestelman, Automorphisms of the field of complex numbers, Proc. London Math. Soc. 53(1951), 1-12 [10] 1. Kra, On the ring of holomorphic functions on an open Riemann surface, Trans. Amer. Math. Soc. 132(1968),231-244 [11] S. Lang, Algebm (second edition), Addison-Wesley, California, 1984. [12] M. H. Lebesgue, Sur les tmnsformations ponctuelles, tmnsformaant les plans en plans, qu'on peut definir par des procedes analytiques, Atti della R. Acc. delle Scienze di Torino 42(1907), 532-539
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[13J T. Miura, Star ring homomorphisms between commutative Banach algebrns, Proc. Amer. Math. Soc. 129(2001), 2005-2010 [14J L. Molnar, The rnnge of a ring homomorphism from a commutative C' -algebrn, Proc. Amer. Math. Soc. 124(1996), 1789-1794 [15J L. Molnar, Automatic surjectivity of ring homomorphisms on H* -algebrns and algebrnic differences among some group algebrns of compact groups, Proc. Amer. Math. Soc. 128(2000), 125-134 [16J R. Mortini, Prime ideals in the algebrn An(D), Complex Variables Theory AppJ. 6(1986), 337-345 [17J M. Nakai, On rings of analytic and meromorphic functions, Proc. Japan Acad. 39(1963), 79-84 [18J W. Rudin, An algebrnic charncterization of conformal equivalence, Bull. Amer. Math. Soc. 61(1955), 543 [19J S. de Corrado Segre, Un nuovo campo di ricerche geometriche, Atti della R. Acc. delle Scienze di Torino 25(1889), 276-301 [20J P. Semrl, Non-linear perturbations of homomorphisms on C(X), Quart. J. Math. Oxford (2) 50(1999),87-109 [21J S.-E. Takahasi and O. Hatori, A structure of ring homomorphisms on commutative Banach algebrns, Proc. Amer. Math. Soc. 127(1999),2283-2288 DEPARTMENT OF MATHEMATICAL SCIENCE, GRADUATE SCHOOL OF SCIENCE AND TECHNOL-
950-2181 JAPAN E-mail address:hatorilDmath.se.niigata-u.ae.jp
OGY, NIIGATA UNIVERSITY, NIIGATA
NIIGATA CHUO HIGH SCHOOL, NIIGATA
951-8126 JAPAN
DEPARTMENT OF BASIC TECHNOLOGY, ApPLIED MATHEMATICS AND PHISICS, YAMAGATA UNI-
992-8510 JAPAN E-mail address:miura«lyz.yamagata-u.ae.jp
VERSITY, YONEZAWA
DEPARTMENT OF BASIC TECHNOLOGY, ApPLIED MATHEMATICS AND PHISICS, YAMAGATA UNI-
992-8510 JAPAN E-mail address:sin-eiOemperor.yz.yamagata-u.ae.jp
VERSITY, YONEZAWA
Contemporary :r...1athematics Volume 328. 2003
Carleson Embeddings for Weighted Bergman Spaces Hans Jarchow and Urs Kollbrunner ABSTRACT. We are going to discuss Carleson measures for the standard weighted Bergman spaces A~ (-1 < a < 00, 0 < p < (0). These are finite, positive Borel measures J.L on the unit disk in IC such that, given 0 < q < 00, A~ embeds, as a set, continuously into Lq(J.L). Such measures have been closely investigated by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking [11], [12]. We complement their results, in particular by characterizing compactness, order boundedness and related (absolutely) summing properties of the canonical embedding A~ C is a F'rechet space with respect to the topology of uniform convergence on compact subsets of 1lJ. Let da be normalized area measure on 1lJ. For each a > -1,
dao(z) :=
(a + 1) (1 -lzI 2 )O da(z)
is a probability measure on 1lJ. For each 0 < p < Bergman space is defined to be A~ := A~
00,
the corresponding weighted
1i(1lJ) n P(ao ).
is closed in LP (a 0); it is a Banach space if p 2:: 1 and a p - Banach space if Its (p- ) norm will be denoted by II . Ilo,p. A~ is a Hilbert space and has a reproducing kernel:
o < p < 1.
Ko(z, w) = K(z, w)o+2; here K(z,w) = (1 - ZW)-l is the reproducing kernel for the Hardy space H2. For reasons like this, the scale of Hardy spaces is often considered as the scale of weighted Bergman spaces which corresponds to a = -1. Some of the results below actually remain true for this case, and some can even be extended to analytic Besov spaces B~ (f E B~ {::} f' E A~+p). Nevertheless, in this paper we will only deal with the case -1 < a < 00. 3. Carleson measures All measures on 1lJ will be finite, positive Borel measures. Let -1 < a < 00 and 00 be given. We say that a measure /L on 1lJ is an (a, p, q) - Carleson measure if A~ c Lq(/L) and the embedding A~ '---+ Lq(/L) is continuous: there is a constant C> 0 such that IIfIILq(~) ::; C ·lIfIIA~ "If E A~. Given an (a, p, q) - Carleson measure, the canonical embedding I : A~ -> Lq(/L) will be referred to as a Carleson embedding. As mentioned in the introduction, a number of the results to follow remain true if we just require that f 1-+ f induces a bounded linear map A~ -> Lq(/L). Also, complex measures whose variation is (a, p, q) - Carleson can be incorporated. Moreover, there are extensions to analytic functions of several variables. However, we are not going to discuss such generalizations in this paper. We say that /L is a compact (a,p, q) - Carleson measure if the embedding A~ '---+ Lq(/L) exists and is compact. For example, an a.e. positive function h E Lq(a/3) defines the bounded multiplier Mh : A~ -> Lq(a/3) : f 1-+ fh iff the measure h q da/3 is (a,p, q) - Carleson. Moreover, discrete (a, p, q) - Carleson measures on 1lJ can be defined using appropriate versions of 'sampling sequences', etc.
o < p, q
A~ exists iff a o o'P- 1 is (o:,p,q)-Carleson. More generally, an arbitrary measure J.l on 10 is (0:, p, q) - Carleson if and only if, for every analytic map 'P : 10 ---> 10, CI{J maps A~ boundedly into Aq(J.l) := 1i(1O) n Lq(J.l). In fact, the condition applied to the identity of 10 shows that J.l is (o:,p,q)Carleson. On the other hand, if J.l is (0:, p, q) - Carleson and 'P : 10 ---> 10 is analytic, then CI{J : A~ ---> Aq(J.l) is well-defined and bounded. For non-constant functions 'P, the condition is further equivalent to J.l 0 'P -1 being (0:, p, q) - Carleson. This allows an interpretation of Carleson embeddings, and in particular of multipliers as above, as composition operators. However, in such a general setting the range space of a composition operator might be unpleasent, and desirable properties may not be available. For example, Aq(J.l) embeds continuously into 1i(1O) if and only if Aq(J.l) is a closed subspace of Lq(J.l) and all point evaluations Aq(J.l) ---> C : J f-+ J(z), z E 10, are continuous. (0:, p, q) - Carleson measures have been characterized, even in a more general setting, by V.L. Oleinikov and B.S. Pavlov [15], W.W. Hastings [6] and D.H. Luecking
J
f-+
[11],[12]. The hyperbolic metric on 10 is given by
. lJd(J "I 1 _ J(J
e(z, w) := l~f
where the infimum extends over all smooth curves 'Y in 10 joining z and w. For w E 10 and r > 0, let Br (w ) = {z E 10 : e( z, w) < r} be the corresponding hyperbolic disk. Actually, the particular choice of r > 0 doesn't really matter in our context. Let us agree to write A~ '---+ Lq(J.l) if A~ is a subset of U(J.l) and the embedding is continuous. Similar for other function spaces. The Carleson measures under consideration can be characterized in terms of the function
THEOREM
3.1. Let -1 < 0:
0 such that 1 C . ao(Br(w)) ::; (I-JwJ2)o+2 ::; C· ao(Br(w)) 'Vw E 10 . Therefore we may also say that Theorem 3.1 refers to properties of the function w f-+ J.l(Br (W))I/ q(1 _JWJ2)-(o+2)/p .
220
HANS JARCHOW AND URS KOLLBRUNNER
It also follows that Ha,p,q E L~(A) if and only if J-t(B r (·))/C7 a (B r (·)) is in LP/(p-q) (C7 a ). This will be used in the proof of Theorem 4.3 below. If p ~ q; then the relevant parameter in Theorem 3.1 is q (0: + 2)/p, whereas for p > q and fixed 0:, dependence is on p / q. As a first immediate consequence we may state: COROLLARY
if and only if A~
3.2. For any -1 < 0: < "---> Ltq(J-t).
00,0
< p,q < 00 and t > 0,
A~ "--->
Lq(J-t)
In turn, this leads to: 3.3. Let -1 < 0:,0:' < 00 and 0 < p, pi, q, q' < 00 be given. (a) Ifp ~ q, pi ~ q' and q. (0: + 2)/p = q'. (0: ' + 2)/p', then A~ "---> Lq(J-t) iff A~, "---> Lq' (J-t).
COROLLARY
(b) Ifp> q and p/q = pi /q', then A~
"--->
Lq(J-t) iff A~
"--->
Lq' (J-t).
For a large range of parameters, this allows a reduction to Hilbert spaces as follows: 3.4. Suppose that -1 < 0:,0:' < 00 and 0 < p, q < 00. (a) Ifp ~ q and 0:' + 2 = q. (0: + 2)/p, then A~ "---> Lq(J-t) iff A~,
COROLLARY
(b) If p > q and pi /2 = 2/q' = p/q, then A~ A~ "---> L 2 (J-t). A special known case occurs when we take J-t Horowitz [7]).
"--->
Lq(J-t) iff A~
3.5. Suppose that -1 < 0:, (3 < 00 andO < p, q < (a) If p ~ q, then A~ "---> A~ iff (0: + 2)/p ~ ((3 + 2)/q. "--->
A~ iff (0: + l)/p
"--->
L 2 (J-t).
Lq' (J-t) iff
= C7fJ for some (3 > -1 (see C.
COROLLARY
(b) If p > q, then A~
"--->
00.
< ((3 + l)/q.
There are several ways to modify the domain space of a composition operator. In a systematic fashion, we may proceed as follows; cf. [4]. Each of the kernel functions K a ( Z, .) is bounded (z E lV), and ._ ( 1 - IzI2 ) (a+2)/p
(1 _
ka,p,z(w),has (p-) norm one in representation
A~
(0 < p
-1 by a' + 2 = q(a + 2)/p. Then, with £T = £T(A),
Ho,p,q E Lpq/{p-q) A~
'---+
Lq(J.L)
=? A~/q) = A~, '---+
H O ',l,l E L oo Ho,p,q E L oo
.
L 1 (J.L)
o
HANS JARCHOW AND URS KOLLBRUNNER
222
EXAMPLE 3.8. (a) Let a, p, q be as in Corollary 3.7, let (an) be a sequence in foo \ fP/(p-q) , and let (1]n) be a sequence in llJ such that (!(1]n,1]k) :::: r· 8nk for all n, k, and such that llJ = Un Br(1]n). Consider the measure JL = En bn 811n on llJ where bn = lanl' (1-I1]nI 2 )q("'+2)/p and note that (b n ) E fl. A calculation reveals that H""p,q is in LOO(A) but not in uq/(p-q) (A). We conclude that the converse in Corollary 3.7 doesn't hold. (b) In Proposition 3.6, (ii) {::} (iii) is true for arbitrary 0 < p, q < 00, and (i) ~ (ii) holds trivially whenever q :::: 1. But (iii) ~ (i) fails for 1 ~ q < p and a + 2 > p. In fact, if JL is as in (a), then A~ 'f+ Lq(JL) since H""p,q tfLpq/(p-q)(A), but H-y,l,q = H""p,q E LOO(A) if we put "I = (a + 2)/p - 2. From A~) = A~l) = A~ we conclude that A~ '----> Lq(JL). 4. Compactness We shall frequently make use of the following classical result: THEOREM 4.1 (Pitt's Theorem). If 0 < p < q < 00 then every operator f q --+ fP is compact. This was obtained in 1936 by H.R. Pitt [16] for p :::: 1. For an extension see H.P. Rosenthal [17]. The result as stated was proved recently by E. Oja [14]. By atomic decomposition, A~ and fP are isomorphic (see [10], [1]). Combining this with Pitt's theorem we see that in particular the embedding in Corollary 3.5.(b) is compact. It will follow from the next theorem that the embedding in 3.5.(a) is compact iff (a + 2)/p < (f3 + 2)/q. Our characterization of compactness of Carleson measures splits into two parts. We consider the case p ~ q first. Here the characterization is as expected: THEOREM 4.2. Let -1 < a < 00 and 0 < p ~ q < statements are equivalent: (i) JL is a compact (a,p, q) - Carles on measure. (ii) A~)
'---->
with 1 ~ q. The following
Lq(JL) compactly.
(iii) z-+ limlllk""p,zIILq(ll) (iv)
00
= O.
lim Ho.,p,q(w) = O.
Iwl-+l
PROOF. (i) ~ (ii): If p :::: 1 then nothing is to prove since A~) '----> A~. If p < 1 ~ q then, by [19], A~) is the Banach envelope of A~, and the convex hull of B A~ is dense in B A!:)' Hence relative compactness of B A~ in the Banach space Lq(JL) entails relative compactness of B A 2, then every operator C(K) r - summing for every r > p.
----7
U(v) is (p, 2) -summing, and
Moreover: • If u : X
Lp(v) is order bounded then u is p - summing. Here v is any measure. In the last statement, the converse fails. But: ----7
• If u* p - summing then u is order bounded.
More precisely, we have the following result due to D.J.H. Garling [5]: • Let 1 ::; p < 00. A Banach space operator u : X ----7 Y has a p - summing adjoint if and only if, for every measure v and operator v : Y ----7 LP(v), the composition v 0 u : X ----7 LP(v) is order bounded.
In particular: • An operator u : L 2 (VI) Schmidt.
----7
L 2 (V2) is order bounded iff it is Hilbert-
We are going to characterize order boundedness of Carleson embeddings
A~ ~
Lq(J.L). To this end we introduce, for s > 0, the Banach space
Xs
:= {f: lU
----7
C: f measurable, sup(1-lzI 2 )Slf(z)1 < oo} . zE1U
and its closed subspace
Xs := Xs n H(lU) . It is easy to see that
A~ ~
X(o.+2)/p and that the index (o.+2)/p is best possible.
THEOREM 5.1. Let -1 < 0. < 00, 0 < p < 00 and 1 ::; q < s := (0. + 2)/p, the following statements are equivalent: (i)
A~ ~
00.
Then, with
U(J.L) order boundedly.
(ii) A~) ~ U(J.L) order boundedly. (iii) (1-lzI 2 )-S E U(J.L). (iv)
XS ~ Lq(J.L)
boundedly.
(v) XS ~ Lq(J.L) order boundedly. (vi) XS ~ Lq(J.L) boundedly.
(vii) XS
~
Lq(J.L) order boundedly.
PROOF. (i)::::} (ii) is obtained as before, by considering separately the cases p ;:::: 1 and p < 1. In order to prove (ii)::::} (iii) it suffices to look at the functions ka,p,q' (iii)::::} (iv) and (iv)::::} (vi) as well as (iv) Y is almost summing,
u E IIas(X, Y) , if there is a constant C such that, for any choice of finitely many vectors from X,
(10[III {; rk(t)UXk 112 dt )1/2 ::; C x.~~x. ( n
n
{;
Xl, ... ,
Xn
)1/2
I(x*, xk)1 2
It is known that each of the operator ideals IIp is properly contained in IIas. Moreover, if 1 ::; p < 00 and r = max{p,2} then IIash X) c II r ,2(', X) whenever X is an £P space, or the Schatten p-class Sp(H) for some Hilbert space H. In addition, it was shown by S. Kwapien [9] that • if H is a Hilbert space and u is in IIas(H, Y) then the adjoint 11,* : Y* -> H
is 1 - summing. See [2], p.255 for details. We have the following application to Carleson embeddings. The argument is the same as for composition operators between weighted Bergman spaces [3]. PROPOSITION 5.6. Let /-L be an (a,p, q) - Carles on measure where 1 ::; q < 00., and 2 ::; p < 00. The embedding] : A~ '--+ Lq(/-L) is almost summing if and only if it is order bounded. PROOF. Define "I > -1 by ("I + 2)/2 = (a + 2)/p. Since p 2 2, A~ :::::} ]: A~ '--+ Lq(/-L) is almost summing :::::} ]* is I-summing (Kwapien) :::::} ] is order bounded (Garling)
'--+
A~.
CARLESON EMBEDDINGS FOR WEIGHTED BERGMAN SPACES
229
Combining the preceding two propositions yields: COROLLARY 5.7. Let -1 < ct < 00 and 1 ::; p, q < 00 be such that p 2: min {q* ,2} and let 11, be an (ct, p, q) - Carleson measure. The embedding I : A~ "---> Lq(/.l) is q - summing iff it is order bounded. PROOF. Only sufficiency requires proof. If p 2: q*, then Proposition 5.5 settles the case. And if p 2: 2, then I, being q - summing, is almost summing, and so order bounded by Proposition 5.6. 0
References [IJ R.R. Coifman, R. Rochberg, G. Weiss, Facto'rization theorems for Hardy spaces in seveml variables. Ann. of Math. (2) 103. (1976),611-635. [2J J. Diestel, H. Jarchow, A. Tonge: Absolutely Summing Opemtors. Cambridge University Press 1995. [3J T. Domenig, CompoS'ition opemtors on weighted Beryman sp(,ces and Hardy spaces. Dissertation University of Zurich 1997. [4J T. Domenig, H. Jarchow, R. Riedl, The domain space of an analytic composition opemtor. Journ. Austral. Math. Soc. 66 (1999), 56-65. [5J D.J.H. Garling, Lattice bounding, Radonifying and summing mappings. Math. Proc. Camb. Phil. Soc. 77 (1975), 327-333. [6J W.W. Hastings, A Carleson measure theQrem for Beryman spaces. Proc. Amer. Math. Soc. 52 (1975), 237-241. [7J C. Horowitz Zeros of functions in the Bergman spaces. Duke Math. Journ. 41 (1974), 693-710. [8J S. Kwapien, On a theorem of L. Schwartz and its applications to absolutely summing opemtors. Studia Math. 38 (1970), 193-201. [9J S. Kwapien, A remark on p - summing opemtors in fr - spaces. Studia Math. 34 (1970), 277278. [lOJ J. Lindenstrauss, A. Pelczynski, Contributions to the theory of classical Banach spaces. Journ. Funct. Anal. 8 (1971), 225-249. [l1J D.H. Luecking, Multipliers of Bergman spaces into Lebesgue spaces. Proc. Edinb. Math. Soc. 29 (1986), 125-131. [12J D.H. Luecking, Embedding theorems for spaces of analytic functions via Khinchine's inequality. Mich. Math. Journ. 40 (1993), 333-358. [13J P. Meyer-Nieberg, Banach Lattices. Springer-Verlag 1991. [14J E. Oja, Pitt Theorem for non-locally convex spaces f p • Preprint. [15J V.L. Oleinik, B.S. Pavlov, Embedding theorems for weighted classes of harmonic functions. Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 22 (1971),94-102. Transl. in Journ. Soviet Math. 2 (1974), 135-142. [16J H.R.Pitt, A note on bilinear forms. Journ. London Math. Soc. 11, 171-174 (1936). [17J H.P. Rosenthal, On quasi-complemented subspaces of Banach spaces with an appendix on compactness of opemtors from LP(p,) to Lr(/I). Journ. Funct. Anal. 4 (1969), 176-214. [18J W. Rudin, Real and Complex Analysis. 3 rd ed., McGraw-Hill 1987. [19J J.H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces. Duke Math. Journ. 43 (1976), 187-202. [20J W. Smith, Composition opemtors between Bergman and Hardy spaces. Trans. Amer. Math. Soc. 348 (1996) 2331-2348. [21J W. Smith, L. Yang, Composition opemtors that impro1Je integmbility on weighted Beryman spaces. Proc. Amer. Math. Soc. 126 (1998) 411-420. [22J K. Zhu, Opemtor Theory in Function Space.~. Marcel Dekker, New York 1990.
HANS JARCHOW AND URS KOLLBRUNNER
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INSTITUT FUR MATHEMATIK, UNIVERSITAT ZURICH, WINTERTHURERSTRASSE
190, CH 8057
ZURICH, SWITZERLAND
E-mail address:jarchowlDmath.unizh.ch INSTITUT FUR MATHEMATIK, UNIVERSITAT ZURICH, WINTERTHURERSTRASSE ZURICH, SWITZERLAND
E-mail address:kollbrunlDmath.unizh.ch
190, CH 8057
Contemporary Mathematics Volume 328, 2003
Weak* -extreme points of injective tensor product spaces Krzysztof Jarosz and T. S. S. R. K. Rao ABSTRACT. We investigate weak* -extreme points of the injective tensor product spaces of the form A ®. E, where A is a closed subspace of C (X) and E is a Banach space. We show that if x E X is a weak peak point of A then f (x) is a weak*-extreme point for any weak*-extreme point f in the unit ball of A ®. E C C (X, E). Consequently, when A is a function algebra, f (x) is a weak*-extreme point for all x in the Choquet boundary of A; the conclusion does not hold on the Silov boundary.
1. Introduction
For a Banach space E we denote by E1 the closed unit ball in E and by BeE1 the set of extreme points of E 1 . In 1961 Phelps [16] observed that for the space C(X) of all continuous functions on a compact Hausdorff space X every point f in Be (C (X))1 remains extreme when C (X) is canonically embedded into its second dual C (X)**. The question whether the same is true for any Banach space was answered in the negative by Y. Katznelson who showed that the disc algebra fails that property. A point x E OeE1 is called weak* -extreme if it remains extreme in BeEi*; we denote by B;E1 the set of all such points in E 1. The importance of this class for geometry of Banach spaces was enunciated by Rosenthal when he proved that E has the Radon-Nikodym property if and only if under any renorming the unit ball of E has a weak* -extreme point [19]. While not all extreme points are weak* -extreme the later category is among the largest considered in the literature. For example we have: strongly exposed S;; denting S;; strongly extreme S;; weak* -extreme. We recall that x E E1 is not a strongly extreme point if there is a sequence Xn in E such that Ilx ± xnll ---t 1 while IIxnll ~ 0 (see [3] for all the definitions). We denote by O;E1 the set of strongly extreme points of E 1. It was proved in [14] that e E O;E1 if and only if e E o;Ei* (see [9], [13], or [17] for related results). Examples of weak* -extreme points that are no longer weak* -extreme in the unit ball of the bidual were given only recently in [6]. In this paper we study the weak* -extreme points of the unit ball of the injective tensor product space A®,E, where A is a closed subspace of C(X). Since C(X)®,E Both authors were supported in part by a grant #0096616 from DST/INT/US(NSFRP041)/2000.
© 231
2003 American Mathematical Society
232
KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
can be identified with the space C(X, E) of E-valued continuous functions on X, equipped with the supremum norm, elements of A ®e E can be seen as functions on X. We are interested in the relations between 1 E a; (A ®£ E) and 1 (x) E a;E1 , for all (some?) x E X. Since any Banach space can be embedded as a subspace A of a C (X) space no complete characterization should be expected in such very general setting. For example if A is a finite dimensional Hilbert space naturally embedded in C (X), with X = Ai, and dim E = 1, then any norm one element 1 of A ®£ E is obviously weak* -extreme however the set of points x where 1 (x) is extreme is very small consisting of scalar multiples of a single vector in Ai. Hence in this note we will be primarily interested in the case when A is a sufficiently regular subspace of C (X) and/or x is a sufficiently regular point of X. It wa.. 1 E a; (A ®e E)I'
where we denote by c5x the functional on A of evaluation at the point x. In this paper we obtain a partial converse of the above result (Theorem 1). Our proof also shows that if 1 E a; (A ®€ E)l then 1 (x) E a;El for any weak peak point x (see Def. 1), extending one of the implications of (1.1). It follows that when A is a function algebra then any weak* -extreme point of Al is of absolute value one on the Choquet boundary ChA (and hence on its closure, the Shilov boundary) and consequently is a strongly extreme point [17]. Since we have concrete descriptions of the set of extreme points of several standard function algebras (see e.g. [12], page 139 for the Disc algebra) one can give easy examples of extreme points that are not weak* -extreme. Recently several authors have studied the extremal structure of the unit ball of function algebras ([1], [15], [18]). It follows from their results that the unit ball has no strongly exposed or denting points. Our description that strongly extreme and weak* -extreme points coincide for function algebras and are precisely the functions that are of absolute value one on the Shilov boundary completes that circle of ideas. We also give an example to show that the weak* -extreme points of (A ®, E)l in general need not map the Shilov boundary into aeE1 • Considering the more general case of the space of compact operators K (E, F) (we recall that under assumptions of approximation property on E or F*, K (E, F) can be identified with E* ®£ F) we exhibit weak*-extreme points T E K (fP)1 for 1 < p =I=- 2 < 00 for which T* does not map unit vectors to unit vectors. Our notation and terminology is standard and can be found in [3], [4], or [11]. We always consider a Banach space as canonically embedded in its bidual. By E(n) we denote the n-th dual of E. By a function algebra we mean a closed subalgebra of a C (X) space separating the points of X and containing the constant functions; we denote the Choquet boundary of A by ChAo
2. The result
a;
As noticed earlier, for A c C (X) and a point 1 E (A ®£ Eh we may not have 1 (x) E a;EI for all x E X even in a finite dimensional case. Hence we need to define a sufficiently regular subset of X in relation to A.
WEAK' -EXTREME POINTS
233
DEFINITION 1. A point Xo E X is called a weak peak point of A C C (X) if for each neighborhood U of Xo and £ > 0 there is a E A with 1 = a (xo) = Iiall and la (xo)1 < £ for x E X\U; we denote by opA the set of all such points in X.
There are a number of alternative ways to describe the set opA. If Xo E X is a weak peak point of A C C (X), /-l is a regular Borel measure on X annihilating A, and al' is a net in A convergent almost uniformly to 0 on X\ {xo} and such that a.., (xo) = 1 then /-l({xo}) = liml'Jxal' = O. Hence if a* E A* and VI,V2 are measures on X representing a* we have VI ({Xo}) = V2 ({xo}), consequently V
~ v ( {xo}) is a well defined functional on A * .
On the other hand if X {xo} E A ** then /-l ( {xo}) = 0, for any annihilating measure /-l, and Xo is a weak peak point. To justify the last claim notice that Al is weak*dense in Ai* so X{xo} is in the weak*-closure of the set K = {f E Al : f (xo) = I}. Let U be an open neighborhood of Xo and Ax\U be the space of all restrictions of the functions from A to X\U. We define the norm on Ax\U as sup on X\U. Lct K X\U be the set of restrictions ofthe functions from K and cl (Kx\U) be the norm closure of Kx\U C Ax\U. If 0 ~ clKx\U then there is G E (Ax\U)* , represented by a mcasure 1] on X\U and separating Kx\U from 0: Re G (h) > a > 0
=
X{xo}
(/-l)
I
for all hE clKx\U·
The measure 1] extends G to a functional on A so K is functionally separated from 0 in A contrary to our previous observation. Hence 0 E clKx\u so there is a function in K that is smaller then £ outside U which means that Xo is a weak peak point. The concept of weak peak points is well known in the context of function algebras where opA coincides with the Choquet boundary ([8]' p. 58). For more general spaces of the form Ao 'f1 {foa E C (X) : a E A} I where A C C (X) is a function algebra and fo a nonvanishing continuous function on X we have ChA ~ opAo. Spaces of these type appear naturally in the study of singly generated modules and Morita equivalence bimodules in the operator theory [2J. THEOREM 1. Let E be a Banach space, X a compact Hausdorff space, and A a closed subspace of C (X). If f E A ®, E is a weak* (strongly) extreme point of the unit ball then f (x) is a weak* (strongly) extreme point of the unit ball of E for any x E opA. In particular if f E (C(X,E))I is a weak*(strongly) extreme point then f(x) is a weak* (strongly) extreme point of EI for all x EX.
We first need to show that for a weak peak point Xo E X there exists a function in A not only peaking at Xo but that is also almost real and almost positive. LEMMA
1. Assume X is
(L
compact Hausdorff space, A is a closed subspace of
C (X), and Xo is a weak peak point of A. Then for each neighborhood U of Xo and £
> 0 there is g
E A such that
Ilgll = 1 = g (xo) , (2.1)
Ig (x)1
IIRe+ g where Re+ z
= max{O,Rez}.
< £, for all x
gil < £,
E X\U, and
KRZYSZTOF JAROSZ AND T. S. S.
234
R. K.
RAO
PROOF. Put U1 = U and let gl E A be such that IIg111 = 1 = gl (xo) and Ig1 (x)1 < e for x
i
U1.
Put U2 = {x E U1 : Ig1 (x) - 11 < e} and let g2 E A be such that IIg211
= 1 = g2 (xo)
and Ig2 (x)1 < e for x
i
U2·
Put U3 = {x E U2 : Ig2 (x) - 11 < e}. Proceeding this way we choose a sequence {gn}n>l _ in A. Fix a natural number k such that k> 1e: and put 1 k
g=
k Lgj· j=l
We clearly have Ilgll = 1 = 9 (xo) and Ig (x)1 < e for x i U. Let x E U, then either x belongs to all of the sets Uj , j k, in which case Ig (x) - 11 < e, or there is a natural number p < k such that x E Up \ Up+!. In the later case we have
:s
Ig(X)_P~II=~
tgj-(P-l) ;=1
u {(sint,cost,O): 0::; t::; 11"},
fo : X
-+
df
El, fo (x) = x, and A = {h E C (X) : h (0, 1,·) E A (lD>)},
where A (lD» is the disc algebra. We have ChA = {O} x {I} x 8lD> U {(sin t, cos t, 0) : 0 < t < 11"} . The function fo is in A ®e E and takes extremal values on the Choquet boundary of A so it is a weak* -extreme points of (A ®e E)l. However fo (0, ±1, 0) = (0, ±1, 0) are not extreme points of Ei while (0, ±1, 0) are in the Shilov boundary of A. Since E is finite dimensional clearly the function fo maps the Choquet boundary of A into the set of strongly extreme points of E 1 • We next show that f is not a strongly extreme point. Let gn E A be such that Ilgnll
gn (sint, cost, 0)
= 1 = gn
(sin
~,cos ~,o) ,
= 0, for ~ < t::; 1, and n
gn (0, 1, z) = 0, for z E lD>. Put fn = (O,O,gn) E A ®e E. We have (fo ± fn)(a, b, c) Hence IIf ± fnll
-+
(0,1, c) for = { (a, b, ±gn) for
(a,b,c) E {O} x {I} x 8lD> (a, b, c) E {(sin t, cos t, 0) : 0 ::;
t ::; 11"} .
1 but Ilfnll ~ 0 so f is not a strongly extreme point.
In the next Proposition we consider a more general setting of compact operators. For a Banach space E we denote by C(E) the space of all linear bounded maps on E, by K(E) the set of all compact linear maps, and by S(E) the set of
KRZYSZTOF JAROSZ AND T. S. S. R. K. RAO
236
unit vectors in E. Since K (E, C (X)) can be identified with C (X, E*) our result on weak* -extreme points taking weak* -extremal values can be interpreted as follows T E a;K(E,C(X))l
==}
T* (a;C(X);) c a;Er.
Thus more generally one can ask whether T* (a; Ft) c a; Ei for any TEa; K(E, Fh. We give a class of counter examples with the help of the following proposition. PROPOSITION 1. Let E be an infinite dimensional Banach space such that K(E) is an M-ideal in C(E). 1fT E K(Eh then T*(aeEi) ct. S(E*).
We recall that a closed subspace M of a Banach space E is an AI-ideal if there is a projection P E C (E*) such that ker P = Ml. and liP (e*)II+lle* - P (e*)11 = Ile*ll, for all e* E E* (see [11] for an excellent introduction to 1\/-ideals). PROOF. Since qE) is an AI-ideal it follows from Corollary V1.4.5 in [11] that E* has the Radon-Nikodym property and hence the IP (see [10]). Also since qE) is a proper M-ideal it fails the IP. It therefore follows from Theorem 4.1 in [10] that there exists a net {x~} c e Ei such that x~ ---> Xu in the weak* -topology with Ilxoll < 1. Suppose T*(aeEi) c S(E*). Since T* is a compact operator by going through a subnet if necessary we may assume that T*(x~) ---> T*(xo) in the norm. Thus 1 = IIT*(xo)11 < 1 and the contradiction gives the desired conclusion. 0
a
EXAMPLE 2. Banach spaces E for which K(E) is an AI ideal in C(E) have been well extensively studied. Chapter VI of [11] provides seveml examples including E = p , 1 < p < 00, as well as properties of these spaces. It was observed in [6] that for p # 2 there are weak*-extreme points in the space K(ePh. It follows from the last proposition that the adjoint of these weak" -extreme points do not even map extreme points to unit vectors.
e
A strongly extreme point remains extreme in all the dual spaces of arbitrary even order. A weak* -extreme point remains extreme in the second dual but may not be extreme in the fourth dual. Hence the property of remaining extreme in all the duals of even order is placed between the strong and the weak* type of extreme points. It would be interesting to describe that property in terms of the original Banach space alone. A procedure for generating extreme points which have this property but are not strongly extreme was described in [6]. PROPOSITION 2. Let X be a compact Hausdorff space, A a closed subspace of C(X), and E a Banach space. Suppose Xo E X is a weak peak point and f E A®. E is an extreme point in the unit ball of all the duals of even order. Then f (xo) is an extreme point of the unit ball of all the duals of E of even order. PROOF. Since the space A ®. E** can be canonically embedded in (A ®. E)** [7] we have, for any natural number n
A ®. E(2n) C (A ®( E(2n-2))** C (A ®. E)(2n).
If f E A ®. E is an extreme point of (A ®. E)(2n+2) then it is a weak*-extreme point of (A ®. E)(2n), as it also belongs to A ®. E(2n) it is a weak*-extreme point of A ®. E(2n). Hence by our theorem f (xo) is an extreme point of E~2n). 0 The next proposition characterizes strongly extreme points in terms of ultrapowers.
WEAK' -EXTR.EME POINTS
PROPOSITION
of the unit ball
El
237
3. An element e of a Banach space E is a strongly extreme point if and only if (e).:F is an extreme point of (E.:Fh-
PROOF. If e ¢. O;El then there is a sequence {en}n~l eEl with lie ± enll --+ 1 and infnEN Ilenll > o. Thus II (e).:F ± (en).:F11 = 1 and II (en).:F11 =I- 0 so (e).:F is not an extreme point. If (e).:F ¢. oe(E.:Fh then there is 0 =I- (en).:F E (E.:Fh with 1 = II (e).:F ± (en).:F11 = lim.:F lie ± enll· Thus for every € > 0 the set {n E N: lie ± enll ~ 1 + €} is none empty as an element of F. Hence there exists a sequence {k n } such that lie ± ek n II --+ 1 but Ilek" II ---A- 0 so e is not a strongly extreme point. 0
References [1] P. Beneker and J. Wiegerinck, Strongly exposed points in 'Uniform algebras, Proc. Amer. Math. Soc. 127 (1999) 1567-1570. [2] D. Blecher and K. Jarosz, Isomorphisms of function modules, and generalized approximation in modulus, Trans. Amer. Math. Soc. 354 (2002), 3663-3701 [3] R. D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodym property, LNM 993, Springer, Berlin 1983. [4] A. Browder, [ntroduction to Function Algebras, W. A. Benjamin, New York 1969. [5] P. N. Dowling, Z. Hu and M. A. Smith, Extremal structure of the unit ball of C(K, X), Contemp. Math., 144 (1993) 81-85. [6] S. Dutta and T. S. S. R. K. Rao, On weak*-extreme points in Banach spaces, preprint 2001. [7] G. Emmanuele, Remarks on weak compactness of operators defined on injective tensor products, Proc. Amer. Math. Soc., 116 (1992) 473-476. [8] T. Gamelin, Un'iform Algebras, Chelsea Pub. Comp., 1984. [9] B. V. Godun, Bor-Luh Lin and S. L. Troyanski, On the strongly extreme points of convex bodies in separable Banach spaces, Proc. Amer. Math. Soc., 114 (1992) 673-675. [10] P. Harmand and T. S. S. R. K. Rao, An intersection property of balls and relations with M-ideals, Math. Z. 197 (1988) 277-290. [11] P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Springer LNM No 1547, Berlin 1993. [12] K. Hoffman, Banach spaces of analytic functions, Dover 1988. [13] Z. Hu and M. A. Smith, On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals, Trans. Amer. Math. Soc. 349 (1997) 1901-1918. [14] K. Kunen and H. P. Rosenthal, Martingale proofs of some geometric results in Banach space theory, Pacific J. Math. 100 (1982) 153-175. [15] O. Nygaard and D. Werner, Slices in the unit ball of a uniform algebra, Arch. Math. (Basel) 76 (2001) 441-444. [16] R. R. Phelps, Extreme points of polar convex sets, Proc. Amer. Math. Soc. 12 (1961) 291-296. [17] T. S. S. R. K. Rao, Denting and strongly extreme points in the unit ball of spaces of operators, Proc. Indian Acad. Sci. (Math. Sci.) 109 (1999) 75-85. [18] T. S. S. R. K. Rao, Points of weak-norm continuity in the unit ball of Banach spaces, J. Math. Anal. Appl., 265 (2002) 128-134. [19] H. Rosenthal, On the non-norm attaining functionals and the equivalence of the weak' -KMP with the RNP, Longhorn Notes, 1985-86. DEPARTMENT OF MATHEMATICS AND STATISTICS. SOUTHERN ILLINOIS UNIVERSITY, EDWARDSVILLE.
IL 62026-1653, USA E-mail address: kjaroszlDsiue. edu URL: http://www.siue.edu/-kjarosz/
R. V. COLLEGE POST, BANGALORE 560059. INDIA E-mail address:tsslDisibang.ac . in
INDIAN STATISTICAL INSTITUTE,
Contemporary Mathematics Volume 328. 2003
Determining Sets and Fixed Points for Holomorphic Endomorphisms Kang-Tae Kim and Steven G. Krantz The authors study the fixed point sets of a holomorphic endomorphism of a domain in complex space. Sufficient (and necessary) conditions are given-on the number and configuration of the fixed points-for the endomorphism to be forced to be the identity. The proofs depend on certain key ideas from differential geometry, particularly the notions of cut locus and Hadamard ABSTRACT.
length space.
1. Introduction
This article concerns the study of the concept of determining set for a collection of holomorphic mappings. We first give the definition. DEFINITION 1.1. Let M be a complex manifold, and let Aut (M) be the collection of biholomorphic mappings of M into itself. We call a subset Z c M a determining set for Aut (M) (or, equivalently, an Aut (M)-determining set), if any map f E Aut (M) satisfying f(p) = p for every p E Z is in fact the identity map of
M. We observe first that this article is related to the authors' collaboration with Burna Fridman and Daowei Ma (see [FKKM]), which was originally inspired by the following remarkable theorem in complex dimension one. THEOREM 1.2. Let n be a domain in the complex plane C and let f : n --+ n be a biholomorphic (conformal) mapping. If there are three distinct points Pl,P2,P3 in n such that f(pj) = Pj, for j = 1,2,3, then f is the identity map. The higher-dimensional analog of this theorem given in [FKKM] is as follows: THEOREM 1.3. (Fridman-Kim-Krantz-Ma [FKKM]) Let M be a connected, complex manifold of dimension n admitting a complete invariant Hermitian metric. 2000 Mathematics Subject Classification. 32H02, 32H50, 32H99. Key words and phrases. fixed point set, holomorphic mapping, cut locus, Hadamard length
space. K.- T. Kim supported in part by grant ROl-1999-00005 from The Korean Science and Engineering Foundation. Steven G. Krantz supported in part by grant DMS-9988854 from the National Science Foundation.
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Then a determining set consisting of n + 1 points exists for the automorphisms of .M. Furthermore, the choice of such a determining set is generic. Throughout this paper we shall discuss both endomorphisms and automorphisms. If M is a complex manifold then an endomorphism of .I'IJ is any holomorphic mapping
0 fixing po, ... ,Pn must fix every point ofO. PROOF. Notice that the current hypothesis together with the preceding lemma implies that dfpo is the identity map. Therefore a theorem of H. Cartan implies that f is in fact the identity mapping. 0 We remark that the choice for Po, ... ,Pn is generic. To formulate this notion more precisely, we consider the cartesian product rrnHo of (n + 1) copies of O. In fact it is shown in [FKKM] that there exists an open dense subset U of rrn+10 such that any element of U gives (n + 1) points that satisfy the sufficiency condition of the preceding lemma. We summarize the result more elegantly in the following statement. THEOREM 2.3. For a bounded, strongly convex domain in en, there exists a collection of n + 1 points such that any holomorphic endomorphism of the domain fixing them must fix every point in the domain. Moreover, the choice of such n + 1 points is generic. REMARK 2.4. We point out that the result of this section concerns the class of bounded, strongly convex domains, which is a rather special collection of objects. However, in compensation, we emphasize that we have treated general holomorphic endomorphisms, rather than just biholomorphic self-maps.
3. Hadamard Spaces In light of the preceding section, we would like to present in this section a description of the underlying metric space principles that we use in the study of determining sets. Let (X, d) be a metric space, equipped with the distance function d : X x X ---> lR. By an isometry we mean a self-mapping f : X ---> X satisfying the condition:
d(J(p), f(q)) = d(p, q), Vp, q E X. We denote by Isom(X) the collection of isometries of (X, d). Imitating the concept of "length spaces" that is commonly encountered in geometry (cf. [BUS]), we give the following definition. DEFINITION 3.1. Let"(: (a,b)
--->
X be a continuous curve in X. We call it
minimal if d("((x), ,,((y)) = t - x + d("((t), ,,((y)), for every t,x,y with a < x:::; t:::; y < b. DEFINITION 3.2. A metric space (X, d) is called a length space if, for every pair of points p, q EX, there exists a minimal curve "( : [a, b] ---> X such that "((a) = P and "((b) = q. Furthermore, we call (X, d) a Hadamard space if the minimal curve joining each pair of points is unique up to a reversal of parametrization. Notice that any convex subset of Euclidean space is a Hadamard space with respect to the standard Euclidean distance. A strongly convex domain in en, equipped with the Kobayashi distance, is also a Hadamard space. Every complete,
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simply connected Riemannian manifold with non-positive curvature is also an example of a Hadamard space. These are often called Hadamard manifolds; from this derives our terminology of Hadamard (length) space. Now. for the study of determining sets, we present this lemma. LEMMA 3.3. Let (X, d) be a Hadamard space and let p, q E X be two distinct points. If an isometry f : X --> X fixes P and q, tllen f fixes every point on the minimal curve passing through P and q. PROOF. Let'Y : [O,f] --> X be a minimal curve with 'Y(O) = p,'Y(f) = q. It is immediate to see that the isometry f of (X, d) has the property that f 0 'Y is also a minimal curve. Since P and q are fixed by f, and since (X, d) is Hadamard, it follows that f 0 'Y(t) = 'Y(t) for every t E [0, fl. Now consider the minimal curve passing through P and q. So far, we have shown that the portion of this minimal curve between P and q is pointwise fixed by f. We still must show that the other portion of the minimal curve is fixed pointwise by f. But this is a simple matter using the uniqueness of the minimal curve passing through any two given points and the minimal-curve-preserving property of isometries. This completes the proof. D We remark that it is possible to derive the same conclusions for locally Hadamard spaces but, in order to keep our exposition concise, we do not introduce the concept here. LEMMA 3.4. Let (X, d) be a Hadamard space and let U be an open subset of X. Then any isometry f fixing every point in U must be the identity map. PROOF. Let P E U. Then, for every minimal curve 'Y emanating from p, f fixes a point in 'Y n (U \ p). The preceding lemma implies now that f fixes every point of 'Y. Since every point in X can be joined to P by a minimal curve, this completes the proof. D Now we consider the concept of convex hull in a Hadamard space. We say that a set Q in a Hadamard space X is convex if every minimal curve joining P and q in X is contained in Q. For a subset A of a Hadamard space X, its convex hull W (A) is the smallest convex subset of X containing A. DEFINITION 3.5. Let Po, . .. ,Pm be points in a Hadamard space (X, d) with minimal geodesics 'Yl .... ,'Ym such that 'Yj passes through Po and Pi for every j = 1, .... m. We call the points Po, ... ,Pm spanning if the convex hull Whl U ... U'Ym) has non-empty interior. Now we have the following general result. PROPOSITION 3.6. If a Hadamard space (X, d) admits m + 1 points Po, PI, ... , Pm ill X that are spanning, then these m + 1 points constitute a determining set
for the isometries of( X, d). PROOF. Notice that the convex hull we obtain from the minimal curves through Po and Pi is fixed pointwise by any isometry that fixes the points Po,··· ,Pm. Then
the preceding lemma finishes the proof.
D
Observe that the full isometry condition is not really needed to prove the conclusion of the proposition. In fact, any distance-decreasing map will satisfy the
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same conclusion (just use the unique continuation principle). Notice that this offers an underlying principle for the determining set theorem for the holomorphic endomorphisms of strongly convex domains in the preceding section.
4. CH-Subsets and Automorphisms In this section we demonstrate how the principles of the preceding section reflect upon the main theorem of [FKKM] and its proof. Let M be a connected, complex manifold that admits a smooth invariant Hermitian metric. Here the invariance refers to the property that every holomorphic automorphism of M is an isometry with respect to the Hermitian metric. For a moment, we take the real part of the Hermitian metric, and consider everything in terms of lliemannian geometry. Let P EM. Then we call q E M a cut point of P if there are at least two distinct geodesics joining P and q with the same minimal length. The collection of cut points for P is called the cut locus of p, which we denote by Cpo In [FKKMJ, a subset X of M was called Carlan-Hadamard ('CH' for short) if there exists Xo E X so that X does not intersect the cut locus C(xo) of Xo in M. Furthermore, we call such a CH-set X generating if the set
Ip(X) := {'Y~(O) I 'Y is the unique normal geodesic from Xo to p, Vp
E
Z}
spans TxoM over C. Suppose now that X is a set of finitely many points, and that a certain holomorphic automorphism f fixes every point of X. Then, by complex differentiability, one picks up more geodesics than just the geodesics joining Xo and the other points of X. [That is to say, each geodesic tangent may be multiplied by i.] If x E X \ Xo and if 'Yx is the unit speed geodesic from Xo to x, then 9x == expxo(i"(~(O)) is also fixed, point by point, by f. Now it is not hard to see, using the exponential map and the tangent space, that the convex hull W = Wbl U 91 U ... U 'Ym U 9m) has non-empty interior. Notice that every point of the hull W is fixed by f pointwise. We obtain the following result as a consequence of Proposition 3.6. PROPOSITION 4.1. (Fridman/Kim/Krantz/Ma [FKKM]) Let M be a connected, complex manifold with an invariant Hermitian metric. Let X be a generating CHsubset of M. Then, whenever an automorphism fixes every point of X, it is in fact the identity map. In other words, every generating CH-subset is a determining set for automorphisms. The method of choosing a smallest (in the sense of inclusion of sets) generating CH-subset in a complex manifold with a complete invariant Hermitian metric has been explained in detail in [FKKM]. We briefly describe the paradigm. Choose an arbitrary p E M. Then the cut locus C(p) is a nowhere dense subset of M. Thus choose PI E M \ (C (p) U {p}). Then choose P2 away from C (P) and the complete geodesic through p and Pl. Then P3 will be chosen away from C(p) and the geodesic cone generated by P,Pl and P2. An inductive construction lets us choose p, PI, ... ,Pn which compose a spanning CH-subset of M. Thus we arrive at THEOREM 4.2. (Fridman-Kim-Krantz-Ma [FKKM]) Let M be a connected ndimensional complex manifold admitting a complete invariant Hermitian metric. Then a determining set, consisting of n + 1 points, exists for the automorphisms of M. Furthermore, the choice of such a determining set is generic.
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Here, by "generic", we mean that the the collection of (n + I)-tuples of points in Al x ... x M that satisfy our conclusion form a dense, open set. From the discussion above, if the metric happens to be distance-decreasing, in the sense that all holomorphic endomorphisms are distance-decreasing with respect to the given metric, then this theorem will hold for holomorphic endomorphisms. This result of course uses the idea developed in the preceding section about the distance-decreasing property together with the unique continuation property. We remark at this point that the collection of complex manifolds admitting a complete invariant Hermitian metric is rather large. For instance, every bounded pseudoconvex domain in is equipped with the complete Kiihler-Einstein metric. See [MOY] (also [CHY], [OHS], [YAU]) for instance.
en
5. Examples, Counterexamples, and the Cut Locus One might have the impression that some transversality condition for m + 1 points might be sufficient for the determining set problem for holomorphic automorphisms. However, it is shown in [FKKM] that a simplistic topological transversality assumption will not be sufficient; consider the following statement. THEOREM 5.1. ([FKKM]) Fix a finite set K = {PI, ... , Pk} in n > 1. There exists a bounded domain containing K, and a subgroup H C Aut(D) isomorphic to the unitary group U(n - 1) of en-I, such that each element of H fixes every point of K. Moreover, unlike the one-dimensional planar domain case, the consideration of the cut locus seems essential even for one-dimensional Riemann surfaces. If one considers the torus coming from the lattice generated by {I, i}, then the map z --+ - z of e generates an automorphism on the torus. It is easy to see that it has 4 fixed points, and yet is different from the identity map. If one considers a two-holed torus with a well-balanced fundamental domain centered at 0 in the Poincare disc, then the same map z --+ -z of the disc will generate a non-trivial automorphism with 6 fixed points. In this way, one can generate arbitrarily many fixed points for a non-trivial automorphisms of compact Riemann surfaces of high enough genus. Since our discussion has not depended upon the completeness of manifolds, simple puncturing will create an arbitrary number of fixed points. Notice that all these examples have fixed points in the cut loci.
en,
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References [ALK] G. Aladro and S. G. Krantz, A criterion for normality in Cn, Jour. Math. Anal. and Appl. 161(1991), 1-8. [BED] E. Bedford and J. Dadok, Bounded domains with prescribed group of automorphisms., Comment. !'v'lath. Helv. 62 (1987), 561-572. [BUS] H. Busemann, The geometry of geodesics, Academic Press, New York, NY, 1955. [CHY] S.Y. Cheng and S.T. Yau, On the existence of a compact Kahler metric, Comm. Pure App!. Math., 33 (1980), 507-544. [FIF] S. D. Fisher and John Franks, The fixed points of an analytic self-mapping, Proc. AMS, 99(1987), 76-78. [FKKM] B. Fridman, K-T. Kim, S. G. Krantz, and D. Ma, On Fixed Points and Determining Sets for Holomorphic Automorphisms, Michigan Math. Jour., to appear. [FP] B. L. Fridman and E. A. Poletsky, Upper semicontinuity of automorphism groups, Math. Ann., 299(1994), 615-628. [GRK] R. E. Greene and S. G. Krantz, Stability properties of the Bergman kernel and curvature properties of bounded domains, Recent Developments in Several Complex Variables (J. E. Fornress, ed.), Princeton University Press (1979),179-198. [GRW] R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, 699, Springer, Berlin, 1979. [GKM] D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, 2nd ed., Lecture Notes in Mathematics, v. 55, Springer-Verlag, New York, 1975. [ISK] A. Isaev and S. G. Krantz, Domains with non-compact automorphism group: A survey, Advances in Math. 146(1999), 1-38. [KLI] W. Klingenberg, Riemannian Geometry, 2nd ed., de Gruyter Studies in Mathematics, Berlin, 1995. [KOB] S. Kobayashi, Hyperbolic complex spaces, Springer, 1999. [LEM] L. Lempert, La metrique Kobayashi et las representation des domains sur la boule, Bull. Soc. Math. Prance 109(1981), 427-474. [LES] K Leschinger, Uber fixpunkte holomorpher Automorphismen, Manuscripta Math., 25 (1978), 391-396. [MA] D. Ma, Upper semicontinuity of isotropy and automorphism groups, Math. Ann., 292(1992), 533-545. [MAS] B. Maskit, The conformal group of a plane domain, Amer. J. Math., 90 (1968), 718-722. [MOY] N. Mok and S. T. Yau, Completeness of the Kahler-Einstein metric on bounded domains and the characterization of domains of holomorphy by curvature conditions, Symposia in Pure Math. The mathematical heritage ofH. Poincare, Amer. Math. Soc., 39, Part I. (1983),41-60. [OHS] T. Ohsawa, On complete Kahler domains with C 1 boundary, Pub!. Res. Inst. Math. Sci., RIMS (Kyoto), 16 (1980), 929-940. [PEL] E. Peschl and M Lehtinen, A conformal self-map which fixes 3 points is the identity, Ann. Acad. Sci. Fenn., Ser. A I Math., 4 (1979), no. 1, 85-86. [SUI] N. Suita, On fixed points of conformal self-mappings, Hokkaido Math. J., 10(1981), 667-671. [Vll] J.-P. Vigue, Fixed points of holomorphic mappings in a bounded convex domain in Cn, Proceedings of Symposia in Pure Mathematics, 52(1991), Amer. Math. Soc., 579-582. [VI2] J.-P. Vigue, Fixed points of holomorphic mappings, Complex Geometry and Analysis (Pisa, 1988), Lecture Notes in Mathematics, v. 1422, Springer, Berlin, 1990, pp. 101-106. [YAU] S. T. Yau, A survey on Kahler-Einstein metrics. Complex analysis of several variables (Madison, Wis., 1982), 285-289, Proc. Sympos. Pure Math., 41, Amer. Math. Soc., Providence, RI, 1984. KANG-TAE KIM, DEPARTMENT OF MATHEMATICS, POHANG UNIVERSITY OF SCIENCE AND TECHNOLOGY, POHANG 790-784, KOREA STEVEN G. KRANTZ, DEPARTMENT OF MATHEMATICS, CAMPUS Box 1146, WASHINGTON UNIVERSITY, ST. LOUIS, MISSOURI 63130 U.S.A. E-mail address:kimkt«lpostech.ac.kr E-mail address: sklDmath. wustl. edu
Contemporary Mathenlatics Volume 328, 2003
Localization in the Spectral Theory of Operators on Banach Spaces T. L. Miller, V. G. Miller, and M. M. Neumann ABSTRACT. In the first two sections of this article, we survey some of the recent progress in the local spectral theory of operators on Banach spaces with emphasis on the local spectrum and on restrictions and quotients of decomposable operators. In particular, the problem of characterizing restrictions and quotients of generalized scalar operators with spectrum in the unit circle in terms of suitable growth conditions is addressed in detail, with emphasis on [11], [22], and [23]. The last two sections center around certain localized versions of the single-valued extension property, Bishop's property (13), and the decomposition property (8), mainly in the spirit of [2], [5], [6], and [13]. For each of these properties, we find a smallest closed set modulo which it holds. For these residual sets, we establish a spectral mapping theorem with respect to the Riesz functional calculus. We also obtain precise information about the extent to which Bishop's property «(3) holds on the essential or the Kato resolvent set. Our results are exemplified in the case of weighted shifts. Moreover, several of the outstanding open questions of the field are mentioned in their natural context.
1. Decomposable operators and the local spectrum
Among the various aspects and levels of localization in spectral theory, we choose decomposability as our starting point. Let X be a complex Banach space, and let L(X) denote the Banach algebra of all bounded linear operators on X. For T E L(X), let, as usual, a(T), ap(T), aap(T), r(T), and p(T) denote the spectrum, point spectrum, approximate point spectrum, spectral radius, and resolvent set of T, and let Lat(T) stand for the collection of all T-invariant closed linear subspaces of X. From [18J and [29J we recall that an operator T E L(X) is said to be decomposable provided that, for each open cover {U, V} of C, there exist Y, Z E Lat(T) for which X = Y + Z, a(T Iy) ~ U, and a(T I Z) ~ V. By [18, 1.2.23J or [29, 4.4.28J, this simple definition is equivalent to the original notion of decomposability, as introduced by Foi~ in 1963 and discussed in the classical book by Colojoara and Foi~ [IOJ. 2000 Mathematics Subject Classification. Primary 47All, 47B40; Secondary 47B37. © 247
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As witnessed, for instance, by the monographs [10], [16], [18], and [29], the theory of decomposable operators is now richly developed with many interesting applications and connections. Evidently, the class of decomposable operators contains all normal operators on Hilbert spaces and, more generally, all spectral operators in the sense of Dunford on Banach spaces. Moreover, a simple application of the Riesz functional calculus shows that all operators with totally disconnected spectrum are decomposable. In particular, all compact and all algebraic operators are decomposable. An important subclass of the decomposable operators is formed by the generalized scalar operators, defined as those operators T E L(X) for which there exists a continuous unital algebra homomorphism
. for which Tuf = x. Clearly, f(J-t) = (T - J-t)-lX for all J-t E Un p(T), so that PT(X) is open and contains p(T). Hence the local spectrum aT(x) := C \ pT(X) is a closed subset of a(T). In general, the various analytic functions that occur in the definition of pT(X) need not be consistent. This issue is addressed by the following definition. The operator T E L(X) is said to possess the single-valued extension property (SVEP), if Tu is injective for all open sets U £; IC. By [18, 3.3.2], T has SVEP precisely when, for each x E X, there exists a unique function f E H(PT(X), X) for which
(T - J-t)f(J-t) = x
for all J-t E PT(X).
This function is then called the local resolvent function for T at x. In remarkable contrast to the usual resolvent function, such functions may well be bounded; this recent discovery of Bermudez and Gonzalez will be exemplified below. One might expect a(T) to be the union of the local spectra aT(x) over all x E X, but this is not true in general. In fact, this union coincides with the surjectivity spectrum asu(T) of T, the set of all >. E C for which T - >. fails to be surjective. However, if T has SVEP, then asu(T) = a(T), and aT(x) is non-empty for all non-zero x E X, [18, 1.2.16 and 1.3.2]. As a powerful application, we obtain
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that every surjective operator with SVEP is actually bijective, [18, 1.2.10]. A more precise version of this result will be discussed in Section 3. For arbitrary T E L(X) and F ~ C, let XT(F) := {x EX: aT(x) ~ F} denote the corresponding local spectml subspace. Evidently, XT(F) is a linear subspace of X, but need not be closed. The following classical result illustrates that these subspaces playa basic role for spectral decompositions, see [18, 1.2] and [29,4.4]. THEOREM 1. Suppose that T E L(X) is decomposable. Then T has SVEP, and, for each closed set F ~ C, the space XT(F) is closed and satisfies a(T I XT(F)) ~ F. In fact, XT(F) is the largest among all spaces Y E Lat(T) for which aT(T IY) ~ F. Moreover, XT(F) ~ XT(Ut}+·· +XT(U n ) for every finite open cover {U1 , ... , Un} ~F
0
The following examples may illuminate how spectral decompositions work in some important cases; for details see [18] and, for the last assertion of Example 4, also [26, Th.16]. The extent to which the compactness of the group is essential here remains a challenging open problem. EXAMPLE 2. Let T E L(X) be a normal opemtor with spectml measure ~ on a complex Hilbert space X. Then T is decomposable, and XT(F) = ran ~(F) for every closed set F ~ Co Moreover, there exists a non-zero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 EXAMPLE 3. Let X := C(O) be the space of continuous functions on a compact Hausdorff space 0, and let T E L(X) denote the opemtor of multiplication by a given function g E X. Then T is decomposable, and XT(F) = {f E C(O) : g(supp f) ~ F} for every closed set F ~ Co Also in this case, there exists a non-zero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 EXAMPLE 4. Let X := Ll(G) be the group algebm of a locally compact abelian group G, and let T E L(X) denote the opemtor of convolution by a given function g E X. Then T is decomposable, and XT(F) = {f E Ll(G) : g(suppj) ~ F} for every closed set F ~ C, where j denotes the Fourier tmnsform. Moreover, at least when G is compact, there exists a non-zero bounded local resolvent function for T precisely when int a(T) =f:. 0. 0 On the other hand, there are important classes of operators which are not covered by decomposability. For instance, by [18, 1.6.14] and [22], a unilateral weighted right shift on the sequence space fP(N o) for arbitrary 1 ::::; p < 00 is decomposable, or, equivalently, the quotient of a decomposable operator, only in the trivial case when it is quasi-nilpotent, while unilateral weighted right shifts are never generalized scalar. Moreover, as we shall see, there are many examples of unilateral and bilateral weighted left shifts without SVEP. Another illuminating case is that of isometries. By [18, 1.6.7], an arbitrary Banach space isometry is decomposable, or, equivalently, the quotient of a decomposable operator or generalized scalar, precisely when it is invertible. On the other hand, every isometry may be extended, by a classical result due to Douglas, recorded in [18, 1.6.6], to an invertible isometry, and hence has a decomposable extension. In the next section, we shall discuss a more general version of this result.
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2. Moving beyond decomposability Several years before decomposability was formally introduced by Foi~, Bishop [9] investigated a number of spectral decomposition properties in an attempt to extend some of the features of the theory of normal operators to the general setting of Banach spaces. Among these properties, one turned out to be particularly important. We now say that an operator T E L(X) on a complex Banach space X has Bishop's property ((J) provided that, for each open set U ~ C, the operator Tu on H(U, X) is injective with closed range, equivalently, if, for each sequence (fn)nEN in H(U, X) with (T - )..)fn(>\) -> 0 asn->oo, uniformly on each compact subset of U, it follows that fn()..) -> 0 as n -> 00, again uniformly on the compact subsets of U, [18, 1.2.6]. Actually, by [18, 3.3.5], the injectivity condition in this definition is redundant. Obviously, property ({J) implies SVEP. It was shown a long time ago by Foi~ that all decomposable operators share property ((J), but the precise relationship was discovered only recently by Albrecht and Eschmeier [6]. THEOREM 5. An operator T E L(X) has Bishop's property ((J) precisely when T is similar to the restriction of a decomposable operator to a closed invariant subspace. Moreover, in this case, there exists a decomposable extension 8 for which aCT) ~ a(8). 0
The result was, in part, inspired by the work of Putinar [27] who proved that every hyponormal operator is subscalar, in the sense that it has a generalized scalar extension. Thus all hyponormal operators have property ((J). In particular, all unilateral weighted right shifts on f2(N o) with an increasing weight sequence w have property ({J), but a characterization of ({J) in terms of w seems to be an intriguing open problem. For partial results, see [11], [18], [22], and [23]. To discuss the dual notion of Bishop's property ((J), we need a slight variant of the local spectral subspaces. For arbitrary T E L(X) and a closed subset F of 0 such that 1
-s ~
K(Tn) ~ IITnl1 ~ cn s for all n E Nj cn indeed, in this case, a functional calculus cP for T is given by the formula 00
cpU):=
2:
!(n) Tn
for all
f
E £ (T) ,
n=-oo
where !(n) denotes the nth Fourier coefficient of f. Evidently, all invertible isometries have property (P), and hence are £ (T)scalar. Also, it follows from the preceding characterization that an operator T E L(X) is £(T)-scalar precisely when its adjoint T* is £(T)-scalar. Moreover, since property (P) implies that i(T) = r(T) = 1 and consequently aap(T) ~ T, and since aap(T) = a(T) when T has property (6), we are led to the following result. PROPOSITION
7. For every T E L(X) with property (P), the following equiva-
lences hold: T is invertible ¢:} a(T)
~
T
¢:}
T has (6)
¢:}
T is decomposable
¢:}
T is £(T)-scalar.
Moreover, ifT is not invertible, then aap(T) = T and a(T) is the closed unit disc.o
Evidently, every restriction of an £(T)-scalar operator has property (P), but the converse is open in general. The preceding proposition shows that this problem is equivalent to the problem of extending an arbitrary operator with property (P) to an invertible operator with property (P) for possibly larger constants c, s > O. Since the extension provided by the Albrecht-Eschmeier functional model in Theorem 5 increases the spectrum, a different approach is needed here. As noted above, for isometries, the desired extension is possible by a result of Douglas. Also, for a certain class of operators that includes all unilateral weighted right shifts, a positive solution was recently provided by Didas [11] and the authors [23]. While Didas exploits the theory of topological tensor products in the spirit of Eschmeier and Putinar [16], the more elementary approach from [23] uses a modification of a construction provided by Bercovici and Petrovic [8] to characterize compressions of £(T)-scalar operators. For unilateral weighted right shifts on fP(N o), the method developed in [23] leads to extensions as bilateral weighted shifts on fP(Z) with sharp growth estimates. To reduce the case of quotients of £(T)-scalar operators to that of restrictions, we recall that the minimum modulus 'Y(T) of a non-zero operator T E L(X) is defined as 'Y(T):= inf{IITxll/dist(x,kerT): x (j kerT}. Clearly, 'Y(T) = K(T)
THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES
253
when T is injective. It is also well known that 'Y(T) = 'Y(T*), and that 'Y(T) > 0 precisely when T has closed range. Standard duality theory now leads to the following result. PROPOSITION 8. An operator T E L(X) is the restriction of an £(1l')-scalar operator if and only if its adjoint T* is the quotient of an £(1l')-scalar operator. Moreover, ifT is the quotient of an £(1l')-scalar operator, then T* is the restriction of an £(1l')-scalar operator, and hence there exist constants c, s > 0 for which
~::;'Y(Tn)::;IITnll::;cns cn
0
forallnEN.
In general, it is not known if the last growth condition characterizes the quotients of £(1l')-scalar operators, but, by Proposition 8 and [23, Prop.5], this is the case for the class of all unilateral weighted left shifts on fP(N o ) for arbitrary 1 < p < 00. More precisely, a unilateral weighted left shift on fP(N o) satisfies the growth condition of Proposition 8 if and only if it admits a bilateral weighted shift lifting on fP(Z) that is £(1l')-scalar. Similar results hold for more general growth conditions; see [22] and [23]. For instance, by another classical result due to Colojoara and Foi~, an invertible operator T E L(X) is decomposable provided that T satisfies Beurling's condition (B), in the sense that
L 00
1 n 2 (llogK(Tn)1
+ IlogliTnll1) < 00,
n=l
[10, 5.3.2] and [18,4.4.7]. Clearly, property (B) is inherited by restrictions, but it remains open, if every operator with property (B) has an invertible extension with property (B). In fact, it is not known, if property (B) implies property ((3). For certain unilateral weighted right shifts, a positive answer was recently given in [22] and [23].
3. Localization of the single-valued extension property For an arbitrary operator T E L(X) on a complex Banach space X, here the spaces K(T) := XT(C\ {O}) and Ho(T) := X T ( {O}) will be of particular importance. Both spaces were, in some disguise, studied by Mbekhta and also by Vrbova; see [19], [20], and [30]. By [18, 3.3.7], K(T) coincides with the analytic core of T, defined to consist of all x E X for which there exist a constant c > 0 and elements Xn E X such that for all n E N. By this characterization and the open mapping theorem, K(T)
=X
if and only if
T is surjective. In terms of local spectral theory, this follows also from the fact that asu(T) is the union of all local spectra of T. On the other hand, by [18, 3.3.13], Ho(T) is the quasi-nilpotent part of T, defined as the set of all x E X for which
IITnx11 1/ n _ 0
as n -
00.
In general, neither K(T) nor Ho(T) need to be closed, but, if 0 is isolated in a(T), then, by [19, 1.6], both spaces are closed and X = K(T) EB Ho(T). For more on operators with closed K(T) and Ho(T), see [1], [2], and [24]. For instance, by [24, Cor.6], for any non-invertible decomposable operator T, the point 0 is isolated in
T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
254
a(T) precisely when K(T) is closed. In particular, the analytic core of a compact or, more generally, a Riesz operator T is closed exactly when T has finite spectrum, [24, Cor.9]. The spaces Ho(T) and K(T) are related to the kernel N(T) and the range R(T) of T as follows. With the notation oc
N(T) :=
U N(Tn)
n 00
and
R(T):=
R(Tn)
n=l
n=l
for the generalized kernel and range of T, there is an increasing chain of kernel-type spaces
N(T) ~ N(Tn) ~ N(T) ~ Ho(T) and a decreasing chain of range-type spaces
~ X T ({O})
R(T) 2 R(Tn) 2 R(T) 2 K(T) 2 X T (0) for arbitrary n E N. [18, 1.2.16 and 3.3.1]. The geometric position of the kerneltype spaces versus the range-type spaces turns out to be intimately related to a certain localized version of SVEP for the operator T and its adjoint T*. An operator T E L(X) is said to have SVEP at a point A E C, if, for every open disc U centered at A, the operator Tu is injective on H(U, X). This notion dates back to Finch [17], and was pursued further, for instance, in [1], [2], [3], [4], [5], and [20]. Evidently, T has SVEP at A precisely when T - A has SVEP at 0, while SVEP for T is equivalent to SVEP for T at A for each A E C. Local spectral theory leads to a variety of characterizations of this localized version of SVEP that involve the kernel-type and range-type spaces introduced above. Our starting point is the following characterization from [3, 1.9]. The result shows, in particular, that every injective operator T E L(X) has SVEP at 0, and may be viewed as a local version of the classical fact that T has SVEP if and only if X T (0) = {O}, [18, 1.2.16]. For completeness, we include a short new proof that uses nothing but local spectral theory. THEOREM
9. For every operator T
T has SVEP at 0 . + a3 >.2 + ... ) = 0 first for all non-zero>. E U, and then, by continuity, also for>. = O. Exactly as before, we conclude that at = 0 and hence, by induction, an = 0 for all n ~ O. Thus f == 0 on U, as desired. 0 Since N(T) n K(T) ~ X T ( {O}) n XT(C \ {O}) = X T (0), it clearly follows that N(T) n K(T) = N(T) n X T (0) for every T E L(X). Thus, by Theorem 9, T has SVEP at 0 if and only if N(T) n K(T) = {O}. In particular, if T is surjective, then, as noted above, K(T) = X, so that T has SVEP at 0 precisely when T is injective. This characterization from [3, 1.11] extends a classical result due to Finch [17]. As another immediate consequence of Theorem 9, we obtain the following result. COROLLARY 10. An operator T E L(X) has SVEP at 0 provided that either Ho(T) n K(T) = {O} or N(T) n R(T) = (0). 0 Recent counter-examples in [2] show that, in general, none of the latter conditions is equivalent to SVEP of T at 0, thus disproving a claim made in [20, 1.4]. However, by [1, 2.7], [5, 1.3], and Theorem 12 below, equivalences do hold for certain classes of operators. We now describe how the localized SVEP behaves under duality. For a linear subspace M of X, let Ml. := {cp E X* : cp(x) = 0 for all x E M}, and for a linear subspace N of X*, let l.N := {x EX: cp(x) = 0 for all cp E N}. By the bipolar theorem, l.(Ml.) is the norm-closure of M, and (l.N)l. is the weak-*-closure of N. Moreover, for every T E L(X), it is well known that N(T*) = R(T)l. and N(T) = l.R(T*), while R(T) is a norm-dense subspace of l.N(T*), and R(T*) is a weak-*-dense subspace of N(T)l.. An elementary short proof of the following result may be found in [2,4.1]. PROPOSITION 11. For every operator T E L(X), the following assertions hold: (a) K(T) ~ l.Ho(T*) and K(T*) ~ Ho(T)l.; (b) if Ho(T) + R(T) is norm-dense in X, then T* has SVEP at 0; (c) if Ho(T*) + R(T*) is weak-*-dense in X*, then T has SVEP at o. 0 Even in the Hilbert space setting, the inclusions in part (a) of Proposition 11 need not be identities, and the implications of parts (b) and (c) cannot be reversed in general; see [2] for counter-examples in the class of weighted shifts. However, for suitable classes of operators, the results can be improved. As usual, an operator T E L(X) is said to be a semi-Fredholm operator, if either N(T) is finite-dimensional and R(T) is closed, or R(T) is of finite codimension in X. Also, an operator T E L(X) is said to be semi-regular, if R(T) is closed and N(T) ~ R(T); see [18], [19], and [21] for a discussion of these operators. THEOREM 12. Suppose that the operator T E L(X) is either semi-Fredholm or semi-regular. Then the following assertions hold: (a) R(T) = K(T) = l.Ho(T*) = l.N(T*); (b) R(T*) = K(T*) = Ho(T)l. = N(T)l.; (c) N(T) n R(T) = {O} T has SVEP at 0; (d) N(T*) n R(T*) = {O} T* has SVEP at 0; (e) N(T)
+ R(T) = X
(f) N-;-;:-;:(T=*""7"")-+-:R~(T=*"-;-) w'
+ R(T) = X T* has SVEP at 0; Ho(T*) + R(T*) w' = X* T has SVEP
Ho(T)
= X*
where w* indicates the closure with respect to the weak-*-topology.
at 0,
o
256
T. L. MILLER,
V. G.
MILLER, AND M. M. NEUMANN
Theorem 12 was recently obtained in [2], see also [5]. An important ingredient of the proof is the fact that T has SVEP at 0 if and only Tn has SVEP at 0 for arbitrary n E N. This equivalence is a special case of a spectral mapping formula for the set 6(T) of all A E C at which T fails to have SVEP, namely 6(f(T)) = f(6(T)) for every analytic function f on some open neighborhood of a(T); see [2, 3.1] and also Theorem 18 below. Further developments may be found in [1], [3], [4], [5], [17], and [20]. Here we mention only one simple consequence of Theorem 12 for semi-regular operators from [3, 2.13]. For T E L(X), let PK(T) consist of all A E C for which T - A is semi-regular. The Kato spectrum aK(T) := C \ PK(T) is a closed subset of a(T) and contains oa(T); see [18, 3.1] and [21] for details. We include a short proof of the following result, since the dichotomy for the connected components of the Kato resolvent set PK (T) with respect to the localized SVEP will play an essential role in Section 4. THEOREM 13. Let T E L(X) be semi-regular. Then T has SVEP at 0 precisely when T is injective, or, equivalently, when T is bounded below, while T* has SVEP at 0 precisely when T is surjective. Moreover, for arbitrary T E L(X), each connected component n of PK(T) satisfies either n ~ 6(T) or n n 6(T) = 0. The inclusion n ~ 6(T) OCC1J.rs precisely when n ~ ap(T), or, equivalently, when n n aap(T) i=- 0, while the identity n n 6(T) = 0 occurs precisely when n n ap(T) = 0, or, equivalently, when n \ aap(T) i=- 0.
Proof. If T is semi-regular, then N(T) n n(T) = N(T) and N(T) + R(T) = R(T) = R(T). Hence the first assertions follow from parts (c) and (e) of Theorem 12. For the last claim, it suffices to see that injectivity of T - A for some A E n entails that T - f.J. is injective for every f.J. E n. But this is clear, since, by part (b) of Theorem 12, N(T - f.J.) = J..n(T* -f.J.) and, by [18, 3.1.6 and 3.1.11], n(T* - f.J.) = n(T* -A) for all f.J. E n. 0 It is well known that the approximate point spectrum and the surjectivity spectrum of an arbitrary operator T E L(X) are related by the duality formulas aap(T) = asu(T*) and asu(T) = aap(T*), [18, 1.3.1]. Moreover, by [18, 1.3.2 and 3.1.7], asu(T) = a(T) and aap(T) = aK(T) if T has SVEP, and aap(T) = a(T) and asu(T) = aK(T) if T* has SVEP. The following local version of these results is immediate from Theorem 13. PROPOSITION 14. For every operator T E L(X), the following assertions hold: (a) If A E a(T) \ aap(T), then T has SVEP at A, but T* fails to have SVEP at A; (b) if A E a(T) \ asu(T), then T* has SVEP at A, but T fails to have SVEP at A.
o The next result from [2, 5.2] is a straightforward consequence of Proposition 14. For instance, it follows that 6(T*) is the open unit disc for every non-invertible operator T with property (P) or (B). Further examples including analytic Toeplitz operators, composition operators on Hardy spaces, and weighted shifts may be found in [2]. COROLLARY 15. If aap(T) ~ oa(T), then T has SVEP and 6(T*) = int a(T). Similarly, if asu(T) ~ oa(T), then T* has SVEP and 6(T) = int a(T). 0
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4. Localization of the properties ((:J) and (8) There is a natural extension of the class of decomposable operators for which spectral decompositions are only required with respect to a given open subset U of the complex plane. These operators were introduced by Vasilescu as residually decomposable operators in 1969, shortly after the publication of the seminal monograph [10]. They became also known as S-decomposable operators with S = c \ U, and were studied by Bacalu, Nagy, Vasilescu, and others; see [29, eh.4]. As in [6] and [13], we now say that an operator T E L(X) on a complex Banach space X is decomposable on an open subset U of C provided that, for every finite open cover {Vl , ... , Vn } of C with C \ U ~ Vl , there exist Xl,"" Xn E Lat(T) with the property that X
= Xl + ... + Xn and
a(T I X k ) ~
Vk
for k
= 1, ... , n.
It is known, although certainly not obvious, that, in this definition, it suffices to consider the case n = 2; see [6] and [29]. Evidently, classical decomposability occurs when U = C. On the other hand, every operator T E L(X) is at least decomposable on its resolvent set p(T). Among the remarkable early accomplishments of the theory is the following result due to Nagy [25] from 1979: For every T E L(X), there exists a largest open set U ~ C on which T is decomposable. The complement of this set is Nagy's spectral residuum Sr(T), a closed, possibly empty, subset of a(T). In the present section, we shall employ the recent results of Albrecht and Eschmeier [6] to obtain a short proof for the existence and a useful description of Nagy's spectral residuum. In particular, we shall see how Sr(T) is related to the Kato spectrum aK(T) and the essential spectrum ae(T). For this, we shall work with certain localized versions of property ((:J) and property (8) from [6]. An operator T E L(X) is said to possess Bishop's property ((:J) on an open set U ~ C, if, for every open subset V of U, the operator Tv is injective with closed range, equivalently, if, for every sequence of analytic functions In: V ---+ X for which (T->')In(>\) ---+ 0 as n ---+ 00 locally uniformly on V, it follows that In(>') ---+ 0 as n ---+ 00, again locally uniformly on V. It is straightforward to check that this condition is preserved under arbitrary unions of open sets. This shows that there exists a largest open set on which T has property ((:J), denoted by U{3(T). Its complement S{3(T) := C \ U{3(T) is a closed, possibly empty, subset of a(T). In fact, T satisfies Bishop's classical property ((:J) precisely when S{3(T) = 0. Moreover, the operator T is said to have property (8) on U, if X
= XT(C \ V) + XT(W)
for all open sets V, W ~ C for which C \ U ~ V ~ V ~ W; see [6] and [13]. Quite remarkably, as shown in [6, Th.3], this condition holds precisely when, for each closed set F ~ C and every finite open cover {VI"'" Vn } of F with F \ U ~ VI, it follows that XT(F) ~ XT(V d + ... + XT(V n); see also [18, 2.2.2] for the case U=C, These localized versions of ((:J) and (8) already proved to be useful in the theory of invariant subspaces for operators on Banach spaces, [14]. The following result summarizes the main accomplishments from [6].
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T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
THEOREM 16. For every operator T following equivalences hold:
E
L(X) and every open set U
~
C, the
(a) T has (13) on U {::} T* has (8) on U; (b) T has (8) on U {::} T* has (/3) on U;
(c) T is decomposable on U {::} T has both (13) and (8) on U; (d) T has (13) on U {::} T is the restriction of a decomposable operator on U; (e) T has (15) on U {::} T i,~ the quotient of a decomposable operator on U.
0
It is not at all obvious from the definition of (8) that there exists a largest open set, say Uc5(T), on which the operator T has property (8), but this now follows from the corresponding result for (13) by duality. In fact, Uc5(T) = U{3(T*) by part (b) of the preceding result. More precisely, Theorem 16 leads to the following result. COROLLARY 17. For every operator T E L( X), there exists a smallest closed set Sc5(T) so that T has property (8) on its complement. Moreover, Sc5(T) = S{3(T*), S{3(T) = Sc5(T*), and Sr(T) = S{3(T) U Sc5(T) = S{3(T) U S{3(T*) = Sr(T*). 0
Perhaps somewhat surprisingly, it will be possible to obtain general information about the location of S{3(T), and hence of Sc5(T) and Sr(T). For this, another localized version of SVEP will play a crucial role. For consistency, we say that the operator T E L(X) has SVEP on an open set U ~ C, if, for every open subset V of U, the operator Tv is injective, [13]. It is straightforward to check that T has SVEP on U precisely when T has SVEP at each point >. E U, as defined in the previous section. Obviously, there exists a largest open set on which T has SVEP, and the analytic spectral residuum S(T) is defined to be the complement of this set; see [29, 4.3.2] and [30]. Clearly, 6(T) ~ S(T), but, since 6(T) is open and S(T) is closed, equality occurs only in the trivial case when T has SVEP. Nevertheless, as noted in [2], a simple verification shows that 6(T) = S(T) ~ S{3(T). It is interesting to observe that all these residual sets behave canonically with respect to the Riesz functional calculus. As usual, for T E L(X) and any analytic complex-valued function f on an open neighborhood 0 of a(T), the operator f(T) E L(X) is defined by
~
r
f(>.)(>. - T)-l d>', 2m where r denotes an arbitrary contour in 0 that surrounds a(T), [12] or [18, A.2]. The standard spectral mapping theorem asserts that a(f(T)) = f(a(T)). The next result has a similar flavor, and may be viewed as an extension of the fact that the classical versions of SVEP, property (/3), property (8), and decomposability are all preserved under the Riesz functional calculus, [18, 3.3.6 and 3.3.9]. The fact that the Riesz functional calculus respects Bishop's classical property (13) was established by Eschmeier and Putinar [15]. The following proof involves a different approach to this result. f(T) :=
lr
THEOREM 18. Let T E L(X) be an arbitrary operator, let f : 0 -+ C be an analytic function on an open neighborhood 0 of a(T), and suppose that f is nonconstant on each connected component ofO. If E denotes any of the symbols 6, S, S{3, Sc5, or S., then E(f(T)) = f(E(T)).
THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES
259
Proof. In the case of 6(T), the spectral mapping formula was recently obtained in [2, 3.1]. The result for S(T) is a standard fact that may be found in [29, 4.3.14] and [30, 1.6]. Note, however, that the formula for S(T) is also an immediate consequence of that for 6(T), because 6(T) = S(T). While the existence of the residual set So(T) was most conveniently established by using S{3(T) and the Albrecht-Eschmeier duality between the localized versions of (f3) and (8), for the issue at hand it seems appropriate to switch the order. Indeed, since f(T*) = f(T) *, Corollary 17 ensures that it suffices to prove the claim for So(T). For this, fortunately, we may proceed as in the proof of [18,3.3.6 and 3.3.9], where property (8) is shown to be stable under the Riesz functional calculus. First, consider arbitrary open sets V, W ~ C for which f(So(T)) ~ V ~ V ~ W. Then {f-l(C \ V), f- 1(W)} is an open cover of a(T) for which So(T) ~ f-l(W). Thus, by the characterization of the localized property (8) mentioned above, we obtain that
X = XT(a(T))
= XT
(1-1(C \ V)
n a(T)) + XT (1-1(W) n a(T)) .
Clearly, f- 1(C \ V) n a(T) ~ f- 1(C \ V) n a(T) and, similarly, f-l(W) n a(T) ~ f-l(W) n a(T). Since, by [18,3.3.6], the formula XT (f-l(F) n a(T)) = Xf(T) (F) holds for every closed set F ~ C, we conclude that X = Xf(T) (C \ V)
+ Xf(T) (W).
This shows that f(T) has (8) on C \ f(So(T)), thus C \ f(So(T)) ~ Uo(f(T)), and hence So(f(T)) ~ f(So(T)). Note that this inclusion even holds without the requirement that f be non-constant on each of the connected components of its domain. The reverse inclusion is less obvious, but may be obtained by a suitable modification of the proof of [18, 3.3.9]. Let S := f- 1 (So(f(T))) n a(T). Then the desired inclusion f(So(T)) ~ So(f(T)) means precisely that the decomposition X = XT(C \ V) + XT(W) holds for all open sets V, W ~ C for which S ~ V ~ V ~ W. Evidently, it suffices to show that X = XT(G) + XT(H) for every open cover {G,H} of a(T) for which S ~ G, SnH = 0, and both G and H are compact subsets of n. Ignoring momentarily the exceptional set S, we may proceed word by word along the lines of the proof of [18, 3.3.9] to obtain a finite open cover {WI, ... , W n } of a(T) in n for which fork=I, ... ,n. To handle the residual set, we note that the identity SnH = 0 may be reformulated in the form So(f(T)) n f(a(T) n H) = 0. Hence, by continuity and compactness, there exists an open neighborhood V of So (f(T)) for which V n f(a(T) n H) = 0. This implies that f- 1 (V) n a(T) n H = 0, hence f- 1 (V) n a(T) ~ G, and therefore, by [18, 3.3.6], Xf(T)(V) = XT (J-l(V)
n a(T))
~ XT(G).
Now, since f(T) has (8) on C \ So(f(T)), and since {V, f(W1), ... , f(Wn )} is an open cover of a(f(T)) = f(a(T)) for which So(f(T)) ~ V, we conclude that X
= Xf(T)(a(T)) = Xf(T)(V) + Xf(T)
(f(Wd)
+ ... + Xf(T)
(f(Wn)) '
T.
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L. MILLER, V. G. MILLER, AND M. M. NEUMANN
again by the basic characterization of the localized version of (8) provided in [6, Th.3]. Thus X = XT(G) + XT(H), as desired. 0 In the following results, we show how to verify property ({3) on the Kato resolvent set and its Fredholm counterpart. THEOREM 19. For an arbitrary operator T E L(X) and every connected component n of PK(T), the following equivalences hold:
T has ({3) on
n {:}
T has SVEP on
n {:} n n O'p(T) = 0 {:} n \ O'ap(T) =F 0j
in particular, T has property ({3) on PK(T) precisely when T has SVEP on PK(T). Proof. Clearly, the first of the displayed conditions implies the second one, and the equivalence of the last three conditions follows from Theorem 13. Conversely, if these three conditions hold, then T - A is injective with closed range for all A E n. Thus f'i,(T - A) = "((T - A) > 0 for all A E n. Moreover, by [18, 3.1.10], for every compact subset K of n, there exists a constant c > 0 such that f'i,(T - A) > 0 for all A E Kj in fact, as shown, for instance, in [21,4.1], the function A t-+ "((T - A) is continuous and strictly positive on PK(T). From this it is immediate that T has ({3) on n. 0 The next result is clear from Corollary 17, Theorem 19, and the well-known identity PK(T) = PK(T*), [18, 3.1.6]. Part (c) of Corollary 20 may be viewed as an extension of the fact that, by [18, 3.1.7], p(T) = PK(T) whenever both T and T* have SVEP. COROLLARY 20. For every operatorT E L(X), the following assertions hold:(a) T has SVEP on PK(T) {:} S{3(T) ~ O'K(T)j (b) T* has SVEP on PK(T) {:} SIi(T) ~ O'K(T)j (c) T and T* have SVEP on PK(T) {:} Sr(T) ~ O'K(T). 0 To derive the companion result for the essential spectrum, we employ the fact that, for every operator T E L(X) and every open subset V of the essential resolvent set Pe(T) := C\O'e(T), the operator Tv has closed range (but need not be injective). This interesting result was recently obtained by Eschmeier [13, 3.1], based on sheaftheoretic methods developed by Putinar [28] to show that quasi-similar operators with property ({3) have the same essential spectrum. In tandem with Corollary 17, we obtain the following extension of [13, 3.9]. COROLLARY 21. For every operator T E L(X), the following assertions hold: (a) T has SVEP on Pe(T) {:} S{3(T) ~ O'e(T)j (b) T* has SVEP on Pe(T) {:} SIi(T) ~ O'e(T)j (c) T and T* have SVEP on Pe(T) {:} Sr(T) ~ O'e(T).
o
We close with an application to the spectral theory of weighted shifts. EXAMPLE 22. Let W := (Wn)nEN"o be a bounded sequence of strictly positive real numbers, and let T E L(X) denote the corresponding unilateral weighted right shift on the sequence space X := fP(N o) for some 1 ::; p < 00. Clearly,
i(T) = lim inf
n-+oo k~O
(Wk· .. Wk+n_d 1 / n
and
r(T) = lim sup (Wk· n-+oo k~O
.. Wk+n_d 1 / n .
THE SPECTRAL THEORY OF OPERATORS ON BANACH SPACES
Since T has no eigenvalues, T has SVEP and asu(T) = a(T) = {A E C : IAI by [18, 1.3.2 and 1.6.15J. Moreover, as noted in [18,3.7.7],
261
s r(T)},
siAl s r(T)} , and therefore, by Corollaries 20 or 21, the annulus {A E C : i(T) siAl s r(T)} contains S(3(T). On the other hand, by [2, 6.1J, 6(T*) = {A E C : IAI < c(T)} , where ae(T) = aK(T) = aap(T) = {A E C : i(T)
c(T):= liminf(wl" n->oo
'Wn)l/n.
Thus, by Corollary 17, it follows that S5(T) = S(3(T*) 2 {A E C : IAI S c(T)}. We finally note that, by [22, 2.7J, condition (f3) on T implies that i(T) = r(T) and aT(x) = a(T) for all non-zero x E X, while, by [22, 3.3J or [23, Prop.5], a certain growth condition of exponential type for the weight sequence w suffices to ensure that T has (f3). 0
References [IJ P. Aiena, M. L. Colasante, and M. Gonzalez, Opemtors which have a closed quasi-nilpotent part, Proc. Amer. Math. Soc. 130 (2002), 2701-2710. [2J P. Aiena, T. L. Miller, and M. M. Neumann, On a localized single-valued extension property, to appear in Proc. Royal Irish Acad. [3J P. Aiena and O. Monsalve, Opemtors which do not have the single valued extension property, J. Math. Anal. Appl. 250 (2000),435-448. [4J P. Aiena and O. Monsalve, The single valued extension property and the genemlized Kato decomposition property, Acta Sci. Math. (Szeged) 67 (2001), 791-807. [5J P. Aiena and F. Villafane, Components of resolvent sets and local spectml theory, submitted to this volume. [6J E. Albrecht and J. Eschmeier, Analytic functional models and local spectml theory, Proc. London Math. Soc. (3) 75 (1997), 323-348. [7J E. Albrecht and W. J. Ricker, Local spectml theory f01· opemtors with thin spectrum, preprint, University of Saarbriicken, 2002. [8J H. Bercovici and S. Petrovic, Genemlized scalar opemtors as dilations, Proc. Amer. Math. Soc. 123 (1995), 2173-2180. [9J E. Bishop, A duality theory for an arbitmry opemtor, Pacific J. Math. 9 (1959), 379-397. [lOJ I. Colojoara and C. Foi~, Theory of Genemlized Spectml Opemtors, Gordon and Breach, New York, 1968. [11J M. Didas, E(]"n )-subscalar n-tuples and the Cesaro opemtor on HP, Annales Universitatis Saraviensis, Series Mathematicae 10 (2000), 285-335. [12J N. Dunford and J. T. Schwartz, Linear Opemtors III, Wiley-Interscience, New York, 1971. [13J J. Eschmeier, On the essential spectrum of Banach-space opemtors, Proc. Edinburgh Math. Soc. (2) 43 (2000), 511-528. [14J J. Eschmeier and B. Prunaru, Invariant subspaces for opemtors with Bishop's property ({3) and thick spectrum, J. Funct. Anal. 94 (1990), 196-222. [15J J. Eschmeier and M. Putinar, Bishop's condition ({3) and rich extensions of linear opemtors, Indiana Univ. Math. J. 37 (1988), 325-348. [16J J. Eschmeier and M. Putinar, Spectml Decompositions and Analytic Sheaves, Clarendon Press, Oxford, 1996. [17J J. K. Finch, The single valued exten.9ion property on a Banach space, Pacific J. Math. 58 (1975),61-69. [18J K. B. Laursen and M. M. Neumann, An Introduction to Local Spectml Theory, Clarendon Press, Oxford, 2000. [19J M. Mbekbta, Genemlisation de la decomposition de Kato aux opemteurs pamnormaux et spectmux, Glasgow Math. J. 29 (1987), 159-175. [20J M. Mbekhta, Sur la theorie spectmle locale et limite des nilpotents, Proc. Amer. Math. Soc. 110 (1990), 621-631.
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T. L. MILLER, V. G. MILLER, AND M. M. NEUMANN
[21] M. Mbekhta and A. Ouahab, Operateurs s-regulier dans un espace de Banach et theorie spectrale, Acta Sci. Math. (Szeged) 59 (1994), 525-543. [22] T. L. Miller, V. G. Miller, and M. M. Neumann, Local spectral properties of weighted shifts,
to appear in J. Operator Theory. [23] T. L. Miller, V. G. Miller, and M. M. Neumann, Growth conditions and decomposable exten-
sions, to appear in Contemp. Math. [24] T. L. Miller, V. G. Miller, and M. M. Neumann, On operators with closed analytic core, to appear in Rend. Cire. Mat. Palermo (2) 51 (2002). [25] B. Nagy, On S-decomposable operators, J. Operator Theory 2 (1979),277-286. [26] M. M. Neumann, Recent developments in local spectral theory, Rend. Circ. Mat. Palermo (2) Suppl. 68 (2002), 111-131. [27] M. Putinar, Hyponormal operators are subsealar, J. Operator Theory 12 (1984), 385-395. [28] M. Putinar, Quasi-similarity of tuples with Bishop's property (,8), Integral Equations Operator Theory 15 (1992), 1047-1052. [29] F.-H. Vasilescu, Analytic FUnctional Calculus and Spectral Decompositions, Editura Aeademiei and D. Reidel Publishing Company, Bucharest and Dordreeht, 1982. [30] P. Vrbova., On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23 (98) (1973),483-492. DEPARTMENT OF MATHEMATICS AND STATISTICS, MISSISSIPPI STATE UNIVERSITY, MISSISSIPPI STATE, MS 39762, USA E-mail address: neumannOmath.msstate.edu
Contemporary Mathematics Volume 328, 2003
Abstract harmonic analysis, homological algebra, and operator spaces Volker Runde ABSTRACT. In 1972, B. E. Johnson proved that a locally compact group G is amenable if and only if certain Hochschild cohomology groups of its convolution algebra Ll(G) vanish. Similarly, G is compact if and only if Ll(G) is biprojective: In each case, a classical property of G corresponds to a cohomological propety of Ll(G). Starting with the work of Z.-J. Ruan in 1995, it has become apparent that in the non-commutative setting, i.e. when dealing with the Fourier algebra A(G) or the Fourier-Stieltjes algebra B(G), the canonical operator space structure of the algebras under consideration has to be taken into account: In analogy with Johnson's result, Ruan characterized the amenable locally compact groups G through the vanishing of certain cohomology groups of A(G). In this paper, we give a survey of historical developments, known results, and current open problems.
1. Abstract harmonic analysis, ...
The central objects of interest in abstract harmonic analysis are locally compact groups, i.e. groups equipped with a locally compact Hausdorff topology such that multiplication and inversion are continuous. This includes all discrete groups, but also all Lie groups. There are various function spaces associated with a locally compact group G, e.g. the space Co(G) of all continuous functions on G that vanish at infinity. The dual space of Co(G) can be identified with M(G), the space of all regular (complex) Borel measures on G. The convolution product * oftwo measures is defined via (1,11-* v):= LLf(XY)dJ.L(X)V(Y)
(J.L,V E M(G), f E Co(G))
and turns M(G) into a Banach algebra. Moreover, M(G) has an isometric involution given by
(I,J.L*):= Lf(x-1)dJ.L(X)
(J.L E M(G), f E Co(G)).
1991 Mathematics Subject Classification. 22D15, 22D25, 43A20, 43A30, 46H20 (primary), 46H25, 46L07, 46M18, 46M20, 47B47, 47L25, 47L50. Key words and phrases. locally compact groups, group algebra, Fourier algebra, FourierStieltjes algebra, Hochschild cohomology, homological algebra, operator spaces. Financial support by NSERC under grant no. 227043-00 is gratefully acknowledged. © 263
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The most surprising feature of an object as general as a locally compact group is the existence of (left) Haar measure: a regular Borel measure which is invariant under left translation and unique up to a multiplicative constant. For example, the Haar measure of a discrete group is simply counting measure, and the Haar measure of ]RN, is N-dimensional Lebesgue measure. The space Ll(G) of all integrable functions with respect to Haar measure can be identified with a closed *-ideal of M(G) via the Radon-Nikodym theorem. Both M(G) and Ll(G) are complete invariants for G: Whenver Ll(G 1 ) and £1(G 2 ) (or M(Gt} and M(G 2 )) are isometrically isomorphic, then G 1 and G 2 are topologically isomorphic. This means that all information on a locally compact group is already encoded in Ll (G) and M(G). For example, Ll(G) and M(G) are abelian if and only if G is abelian, and Ll (G) has an identity if and only if G is discrete. References for abstract harmonic analysis are [Fol], [H-R], and [R-St]. The property of locally compact groups we will mostly be concerned in this survey is amenability. A a mean on a locally compact group G is a bounded linear functional m: LOC(G) ---+ C such that (1, m) = Ilmil = 1. For any function I on G and for any x E G, we write LxI for the left translate of I by x, i.e. (Lxf)(y) := I(xy) for y E G. DEFINITION 1.1. A locally compact group G is called amenable if there is a (left) translation invariant mean on G, i.e. a mean m such that
(¢, m) = (L x ¢, m)
(¢ E LOC(G), x E G).
EXAMPLE 1.2. (1) Since the Haar measure of a compact group G is finite, LOC(G) C £1(G) holds. Consequently, Haar measure is an invariant mean on G. (2) For abelian G, the Markov-Kakutani fixed point theorem yields an invariant mean on G. (3) The free group in two generators is not amenable ([Pat, (0.6) Exanlple]). Moreover, amenability is stable under standard constructions on locally compact groups such as taking subgroups, quotients, extensions, and inductive limits. Amenable, locally compact groups were first considered by J. v. Neumann ([Neu]) in the discrete case; he used the term "Gruppen von endlichem MaB". The adjective amenable for groups satisfying Definition 1.1 is due to M. M. Day ([Day]), apparently with a pun in mind: They are amenable because they have an invariant mean, but also since they are particularly pleasant to deal with and thus are truly amenable - just in the sense of that adjective in everyday speech. For more on the theory of amenable, locally compact groups, we refer to the monographs [Gre], [Pat], and [Pie].
2. homological algebra, ... We will not attempt here to give a survey on a area as vast as homological algebra, but outline only a few, basic cohomological concepts that are relevant in connection with abstract harmonic analysis. For the general theory of homological algebra, we refer to [C-E], [MacL], and [Wei]. The first to adapt notions from homological algebra to the functional analytic context was H. Kamowitz in [Kam]. Let 2l be a Banach algebra. A Banach 2l-bimodule is a Banach space E which is also an 2l-bimodule such that the module actions of 2l on E are jointly continuous.
ABSTRACT HARMONIC ANALYSIS A derivation from 2l to E is a (bounded) linear map D: 2l D(ab) = a . Db + (Da) . b
265
--+
E satisfying
(a, bE !2l);
the space of all derivation from 2l to E is commonly denoted by ZI(!2l, E). A derivation D is called inner if there is x E E such that Da = a·x-x·a
(a E !2l).
The symbol for the subspace of ZI (!2l, E) consisting of the inner derivations is B 1 (2l,E); note that B 1 (!2l,E) need not be closed in ZI(!2l,E). DEFINITION 2.1. Let !2l be a Banach algebra, and let E be a Banach 2l-bimodule. Then then the first Hochschild cohomology group 'HI (2l, E) of 2l with coefficients in E is defined as 'H 1 (!2l, E) := ZI(!2l, E)/B 1 (2l, E). The name Hochschild cohomology group is in the honor of G. Hochschild who first considered these groups in a purely algebraic context ([Hoch 1] and [Hoch 2]). Given a Banach !2l-bimodule E, its dual space E* carries a natural Banach 2l-bimodule structure via (x,a· ¢) := (x· a,¢)
and
(x,¢· a) := (a· x,¢)
(a E !2l, ¢ E E*, x E E).
We call such Banach !2l-bimodules dual. In his seminal memoir [Joh 1], B. E. Johnson characterized the amenable locally compact groups G through Hochschild cohomology groups of Ll(G) with coefficients in dual Banach £l(G)-bimodules ([Joh 1, Theorem 2.5]): THEOREM 2.2 (B. E. Johnson). Let G be a locally compact group. Then G is amenable if and only if'Hl(Ll(G),E*) = {O} for each Banach Ll(G)-bimodule E. The relevance of Theorem 2.2 is twofold: First of all, homological algebra is a large and powerful toolkit - the fact that a certain property is cohomological in nature allows to apply its tools, which then yield further insights. Secondly, the cohomological triviality condition in Theorem 2.2 makes sense for every Banach algebra. This motivates the following definition from [Joh 1]: DEFINITION 2.3. A Banach algebra 2l is called amenable if 'Hl(!2l, E*) = {O} for each Banach 2l-bimodule E. Given a new definition, the question of how significant it is arises naturally. Without going into the details and even without defining what a nuclear C* -algebra is, we would like to only mention the following very deep result which is very much a collective accomplishment of many mathematicians, among them A. Connes, M. D. Choi, E. G. Effros, U. Haagerup, E. C. Lance, and S. Wassermann: THEOREM 2.4. A C* -algebra is amenable if and only if it is nuclear. For a relatively self-contained exposition of the proof, see [Run, Chapter 6]. Of course, Definition 2.3 allows for modifications by replacing the class of all dual Banach 2l-bimodules by any other class. In [B-C-D], W. G. Bade, P. C. Curtis, Jr., and H. G. Dales called a commutative Banach algebra !2l weakly amenable if and only if 'HI (2l, E) = {O} for every symmetric Banach !2l-bimodule E, i.e. satisfying (a E !2l, x E E). a·x=x·a
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This definition is of little use for non-commutative 21. For commutative 21, weak amenability, however, is equivalent to 'Jtl(21, 21*) = {O} ([B-C-D, Theorem 1.5]), and in [Joh 2], Johnson suggested that this should be used to define weak amenability for arbitrary 21: DEFINITION 2.5. A Banach algebra 21 is called weakly amenable if 'Jtl(21, 21*) =
{O}. REMARK 2.6. There is also the notion of a weakly amenable, locally compact group ([C-H]). This coincidence of terminology, however, is purely accidental. In contrast to Theorem 2.2, we have: THEOREM 2.7 ([Joh 3]). Let G be a locally compact group. Then £l(G) is weakly amenable. For a particularly simple proof of this result, see [D-Gh]. For M(G), things are strikingly different: THEOREM 2.8 ([D-Gh-H]). Let G be a locally compact group. Then M(G) is weakly amenable if and only if G is discrete. In particular, M (G) is amenable if and only if G is discrete and amenable. Sometime after Kamowitz's pioneering paper, several mathematicians started to systematically develop a homological algebra with functional analytic overtones. Besides Johnson, who followed Hochschild's original approach, there were A. Guichardet ([Gui]), whose point of view was homological rather than cohomological, and J. A. Taylor ([Tay]) and - most persistently - A. Ya. Helemskil and his Moscow school, whose approaches used projective or injective resolutions; Helemskil's development of homological algebra for Banach and more general topological algebras is expounded in the monograph [He} 2]. In homological algebra, the notions of projective, injective, and flat modules play a pivotal role. Each of these concepts tranlates into the functional analytic context. Helemskil calls a Banach algebra 21 biprojective (respectively biflat) if it is a projective (respetively flat) Banach 21-bimodule over itself. We do not attempt to give the fairly technical definitions of a projective or a flat Banach 21-bimodule. Fortunately, there are equivalent, but more elementary characterizations of biprojectivity and biflatness, respectively. We use ®-y to denote the completed projective tensor product of Banach spaces. If 21 is a Banach algebra, then 21 ®-y 21 has a natural Banach 21-bimodule structure via a·(x®y):=ax®y and (x®y)·a=:x®ya (a, x, y E 21). This turns the multiplication operator
A: 21 ®-y 21
-+
21,
a ® b f-+ ab
into a homomorphism of Banach 21-bimodules. DEFINITION 2.9. Let 21 be a Banach algebra. Then: (a) 21 is called biprojective if and only if A has bounded right inverse which is an 21-bimodule homomorphism. (b) 21 is called biftat if and only if A * has bounded left inverse which is an 21-bimodule homomorphism.
ABSTRACT HARMONIC ANALYSIS
267
Clearly, biflatness is a property weaker than biprojectivity. The following theorem holds ([Hell, Theorem 51]): THEOREM 2.10 (A. Ya. Helemskil). Let G be a locally compact group. Then Ll (G) is biprojective if and only if G is compact. Again, a classical property of G is equivalent to a cohomological property of Ll(G). The question for which locally compact groups G the Banach algebra Ll(G) is biflat seems natural at the first glance. However, any Banach algebra is amenable if and only if it is biflat and has a bounded approximate identity ([Hel 2, Theorem Vii.2.20]). Since Ll(G) has a bounded approximate identity for any G, this means that Ll (G) is biflat precisely when G is amenable. Let G be a locally compact group. A unitary representation of G on a Hilbert space jj is a group homomorphism 7r from G into the unitary operators on jj which is continuous with respect to the given topology on G and the strong operator topology on B(jj). A function G-+C,
with
~, TJ
Xf-+(7r(x)~,1J)
E jj is called a coefficient function of 7r.
EXAMPLE 2.11. The left regular representation A of G on L2(G) is given by A(X)~ := LX-l~
(x E G, ~ E L2(G)).
DEFINITION 2.12 ([Eym]). Let G be a locally compact group. (a) The Fourier algebra A(G) of G is defined as A(G) := {f: G
-+
C : f is a coefficient function of A}.
(b) The Fourier-Stieltjes algebra B( G) of G is defined as B( G) := {f: G
-+
C : f is a coefficient function of a unitary representation of G}.
It is immediate that A(G) c B(G), that B(G) consists of bounded continuous functions, and that A(G) C Co(G). However, it is not obvious that A(G) and B(G) are linear spaces, let alone algebras. Nevertheless, the following are true ([Eym]): • Let C*(G) be the enveloping C*-algebra of the Banach *-algebra Ll(G). Then B(G) can be canonically identified with C*(G)*. This turns B(G) into a commutative Banach algebra. • Let VN(G) := A(G)" denote the group von Neumann algebra of G. Then A(G) can be canonically identified with the unique predual of VN(G). This turns A( G) into a commutative Banach algebra whose character space is G. • A(G) is a closed ideal in B(G). If G is an abelian group with dual group r, then the Fourier and FourierStieltjes transform, respectively, yield isometric isomorphisms A( G) ~ Ll (r) and B(G) ~ M(r). Consequently, A(G) is amenable for any abelian locally compact group G. It doesn't require much extra effort to see that A(G) is also amenable if G has an abelian subgroup of finite index ([L-L-W, Theorem 4.1] and [For 2, Theorem 2.2]). On the other hand, every amenable Banach algebra has a bounded approximate identity, and hence Leptin's theorem ([Lep]) implies that the amenability of A(G) forces G to be amenable. Nevertheless, the tempting conjecture that A( G) is amenable if and only if G is amenable is false:
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THEOREM 2.13 ([Joh 4]). The Fourier algebra of SO(3) is not amenable. This leaves the following intriguing open question: QUESTION 2.14. Which are the locally compact groups G for which A(G) is amenable? The only groups G for which A( G) is known to be amenable are those with an abelian subgroup of finite index. It is a plausible conjecture that these are indeed the only ones. The corresponding question for weak amenability is open as well. B. E. Forrest has shown that A( G) is weakly amenable whenever the principal component of G is abelian ([For 2, Theorem 2.4]). One can, of course, ask the same question(s) for the Fourier-Stieltjes algebra: QUESTION 2.15. Which are the locally compact groups G for which B(G) is amenable? Here, the natural conjecture is that those groups are precisely those with a compact, abelian subgroup of finite index. Since A( G) is a complemented ideal in B( G), the hereditary properties of amenability for Banach algebras ([Run, Theorem 2.3.7]) yield that A( G) has to be amenable whenever B( G) is. It is easy to see that, if the conjectured answer to Question 2.14 is true, then so is the one to Question 2.15. Partial answers to both Question 2.14 and Question 2.15 can be found in [L-L-W] and [For 2]. 3. and operator spaces Given any linear space E and n E N, we denote the n x n-matrices with entries from E by Mn(E); if E = C, we simply write Mn. Clearly, formal matrix multiplication turns Mn(E) into an Mn-bimodule. Identifying Mn with the bounded linear operators on n-dimensional Hilbert space, we equip Mn with a norm, which we denote by I· I· DEFINITION 3.1. An operator space is a linear space E with a complete norm II· lin on Mn(E) for each n E N such that (R 1) II
~ I~
Iln+m
= max{llxll n, IIYllm}
(n, mEN, x E Mn(E), Y E Mm(E))
and
(R 2) EXAMPLE 3.2. Let fJ be a Hilbert space. The unique C*-norms on Mn(13(SJ)) 13(fJn) turn 13(SJ) and any of its subspaces into operator spaces.
~
Given two linear spaces E and F, a linear map T: E --+ F, and n E N, we define the the n-th amplification T(n) : Mn(E) --+ Mn(F) by applying T to each matrix entry. DEFINITION 3.3. Let E and F be operator spaces, and let T E 13(E, F). Then: (a) T is completely bounded if IITllcb := sup nEN
IIT(n) IIB(Mn(E),Mn(F))
~ 181",7).
1. Introduction
The purpose of this article is to summarize and explore some of the various constructions of the relative tensor product (RTP) of von Neumann algebra modules. Alternately known as composition or fusion, RTPs are a key tool in subfactor theory and the study of Morita equivalence. The idea is this: given a von Neumann algebra M, we want a map which associates a vector space to certain pairs of a right M-module and a left M-module. If we write module actions with subscripts, we have (XM,M!i)) f-> X 0M!i). This should be functorial, covariant in both variables, and appropriately normalized. Other than this, we only need to specify which modules and spaces we are considering. In spirit, RTPs are algebraic; a ring-theoretic definition can be found in most algebra textbooks. But in the context of operator algebras, the requirement that the output be a certain type of space - typically a Hilbert space - causes an analytic obstruction. As a consequence, there are domain issues in any vector-based construction. Fortunately, von Neumann algebras have a sufficiently simple representation theory to allow a recasting of RTPs in algebraic terms. The analytic study of RTPs can be related nicely to noncommutative £P spaces. Indeed, examination of the usual (£2) case reveals that the technical difficulties 2000 Mathematics Subject Classification. Primary: 46LIO; Secondary: 46M05. Key words and phrases. relative tensor product, von Neumann algebra, bimodule. © 275
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DAVID SHERMAN
come from a "change of density". (We say that the density of an LP-type space is lip.) Once this is understood, it is easy to handle LP modules [JS] as well. Modular algebras ([Y], [S]) provide an elegant framework, so we briefly explain their meaning. The final section of the paper investigates the question, "When is the map (~, 1]) f--+ ~ ®
= 'P(Y* x), and set fJ", to be the closure in the inherited topology, modulo the null space. The left action of M on fJ", = M'P1/2 is bounded and densely defined by left composition. When 'P is faithful (meaning x > 0 => 'P(x) > 0), the vector 'P 1/ 2 = I'P1/2 is cyclic (M'P1/2 = fJ",) and separating (x =f. 0 => X'P1/2 =f. 0). Now all representations with a cyclic and separating vector are isomorphic - a sort of "left regular representation"; we will denote this by ML 2 (M). It is a fundamental fact that the commutant of this action is antiisomorphic to M, and when we make this identification we call ML2(M)M the standard form of M. If 'P is not faithful, the GNS construction produces a vector 'Pl/2 which is cyclic but not separating, and a representation which is isomorphic to ML2(M)s('P) ([T2], Ch. VIII, IX). Now let us examine an arbitrary (separable, so countably generated) module MfJ. Following standard arguments (e.g. [TI] I.9), fJ decomposes into a direct sum of cyclic representations M(M~n), each of which is isomorphic to the GNS representation for the associated vector state w~n (=< ·~n, ~n ». With qn = s(w~J, we have MfJ ~ EBMM~n ~ EBMfJw~n ~ EB M L 2(M)qn. (Here and elsewhere, "~" means a unitary equivalence of (bi)modules.) Since this is a left module, it is natural to write vectors as rows with the nth entry in L 2(M)qn:
We will call such a decomposition a row representation of MfJ. Here e nn are diagonal matrix units in Moo, so (Eqn ®e nn ) is a diagonal projection in Moc(M). The left action of M is, of course, matrix multiplication (by 1 x 1 matrices) on the left. The module (L2(M)L2(M) ... ) will be denoted R2(M) (for "row"). Since the standard form behaves naturally with respect to restriction - L2(q,Nq) ~ qL 2(,N)q as bimodules - it follows that L2(Moo(M)) is built as infinite matrices over L2(M) (see (3.3)). Thus R2(M) can be realized as ellL 2(Moo (M)). PROPOSITION 2.1. Any countably generated left representation of M on a Hilbert space is isomorphic to R 2(M)q for some diagonal projection q E Moc(M). Any projection ,in Moc(M), diagonal or not, defines a module in this way, and two such modules are isomorphic exactly when the projections are equivalent. In fact
(2.2)
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So isomorphism classes correspond to equivalence classes of projections in lVIoo(M), which is the monoid V(Moo(M)) in K -theoretic language [W-O}. The direct sum of isomorphism ciasses of modules corresponds to the sum of orthogonal representatives in V(Moo(M)), giving a monoidal equivalence.
We denote the category of separable left M-modules by Left L 2(M). For us, the most important consequence of (2.2) is that (2.3)
C.(MR2(M)q) = R(qMoo(M)q),
where "c." stands for the commutant of the M-action. (In particular, the case = ell is just the standard form.) The algebra qMoo(M)q is called an amplification of M, being a generalization of a matrix algebra with entries in M. Of course everything above can be done for right modules - the relevant abbreviations are C 2(M), for "column," and Right L 2(M). Example. Suppose M = M3(C). In this case the standard form may be taken as M3L2(M3hf3; L2(M3) ~ (M3, < .,. », where < x, y >= Tr(y*x). Note that this norm, called the Hilbert-Schmidt norm, is just the e2 norm of the matrix entries, and that the left and right multiplicative actions are commutants. (If we had chosen a nontracial positive linear functional, we would have obtained an isomorphic bimodule with a "twisted" right action ... this is inchoate Tomita-Takesaki theory.) The module R2(M3) is M 3xoo , again with the Hilbert-Schmidt norm, and the commutant is Moo(M3) ~ Moo. According to Proposition 2.1, isomorphism classes of left M 3-modules should be parameterized by equivalence classes of projections in Moo. These are indexed by their rank n E (1:+Uoo); the corresponding isomorphism class of modules has representative M 3xn . In summary, we have learned that any left representation of M3 on a Hilbert space is isomorphic to some number of copies of C 3. The same argument shows that V(Moo(Mk)) ~ (1:+ U 00) for any k. Properties of the monoid V ( Moo (M)) determine the so-called type of the algebra. For a factor (a von Neumann algebra whose center is just the scalars), there are only three possibilities: (1:+ U 00), (lR.+ U 00), and {O,+oo}. These are called types I, II, III, respectively; a fuller discussion is given in Section 7. q
3. Algebraic approaches to RTPs When R is a ring, the algebraic R-relative tensor product is the functor, covariant in both variables, which maps a right R-module A and left R-module B to the vector space (A ®alg B)/N, where N is the subspace generated algebraically by tensors of the form ar ® b - a ® rb. In functional analysis, where spaces are usually normed and infinite-dimensional, one obvious amendment is to replace vector spaces with their closures. But in the context of Hilbert modules over a von Neumann algebra M, this is still not enough. Surprisingly, a result of Falcone ([FJ, Theorem 3.8) shows that if the RTP L2(M) ®M L2(M) is the closure of a continuous (meaning III(~ ® 7])11 < ClI~IIII7]11) nondegenerate image ofthe algebraic M-relative tensor product, M must be atomic, Le. M ~ EBnB(f.>n). We will discuss the analytic obstruction further in Section 5. For now, we take Falcone's theorem as a directive: do not look for a map which is defined for every pair of vectors. If we give up completely on a vector-level construction, we can at least make the functorial
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DEFINITION 3.1 (Sa). Given a von Neumann algebra M, a relative tensor product is a junctor, covariant in both variables,
RightL 2(M) x LeJtL2(M)
(3.1)
->
Hilber·t:
(n,.ft)
f-+
n ®M.ft,
which satisfies (3.2)
as bimodules. Although at first glance this definition seems broad, in fact we see in the next proposition that there is exactly one RTP functor (up to equivalence) for each algebra. The reader is reminded that functoriality implies a mapping of intertwiner spaces as well, so it is enough to specify the map on representatives of each isomorphism class. In particular we have the bimodule structure '£:(j) .••ll(n
®M .ft)C(M.ll)·
PROPOSITION 3.2. Let n ~ P C 2(M) E Right L2 M od(M) and.ft ~ R 2(M)q E Left L2Mod(M) Jor some projections p,q E Moo(M). Then
n ®M.ft ~ P L2(Moo(M))q with natural action oj the commutants. PROOF. By implementing an isomorphism, we may assume that the projections are diagonal: p = LPi ®eii, q = Lqj ®ejj. Using (3.2) and functoriality, we have the bimodule isomorphisms
n ®M.ft ~ (EBPi L2(M)) ®M (EBL2(M)qj) ~
E9Pi L2(M) ®M L2(M)qj ~ E9Pi L2(M)qj ~ p L2(Moo(M))q. i,j
i,j D
Visually, (3.3) where of course the £2 sums of the norms of the entries in these matrices are finite. After making the categorical definition above, Sauvageot immediately noted that it gives us no way to perform computations. We will turn to his analytic construction in Section 5; here we discuss an approach to bimodules and RTPs due to Connes. In his terminology a bimodule is called a correspondence. (The best references known to the author are [C2l and [Pl, but there was an earlier unpublished manuscript which is truly the source of Connes fusion.) Consider a correspondence MnN. Choosing a row representation R2(M)q for n, we know that the full commutant of L(M) is isomorphic to R(qMoo(M)q). This gives us a unital injective *-homomorphism p : N '---+ qMoo(M)q, and from the map p one can reconstruct the original bimodule (up to isomorphism) as M(R 2(M)q)p(N)'
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DAVID SHERMAN
What if we had chosen a different row representation R 2 (M)q' and obtained P' : N ---+ q'Moo(M)q'? By Proposition 2.1, the module isomorphism MR 2(M)q ~ MR 2(M)q'
(3.4)
is necessarily given by the right action of a partial isometry v between q and q' in Moo(M). Then P and P' differ by an inner perturbation: p(x) = v*p'(x)v. We conclude that the class of M - N correspondences, modulo isomorphism, is equivalent to the class of unital injective *-homomorphisms from N into an amplification of M, modulo inner perturbation. (These last are called sectors in subfactor theory.) The distinctions between bimodules, morphisms, and their appropriate equivalence classes are frequently blurred in the literature; our convention here is to use the term "correspondence" to mean a representative *-homomorphism for a bimodule. Notice that a unital inclusion N eM gives the bimodule ML 2(M)N. The RTP of correspondences is extremely simple. PROPOSITION 3.3. Consider bimodules MYJN and N.ftP coming from correspondences PI : N '----> qMoo(M)q and P2 : P '----> q'Moo(N)q'. The bimodule M(YJ®N.ft)P is the correspondence PI 0 P2, where we amplify PI appropriately.
We pause to mention that it is also fruitful to realize correspondences in terms of completely positive maps. We shall have nothing to say about this approach; the reader is referred to [P] for basics or [A2] for a recent investigation. 4. Applications to Morita equivalence and index An invertible *-functor from Left L2 M od(N) to Left L2 M od(M) is called a Morita equivalence [R]. Here a *-functor is a functor which commutes with the adjoint operation at the level of morphisms. One way to create *-functors is the following: to the bimodule MYJN, we associate (4.1) FSj: Left L2 Mod(N)
---+
Left L2 Mod(M);
N.ft 1---+ (MYJN) ®N (N.ft).
The next theorem is fundamental. THEOREM 4.1. When L(M) and R(N) are commutants on YJ, the RTP functor FSj is a Morita equivalence. Moreover, every Morita equivalence is equivalent to an RTP functor.
This type of result - the second statement is an operator algebraic analogue of the Eilenberg-Watts theorem - goes back to several sources, primarily the fundamental paper of Rieffel [R]. His investigation was more general and algebraic, and his bimodules were not Hilbert spaces but rigged self-dual Hilbert C*-modules, following Paschke [Pal. From a correspondence point of view, rigged self-dual Hilbert C*-modules and Hilbert space bimodules give the same theory; the equivalence is discussed nicely in [A1]. (And the former is nothing but an L oo version of the latter, as explained in [JS].) Our Hilbert space approach here is parallel to that of Sauvageot [Sa], though modeled more on [R], and is streamlined by our standing assumption of separable preduals. We will need DEFINITION 4.2. The contragredient of the bimodule MYJN is the bimodule NfJM' where fJ is conjugate linearly isomorphic to fj (the image of ~ is written (J, and the actions are defined by n~m = m*~n*.
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LEMMA 4.3. Suppose L(M) and R(N) are commutants on fl. Then
NfJM ®M MflN ~ NL2(N)N. PROOF. If Mfl ~ MR 2(M)q, then N ~ qMoo(M)q by (2.3), and fJM qC 2 (M)M. By Proposition 3.2 and the comment preceding Proposition 2.1,
NfJM ®M MflN ~ N(qL2(Moo(M))q)N ~ NL2(qMoo(M)q)N ~ NL2(N)N. D Lemma 4.3 was first proven by Sauvageot (in another way). In our situation it means
FfJ
0
FSj(Nfi) ~ L2(N) ®N Nfi ~ Nfi.
(Here we have used the associativity of the RTP, which is most easily seen from the explicit construction in Section 5.) We conclude that Fi) 0 FSj is equivalent to the identity functor on Left L2Mod(N), and by a symmetric argument FSj 0 Fi) is equivalent to the identity functor on Left L2Mod(M). Thus FSj is a Morita equivalence, and the first implication of Theorem 4.1 is proved. Now let F be a Morita equivalence from Left L2 M od(N) to Left L2 M od(M). Then F must take NR2(N) to a module isomorphic to MR2(M), because each is in the unique isomorphism class which absorbs all other modules. (This is meant in the sense that NR2(N) ffiNfl ~ NR2(N); R2(N) is the "infinite element" in the monoid V (Moo (N)).) Being an invertible *-functor, F implements a *-isomorphism of commutants - call it a, not F, to ease the notation: (4.2) Apparently we have (4.3) Before continuing the argument, we need an observation: isomorphic algebras have isomorphic standard forms. Specifically, we may write L2(Moo(N)) as the GNS construction for tp E Moo(N)t and obtain the isomorphism (a-1)t : L2(Moo(N)) ..::. L2(Moo(M)), (a-1)t : xtp1/2 ........ a(x)(tp 0 a- 1)1/2. Note that (a- 1 )t(x~y) = a(x)[(a- 1 )t(~)la(y). Now consider the RTP functor for the bimodule
MflN =
u- 1 (M)a- 1(et'{)C 2(N)N.
By Proposition 3.2 and the comment preceding Proposition 2.1, its action is
R2(N)q ........
17-1 (M)a- 1 (et'{)L2(Moo (N))q (17;:)'
Met'{ L 2(Moo (M))a(q)
~ MR 2(M)a(q) ~ F(R 2(N)q).
We conclude that F is equivalent to FSj, which finishes the proof of Theorem 4.1. Notice that (4.2) and (4.3) can also be used to define a functor; this gives us COROLLARY 4.4. For two von Neumann algebras M and N, the following are equivalent: (1) M and N are Morita equivalent;
DAVID SHERMAN
282
(2) Moo(N) ~ Moo(M); (3) there is a bimodule Mf)N where the actions are commutants of each other; (4) there is a projection q E Moo(M) with central support 1 such that
qMoo(M)q
~
N.
(The central support of x E M is the least projection z in the center of M satisfying x = zx.) Example continued. M3 and M5 are Morita equivalent. This can be deduced easily from condition (2) or (4) of the corollary above. It also follows from the (Hilbert) equivalence bimodule M3HS(M3X5)Ms, where "HS" indicates the HilbertSchmidt norm; this bimodule gives us an RTP functor which is a Morita equivalence. Regardless of the construction, the equivalence will send (functorially) n copies of C 5 to n copies of C 3 . Apparently Morita equivalence is a coarse relation on type I algebras - it only classifies the center of the algebra (up to isomorphism). At the other extreme, Morita equivalence for type III algebras is the same as algebraic isomorphism; Morita equivalence for type II algebras is somewhere in the middle ([RJ, Sec. 8). For a bimodule Mf)N where the algebras are not necessarily commutants, the functor (4.1) still makes sense. To get a more tractable object, we may consider the domain and range to be isomorphism classes of modules:
(4.4)
71'1) :
V(Moo(N)) ---. V(Moo(M));
F1)(R 2(N)q) = Mf)N ®N R2(N)q ~ R2(M)7I'1) ([q]). We call this the bimodule morphism associated to f), a sort of "skeleton" for the correspondence. It follows from Proposition 3.3 that if the bimodule is p : N '---+ qMoo(M)q, then 71'1) is nothing but poo, the amplification of p to Moo(N), restricted to equivalence classes of projections. This has an important application to inclusions of algebras. We have seen in M is equivalent to a bimodule ML2(M)N. Section 3 that a unital inclusion N When the correspondence p is surjective, the module generates a Morita equivalence via its RTP functor, and the induced bimodule morphism is an isomorphism of monoids. When N i= M, it is natural to expect that the bimodule morphism gives us a way to measure the relative size, or index, of N in M. (For readers unfamiliar with this concept, the index of an inclusion N c M is denoted [M : N] and is analogous to the index of a subgroup. The kernel of this idea goes back to Murray and von Neumann, but the startling results of Jones [J] in the early 1980's touched off a new wave of investigation. We recommend the exposition [K] as a nice starting point.) For algebras of type I or II, the index can be calculated in terms of bimodule morphisms. (There are also broader definitions of index which require a conditional expectation (=norm-decreasing projection) from M ontoN.) This amounts largely to rephrasing and extension of the paper [Jol] , and we do not give details here. Very briefly, let 71' : V(Moo(M)) ---. V(Moo(M)) be the bimodule morphism for
t.
(4.5)
RELATIVE TENSOR PRODUCTS
283
When M is a factor, 11" is a monoidal morphism on the extended nonnegative integers (type I) or extended nonnegative reals (type II). It must be multiplication by a scalar, and this scalar is the index. If M is not a factor, the index is the spectral radius of 11", provided that V(Moo(M)) is endowed with some extra structure (at present it is not even a vector space). Example. Consider the correspondence
The image of M6L2(M6) under the RTP functor for (4.5) is
~ M6 L2 (M6)M3 ®M3 M3 L2 (M6) (now counting the dimensions of the Hilbert spaces) ~
M6HS(M12X3)M3 ®M3 M3 HS (M3xd ~ M6HS(M12X12) ~ M6HS(M6X24).
We have gone from 6 copies of (:6 to 24 copies; that is, 61-+24. Apparently the index is 4, which is also the ratio of the dimensions of the algebras. 5. Analytic approaches to RTPs
As mentioned in Section 3, we cannot expect the expression ~ ®M T/ to make sense for every pair of vectors ~,T/. In essence, the problem is that the product of two L2 vectors is Ll, and an Ll space does not lie inside its corresponding L2 space unless the underlying measure is atomic. Densities add, even in the noncommutative setting, and so the product in (3.3) "should" be an Ll matrix. To make this work at the vector level, we need to decrease the density by 1/2 without affecting the "outside" action of the commutants ... and the solution by Connes and Sauvageot ([Sa], [C2]) is almost obvious: choose a faithful cp E Mt and put cp-l/2 in the middle of the product. That is, (5.1) This requires some explanation. Fix faithful cp E Mt and row and column representations of 5) and .It as in (2.1). We define
D(S;, 1") ~ {
(::~:;:)
E S; ,
~>~Xn ex;"" in M}
V(5), cp) is dense in 5), and its elements are called cp-left bounded vectors [C1]. Now by (5.1) we mean the following: for ~ E V(5), cp), we simply erase the symbol cpl/2 from the right of each entry, then carry out the multiplication. The natural domain is V(5), cp) x .It. Visually, .
284
(5.2)
DAVID SHERMAN
( ( ::::;:) ,(" "' '" ))
~ (::::;:) (,-"') (" "' '" ) ~ (X;"»,
For cp =I- 1jJ E Mt, we cannot expect € ®cp TJ = € ®,p TJ even if both are defined, although the reader familiar with modular theory will see that (5.3)
€ ®cp TJ = (€cp-l/2)TJ = (€cp-l/21jJl/21jJ-l/2)TJ = €(Dcp : D1jJ)i/2 ®,p TJ.
(An interpretation of the symbols cpl/2, cp-l/2 as unbounded operators will be discussed in the next section.) Now we define S) ®cp .!'t to be the closed linear span of the vectors € ®cp TJ inside L2(Moo(M)). Up to isomorphism, this is independent of cp. (We know this because of functoriality; the "change of weight" isomorphism is densely defined by (5.3).) The given definition for V(S), cp) c S) makes it seem dependent on the choice of column representation. That this is not so can be seen by noting (as in (3.4)) that the intertwining isomorphism is given by L( v) for some partial isometry v E Moo(M). But let us also mention a method of defining the same RTP construction without representing S) and.!'t. First notice that V(S), cp) can also be defined as the set of vectors € for which 7rf(€) : L 2(M)M -+ S)M. cpl/2x 1-+ €x, is bounded. (A more suggestive (and rigorous) notation would be L(€cp-l/2).) Now we consider an inner product on the algebraic tensor product V(S), cp) ®.!'t, defined on simple tensors by (5.4) The important point here is that 7rt(€2)*7rf(6) E .c(L2(M)M) = M. The closure of V(S), cp) ®.!'t in this inner product, modulo the null space, is once again S) ®cp .!'t. (If we do choose a row representation as in (5.2), we have
The paper [F] contains more exposition of this approach, including some alternate constructions. 6. Realization of the relative tensor product as composition of unbounded operators
In this section we briefly indicate how (5.1) can be rigorously justified and extended. Readers are referred to the sources for all details. In his pioneering theory of noncommutative LP spaces, Haagerup [H] estab-. lished a linear isomorphism between Mt and a class of positive unbounded operators affiliated with the core of M. (The core, well-defined up to isomorphism, is the crossed product of M with one of its modular automorphism groups.) We will denote the operator corresponding to the positive functional cp by cp also. These operators are r-measurable (see the next section), where r is the canonical trace
RELATIVE TENSOR PRODUCTS
285
on the core, and so they generate a certain graded *-algebra: positive elements of LP(M) are defined to be operators of the form '{)1/ p . The basic development of this theory can be found in [Tej; our choice of notation is influenced by [Yj, where it is called a modular algebra. The composition of two L2 operators is an Ll operator, and it turns out that (5.1) can be rigorously justified [Sj as an operator equation. (This is not automatic, as operators like ,{)-1/2 are not 7-measurable and require more delicate arguments.) In fact, there is nothing sacred about half-densities. With the recent development of noncommutative LP modules [JS], one can allow relative tensor products to be bifunctors on Right U(M) x Left Lq(M), with range in a certain L'" space. The mapping is densely-defined by
~ ®cp 1/ ~ (~,{)~-*-~)1/. In the case of an RTP of L oo modules (or more generally, Hilbert C*-modules), the middle term is trivial and there is no change of density. This explains why there are no domain issues in defining a vector-valued RTP of Hilbert C*-modules [Rj. Let us mention that the recent theory of operator bimodules, in which vectors can be realized as bounded operators, allows a variety of relative tensor products over C*-algebras [APj. This can be naturally viewed as a generalization of the theory of Banach space tensor products, which corresponds to a C*-algebra of scalars. 7. Preclosedness of the map
(~, 1/)
1-+
~
®cp 1/
Our purpose in this final section is to study when the relative tensor map is preclosed. This is a weaker condition than that of Falcone, who studied (effectively) when the map was bounded. We begin with a base case: a fa.ctor, two standard modules, and a simple product. With the usual notation Bcp for V(L 2 (M), '{)), the relevant map is
Bcp x L2(M) :3 (~, 1/) 1-+ ~ ®cp 1/ E L 2(M). This is bilinear: we take "preclosed" to mean that if ~n -+ ~ E Bcp, 1/n -+ 1/, ~n ®cp 1/n -+ (, then necessarily ( = ~ ®cp 1/. We will also consider several variations: changing the domain to an algebraic tensor product, allowing non-factors, and allowing arbitrary modules. Readers unfamiliar with von Neumann algebras will find this section more technical, and any background we can offer here is sure to be insufficient. Still, we introduce the necessary concepts in hopes that the non-expert will at least find the statements of the theorems accessible. A weight is an "unbounded positive linear functional": a linear map from M+ to [0, +ooj. We will always assume that weights are normal, so Xc< / x strongly => '{)(xc'k X. This series, not necessarily convergent, is called the Dirichlet
k=l series of f(x). If all Dirichlet coefficients of a f E AP(lR.) are zero, then, as it is easy to see, f == O. Consequently, the correspondence between almost periodic functions and their Dirichlet series is injective. E~ery almost periodic function f on lR. can be extended as a continuous function f on the Bohr compactification J3lR. of R The Fourier coefficients c[ of the extended in this way function 1 on J3lR. equal the Dirichlet coefficients A' of f. Moreover, the maximal ideal space MAP(IR) of the algebra of almost periodic functions on lR. is homeomorphic to the Bohr compactification J3lR. of R For every A C lR., by APA(lR.) we denote the space of all almost periodic Ajunctions, namely, almost periodic functions on lR. with spectrum contained in the set A, i.e.
APA(lR.) = {f E AP(lR.) : sp (I) C A}. Note that every f E APA(lR.) can be approximated uniformly on lR. by exponential n
A-polynomials, i.e. by exponential polynomials of type
L akei8kX, Sk k=l
E A.
ANALYTIC FUNCTIONS ON COMPACT GROUPS
301
2. Shift invariant algebras on groups. Let G be a compact abelian group, and let S be an additive subsemigroup of its dual r = 8, containing the origin. Linear combinations over C of functions of type Xa , a E S are called S -polynomials on G. Denote by As the set of all continuous functions on G whose Fourier coefficients c!
=
J
f(g)xa(g) da are zero for any a outside
r\s.
Here a is the normalized Haar
G
measure on G. The functions in As are called S-functions on G. Any S-function on G can be approximated uniformly on G by S-polynomials, and vice versa. The set As is a uniform algebra on the group G. A uniform algebra A on G is G-shift invariant if, given an f E A and 9 E G, the translated function fg(h) = f(gh) belongs to A. Every algebra of S-functions is invariant under shifts by elements of G. Vice versa, every G-shift invariant uniform algebra on G is an algebra of S-functions for some uniquely defined subsemigroup S C 8 (Arens, Singer [1]). Algebras As of S-functions are natural generalization of polydisc algebras A(']['n), n E N. With G = ']['n, = 8 = zn, and S = Z+.' the algebra As in fact coincides with the algebra Azn = A(']['n) on the torus ']['n, and Z+-functions
r
+
are traces on ,][,n of usual analytic functions in n variables in the polydisc continuous up to the boundary ']['n.
W,
The maximal ideal space Ms of As is the set H(S) = Hom (S, ~), and the Shilov boundary BAs is the group G (Arens-Singer [1]). H(S) is a semigroup under the pointwise operation (cp1/1)(a) = cp(a)1/1(a), a E S. The Gelfand transforms of elements f E As are continuous functions on Ms, and the space As = {i: f E As} is a uniform algebra on Ms.
i
As shown by Arens and Singer (e.g. Gamelin [14]), As is a maximal algebra if and only if the partial order generated by the semigroup S in 8 is Archimedean. Note that in this case 8 c JR and there is a natural embedding of the real line JR into G so that the restrictions of S-functions on this embedding are almost periodic functions that admit analytic extension on the upper half-plane II over JR. Moreover, an algebra of type As is antisymmetric if and only if the semigroup S does not contain nontrivial subgroups, i.e. if S n (-S) = {O} (Arens, Singer [1]). A compact group G is said to be solenoidal, if there is an isomorphism of the group JR of real numbers into G with a dense range. Equivalently, a compact group is solenoidal if and only if there is an isomorphism from 8 into R Note that the Stone-Chech compactification (3T = of T is a solenoidal group for every additive subgroup r of JR. If G is a solenoidal group, then its dual group r = 8 is isomorphic to a subgroup of R If r is not dense in JR, then it is isomorphic to Z. In this case G is isomorphic to the unit circle '][', S c Z+, and therefore the elements of the algebra As can be approximated uniformly on '][' by polynomials. Hence they can be extended on the unit disc lJ)) as analytic functions, and therefore Ms = ~, while As ~ A(lJ))). If r is dense in JR, then the maximal ideal space Ms has a more complicated nature.
r;,
In the case when S c JR+ and S u (-S) = r, the S-functions in As, are called analytic, or generalized analytic functions in the sense of Arens-Singer on G. As
302
T. TONEV AND S. GRIGORYAN
mentioned before, if S = R+ the group G coincides with the Bohr compactification ,BR of R In this case the maximal ideal space of the algebra AIR+ is the set ll}a = ([0,1] x G)/( {O} x G), which is called the G-disc, or big disc over G. The algebra Ar+ = Ar+ (lI}a) is called also the G-disc algebra, or the big disc algebra. The points in the G-disc ll}a are denoted by r· g, where r E [0,1] and 9 E G =,BR We identify the points of type 0 . g, 9 E G, and the resulting point we denote by w. Hence, w = O· 9 for every 9 E G. The points of type 1· g, 9 E G, we denote by g. Since R is dense in G, the set (0,1] x R is dense in the G-disc ll}a. Equivalently, the upper half-plane n ~ (0,1] x R can be embedded as a dense subset of the G-disc ll}a. Below we summarize some of the basic properties of the G-disc algebra Ar+ (lI}a), where r+ = r n [0,00) (cf. Gamelin [14]). (i) Mr+ = ll}a. (ii) 8A r +(lI}a) = G. (iii) A local maximum principle holds on Ar+(lI}a), namely, for every analytic r+-function f(r . g) on ll}a, for every compact set U c ll}a, and for each ro . go E U we have If(ro . go)l:::; max If(r. g)l· r·gEbU
(iv) Every f E LP(G, da), 1 :::; p :::; 00 can be approximated in the LP(G, da)norm by sp (f)-polynomials. In particular, every f E As can be approximated uniformly on G by S-polynomials. (v) Ar+(lI}a) is an analytic algebra, i.e. every analytic r+-function which vanishes on a non-void open subset of ll}a vanishes identically on ll}a. (vi) Any real-valued analytic r+-function is constant. (vii) Ar+ (lI}a) is a Dirichlet algebra; (viii) A r + (lI}a) is a maximal algebra. (ix) The upper half-plane n can be embedded as a dense subset of the G-disc ll}a.
Examples 1. (a) Let G be a solenoidal group, and S is an additive subsemigroup ofR, containing the origin. Note that the restriction of a character Xa E Gon R is the function eiax , x E R. As an algebra generated by the characters Xa, a E S on G, the algebra As of analytic S-functions is isometrically isomorphic to the algebra APs(R) of almost periodic S-functions on R, generated by the functions eiax , x E R, a E S. (b) It is easy to see that As, S C R is isometrically isomorphic to the algebra on '][' \ {I} generated by the singular functions ea :~~, a E S via a Mobius transformation. In the case when S C R+, As is isometrically isomorphic to the subalgebra .!.ll He; of H OO generated by the functions ea .- 1 , a E S on II} \ {I}. (c) The portion over Jij \ {O} of the Riemann surface is densely embeddable into the G-disc ll}a.
Slog
of the function log z
Example 1 b) implies the following PROPOSITION 1. Let G be a solenoidal group, such that its dual group r = is a dense subgroup of R, and let S be an additive subsemigroup of r+ = r
G n
ANALYTIC FUNCTIONS ON COMPACT GROUPS
303
[0,00), containing the origin. Then the algebra As of analytic S-functions on G is isometrically isomorphic to the algebra of almost periodic S-functions on R DEFINITION 1. Let S be a semigroup of G. The weak enhancement [S]8 of S is the set of elem ts a E G for which there is a rna E N such that na E S for every n ~ rna. The stron enhancement [S]8 of S is the set of elements a E G for which there is a rna EN su that rnaa E S. S is weakly enhanced, or strongly enhanced if [S]w = S, or [S]8 = respectively. Note that S c [S]w C [S]8 C weakly and strongly enhanced.
G.
If S
c G and S U (-S)
=
G,
then S is both
PROPOSITION 2 [23]. For an a E G\ S by Sa denote the semigroup Sa = S +Na. Then MSa = Ms if and only if a E [S]w. As an immediate consequence we obtain that M[sJw = Ms for every semigroup S c G. Also, if S, reG are two subsemigroups of G such that S + (-S) = r + (-r) = G, and if [S]w = [r]w then Ms = Mr· PROPOSITION 3 [23]. Let S e r e G be two subsemigroups of G such that S + (-S) = G and Mr = Ms. IE ASa is analytic for some a E r \ S, then Ms = MSa (and therefore a E [S]w according to the previous proposition). In particular, [S]w = [r]w if Ms = Mr·
II. Shift invariant algebras on groups 1. Rad6's and Riemann's theorems for analytic functions on groups. Let U be an open set in the maximal ideal space MA of a uniform algebra A. A continuous function on U is said to be A-holomorphic on U if for every x E U there is a neighborhood V of x so that can be approximated uniformly on V by Gelfand transforms of functions in A. A uniform algebra A is said to be analytic on its maximal ideal space MA if whenever a function f E A vanishes on an open subset of MA \ 8A then f vanishes identically on MA. If a G-shift invariant algebra As is analytic, then S does not contain subgroups other than {O}, i.e. Sn( -S) = {O}. Throughout this section we will consider all algebras to be analytic, and that S + (-S) = G.
f
DEFINITION 2. A uniform algebra A satisfies Rad6 's property, if every function continuous on MA and A-holomorphic on MA \ Z(f) belongs to A.
The classical theorem of Rad6 asserts that the disc algebra A(j())) possesses Rad6's property. However, it fails for the algebra Ao(j())) of functions f E A(j())) with vanishing at 0 derivatives. Observe that this algebra is of type As with S = {O, 2, 3, 4 ... }, whose weak enhancenment is Z+ =/:. S. THEOREM 1 (Grigoryan, Ponkrateva, Tonev [23]). The algebra As possesses Rad6's property if and only if the semigroup S is weakly enhanced.
T. TONEV AND S. GRIGORYAN
304
DEFINITION 3. A uniform algebra A C C(MA) is integrally closed in C(MA) if every continuous function on MA satisfying a polynomial equation of type xn + alX n - 1 + ... + an = 0, aj E A belongs to A. Integrally closed in C(MA) uniform algebras were studied extensively by Glicksberg [15]. Examples of integrally closed in C(MA) algebras are the disc algebra, the polydisc algebra, the algebra of analytic S-functions on a G-disc over a group G with ordered dual. THEOREM 2 [23]. The algebra As is integrally closed in C(MA) if and only if the semigroup S is weakly enhanced. DEFINITION 4. A uniform algebra A possesses Riemann's property if, given a function 9 E A with Z(g) n 8A = 0, then every bounded A-holomorphic function on MA \ Z(g) belongs to A. The classical theorem of Riemann asserts that the disc algebra A(lJ))) possesses Riemann's property. Note that single points in the complex plane are zeros of certain analytic functions. DEFINITION 5. The bounded enhancement [S]b of S is the set of elements a E which there are b, c E S with a = b - c, such that Xb/X c is bounded on Ms \ Z(X C ), where Z(X) = {m E Ms : m(x) = o} is the zero-set of X. A semigroup S is said to be boundedly enhanced if [S]b = S.
G for
THEOREM 3 (Grigoryan, Ponkrateva, Tonev [23]). The algebra As possesses Riemann's property if and only if the semigroup S is boundedly enhanced. A uniform algebra A possesses the weak Riemann property if, given a function n 8A = 0, then every bounded A-holomorphic function on MA \ Z(g) can be extended continuously on MA. One can show in a similar way that a G-shift invariant algebra As possesses the weak Riemann property if and only if the weak and the strong enhancements of S coincide [23].
9 E A with Z(g)
2. Extension of linear multiplicative functionals of shift invariant algebras on groups. Let S be an additive semigroup which contains 0, and possesses the cancellation property, i.e. a = c whenever a + b = c + b for some bE S. In this case S is a subsemigroup of a group. Denote by r = S - S the group generated by S. Consider a subsemigroup P ::::> S of r such that P + (-P) = r, and Pn (-P) = {O}. P generates a partial (pseudo-) order on r, by b»- a if and only if b - a E P. Note that every non-negative semicharacter e E Hom (P, [0, 1]) is monotone decreasing on P with respect to the order generated by P. Indeed, if b »- a for some a, bE r+, then b = a + p for some pEP. Therefore, e(b) = e(a)e(p) ::; e(a) since e(c) ::; 1 on P. Consequently, if a non-negative semicharacter e is extendable on + as an element in Hom (P, [0, 1]), then ~!ly is monotone decreasing onSCP. / .
r
ANALYTIC FUNCTIONS ON COMPACT GROUPS
305
PROPOSITION 4 (Grigoryan, Tonev [25]). A positive semicharacter e E H(S) is extendable on r+ as a positive semicharacter if and only if e is monotone decreasing on S with respect to the order generated by P.
Proof. Let the positive semicharacter eon S be monotone decreasing. If bE r+, then b = a - c for some a,c E S, a >- c, and e(b) = e(a)je(c) is a well defined and natural homomorphic extension of (} on r+. Clearly, e(a) ~ e(c) if and only if e(b) ~ 1, i.e. if and only if e is a positive semicharacter on r+. THEOREM 4 [25]. Non-vanishing semicharacters rp on S can be extended as (non-vanishing) semicharacters on r+ if and only if eveIY positive semicharacter e E H+(S) is monotone decreasing on S with respect to the order generated by P.
Proof. Let rp E H (S), rp =I- O. The function ')'(a) = {rp(a)jlrp(a)1 ')'( -a) = Ih(a) can be extended naturally on the group ;Y(b) =
~~:j
for a E S for a E (-S)
r as a character of r by
whenever b = a - c E r, a, c E S.
Thus, rp = Irpl~ = (!')' extends on r+ (as an element of H(r+)) if and only if e = Irpl does. By the above proposition this happens if and only if e is monotone decreasing on S with respect to the order on S generated by P. Let S c JR, and P = JR+. Define eg = Lio} E H+(S) to be the characteristic function of {O} in S, namely eg (0) = 1, eg (a) = 0 for every a E S \ {O}. Note that e~+ is the only vanishing semicharacter on r+. Consequently, if S is an additive subsemigroup of JR+ containing 0 and P = JR+, then a vanishing semicharacter e E H(S) is extendable on r+ if and only if e = eg. Therefore not every vanishing semicharacter e E H (S) possesses a semicharacter extension on a larger semigroup. PROPOSITION 5 [25]. Let S c P = JR+. A non-negative semicharacter e E H(S) is uniquely extendable on r+ as a non-negative semicharacter on r+ if and only if e is monotone decreasing on S with respect to the order generated by P.
Proof. Assume that a semicharacter e is monotone decreasing and e( a) = 0 for some a E S. Then e(na) = 0 for every n E N, and the monotonicity argument shows that e(a) = 0 for all a E S \ {O}. In this case e = eg extends naturally on r + to the semicharacter e = e~+ . Recently by S. Grigoryan, and independently - Sherstnev [31], have generalized Proposition 5 for arbitrary semigroups S with cancellation property. Namely, a nonnegative semicharacter r on S can be extended (non-uniquely) as a (non-negative) semicharacter on a supsemigroup E :::::> S if and only if r is monotone decreasing with respect to the order on S generated by E.
306
T. TONEV AND S. GRIGORYAN
Example 2 (cf. Tonev [32]). Let v > 0 be a positive number. Consider the semigroup rv = {O} U [v,oo) c R Clearly, r = rv - rv = lR., and r+ = lR.+. Since x(a + b) = x(a)x(b) ::; x(a) for every a, b E r v , every semicharacter X on rv is monotone decreasing. Therefore, it is extendable on lR.+, namely as the characteristic function l?~+ of the origin {O}. Example 3. Let a be an irrational number. Consider the 2-dimensional semigroup 80. = {n+ma : n, m E Il+} C R Here the group generated by 80. is ro. = 80.-80. = {n+ma: n,m Ell}, while (ro.)+ = ro.nlR.+ = {n+ma ~ 0: n,m Ell}. Clearly, 80. -=I- (ro.)+' For instance the positive number a - [a] E (ro.)+ \ 80.' For a fixed a E (0,1) the function I'(n + ma) = an, n + ma E 80. is a homomorphism from 80. to (0,1] C iTh". However, I' is not monotone decreasing on 8. Indeed, I'(ma) = 0, while I'(n) = an> 0 for every n > ma. The natural (and only possible) homomorphic extension 1 of I' on (ro.)+ is given by 1(n + ma) = an, n,m E Il,n+ma ~ O. However, 1 ¢ H((ro.)+), since, for instance, 1(a- [a]) = a-[o.] > 1. PROPOSITION 6 (Grigoryan, Tonev [25]). The maximal ideal space Ms of the algebra As of analytic 8-functions on G = r = 8 - 8 with spectrum in 8 C lR.+ is homeomorphic to the maximal ideal space Mr+ = iTh"c of the algebra Ar+ of analytic r+-functions on G if and only if all positive semicharacters on 8 are monotone decreasing.
r,
As an immediate consequence we get the following PROPOSITION 7 [25]. The maximalideal space MAPs(JR) ofthe algebra APs(lR.) of almost periodic functions with spectrum in a semigroup 8 C lR.+ is homeomorphic to the G-disc iTh"c, where G = if and only if all positive semicharacters on 8 are monotone decreasing.
r,
Since the upper half plane II = {z E C : 1m Z ~ O} can be embedded densely in the maximal ideal space Ms of the algebra As (and, together, of APs(lR.)) if and only if MAs = iTh"c, then the upper half plane II is densely embeddable in the maximal ideal space MAPs(JR) of the algebra APs(lR.) of almost periodic functions with spectrum in 8 if and only if all positive semicharacters on 8 are monotone her [6], II is densely embedable in MAPs decreasing. Note that, as shown by B if and only if every additive posit' e function 0 on 8 is of type O(a) = yoa for some Yo E [0,00), or O(a) = 00, for a O. . .!.±!.
For an a E 8 let 'Pa E Hoo be the singular function 'Pa(z) = eW 1-. on the unit disc j[)). Recall that HS' is the Banach algebra on j[)) generated by the functions 'Pa(z), a E 8 equipped by the sup-norm on j[)). As mentioned in Example 1 b), HS' is a subalgebra of Hoo, which is isometrically isomorphic to the algebra As of analytic 8-functions on G = (8 - 8)~. PROPOSITION 8 (Grigoryan, Tonev [25]). The unit disc j[)) is dense in the maximal ideal space of the algebra HS' if and only if all positive semicharacters on 8 are monotone decreasing.
Let P be a semigroup of r that generates a partial order on r, and suppose that 8 C E are additive subsemigroups of P that contain the origin, and such that
ANALYTIC FUNCTIONS ON COMPACT GROUPS
307
[S] .. :J E, i.e. Na nSf. 0 for every a E E. Then every non-negative semicharacter E H(S) can be extended naturally on E as a monotone decreasing semicharacter, namely by ~(a) = [(!(na)]l/n. (!
PROPOSITION 9 [25]. If SeE are subsemigroups of P such that E C [S]s, then every semicharacter c.p E H(S) on S is uniquely extendable on E as a semicharacter in H(E), and therefore, Ms = ME.
In particular, if S is a subsemigroup of IR such that [S]s :J r+, then the upper half plane II is densely embedable in the maximal ideal space M APs (lR) of the algebra APs(lR) of almost periodic functions on IR with spectrum in S. PROPOSITION 10 [25]. If S is a subsemigroup of IR such that [S]s :J r+, then the algebra H'S does not have corona, i.e. the unit disc ID> is dense in its maximal ideal space MH:;'. PROPOSITION 11 [25]. Let S be a subsemigroup of R Then ME = Mr+ = ll}c for every semigroup E with SeE c 1R+ if and only if [S]s = r+, i.e. for every a E r+ = r n [0,00) there is an n E N such that na E S.
Note that under the hypotheses of this proposition, the semicharacters on all semigroups E with SeE c 1R+ are uniquely extendable on r+ as semicharacters on r+.
3. Automorphisms of shift invariant algebras on groups. Assume that = {O}, i.e. that S contains no non-trivial subgroups. Under this condition the algebra As is antisymmetric. An element £ E Ms = H(S) is an idempotent homomorphism of S if £2 = £. Let Is be the set of all idempotents in H(S) that are not identically equal to 0 on S. It is easy to see that Is is a subsemigroup of H(S). Clearly, an idempotent homomorphism can take values 0 or 1 only. Denote by Z. the zero set {a E S : £(a) = O}, and by E. - the support set {a E S: £(a) = I} of £ E Is. It is easy to see that if £ is an idempotent homomorphism of S, then E. is a semigroup of S, Z. is a semigroup ideal in S, Z. U E. = S, and Z. n E. = 0.
Sn (-S)
12 [20]. Let As be a G-shift invariant algebra on G, where £ E Is possesses a representing measure supported on a subgroup of G. PROPOSITION
SeC. Every idempotent homomorphism
Note that every idempotent homomorphism of S can be extended un~ely to an idempotent homomorphism on the strong saturation [S]8 of S, i.e. I{, '= IIS]. for every subsemigroup SeC. An automorphism on a shift-invariant algebra As is an isometric isomorphism c.p : As --+ As that maps As onto itself. The conjugate mapping c.p* of c.p defined by ( c.p* (m) ) (f) = m ( c.p(f) ), is a homeomorphism of the maximal ideal space M s onto itself. For instance, the conjugate mapping c.p* of an automorphism c.p of the disc algebra A(ID» = Az+ is a Mobius transformation of the unit disc, i.e. c.p*(z)
=G
z - Zo
1- ZoZ
,
IGI = 1, Izol
onto the maximal ideal space M AAPo (lR) ~ ~G x~. THEOREM
12. The maximal ideal space of any subalgebra of AAPo(lR) of type AAPs(lR) EB B, where S c lR+ and B C Co(lR)n HoI (II), is the set M.4APs (IR)EBB THEOREM
=~G
X MB.
In particular, the upper half-plane II is not dense in the maximal ideal space of any subalgebra AAPs(lR) EB B of AAPo(lR) which contains properly AAPs(lR); The unit disc II} is not dense in the maximal ideal space of any subalgebra of the algebra [ea~ ,a E S] EB B c Hoc n A(~ \ {1}), where S is an additive semigroup in lR, and B =f. {a} is a subalgebra of the space {J E C(1l') : 1(1) = a}. A function 1 E BC(lR) is called weakly almost periodic, if the set of alllR-shifts, It(x) = I(x + t), t E lR is relatively weakly compact in BC(JR) (e.g. Eberline [13], Burckel [7]). If W AP(lR) denotes the set of weakly almost periodic functions onlR, then AP(lR) C APo(lR) c W AP(lR). In fact, W AP(lR) = AP(lR) EB C([-oo, 00])11R. Similarly to Theorem 11, one can show the following THEOREM 13. The maximal ideal space MAWAP(IR) of AW AP(lR) of analytically extendable on II weakly almost periodic functions on lR is homeomorphic to tile Cartesian product ~~IR x {([a, 1] x [0,1])/([0,1] x {a})}.
The space AW AP(lR) orp is isometrically isomorphic to the subalgebra of HOC n =.±! A(II} \ {I}) generated by the functions ea z-1, a E lR+ and the set of continuous functions on 1l' \ {I} that possess both one sided limits at 1. -
III. Inductive limits and shift invariant algebras on solenoidal groups 1. Inductive limits of disc algebras on G-discs. Let A C lR+ be a basis in lR over Q, and lR = lim r h Il)' where --->h.n)EJ
'
Let P h .n ) = r(Y, n)+ = r h .n ) U [0,00). If Ap(-Y.n) is the algebra of analytic P h .n )functions on G, one can show that AIR+ = [ lim A p ("Y,n ) (II}G)]. A similar expression --+ h,n)EJ
holds for the algebra As, S c lR+. Uniform algebras that can be expressed as inductive limits of disc algebras A(II}) are of special interest. Consider the inverse sequence {~kH' T;H }k=I' ~k = ~ and T;H(z) = Zdk on ~k' The limit lim {~k+l,T;+I} of the inverse sequence ......... k ..... oc
{~k+l,T;H}, is the GA-disc ~G,\ = ([0,1] x GA)/({O} x GA) over the group GA = TA . There arises a conjugate inductive sequence {A(~k)' i~H}f of algebras A(~) ~
ANALYTIC FUNCTIONS ON COMPACT GROUPS
A('Jl') with connecting homomorphisms iZ+l: A(Jijk)
---+
311
A(Jijk+d defined by
(iZ+1(f))(Z) = (f(z))d k , i.e. iZ+ 1 = (7:+ 1)*. The elements of the component algebras A(Jijd can be interpreted as continuous functions on G J1. The uniform closure A(Jijc,\) = [ {A(Jijk), iZ+l}] in C(JijCA) of
!!!!!
k ....... oo
the inductive limit of the system {A(Jijk), i~+1 }k=1 and the corresponding restricted {A('Jl' k), iZ+l }] are isometrically isomorphic to the GJ1-disc algebra algebra
[!!!!!
k ....... oo
A rA +, i.e., to the algebra of analytic r,1+-functions on the GJ1-disc (e.g. [21]). Consider an inductive sequence of disc algebras
where the connecting homomorphisms iZ+ 1 : A('Jl'k) ---+ A('Jl'k+l) are embeddings with Mi~+l(A(ll'k)) = Jij and 8(iZ+ 1 (A('Jl'k))) = 'Jl'. There are finite Blaschke products
Bk :
JD) ---+ JD),
Bk(Z)
= eiIJk
IT (
1=1
z-
~~~) )
1 - zl
, Izfk)1 < 1, such that iZ+l
=
Bie for
z
every kEN, i.e. i~+1(f) = ! 0 B k. Let B = {Bdk=l be the sequence of finite Blaschke products corresponding to iZ+ 1 , i.e. (Bk)*(Z) = iZ+1(z). Let A = {dk}~1 be the sequence of orders of Blaschke products {Bdk=l and let rJ1 c IQl be the group generated by l/mk, k = 0,1,2, ... , where mk = I1~=1 dj, mo = 1. Consider the inverse sequence Jijl ~ Jij2 ~ Jij3 ~ Jij4 ~ ...
The inverse limit VB
+--
VB·
= lim {Jijk+1, Bd is a Hausdorff compact space. The limit +k-oo
of the composition system {A(Jijk), ,B~+1 HO of disc algebras A(Jijk) and connecting homomorphisms ,B~+1 = Bie : A(Jijk) ---+ A(Jijk+1): (,BZ+l(f))(zk+d = !(Bk(Zk+d) is an algebra of functions on VB whose closure [lim {A(Jijk),,B~+1}] ---t
= A(VB )
k ....... oo
in C(V B ) we call a Blaschke inductive limit algebra. It is isometrically isomorphic to the algebra [lim {A('Jl' k), ,B~+1 }]. ---t
k ....... oo
PROPOSITION 13 (Grigoryan, Tonev [21]). Let B = {Bdk=l be a sequence of finite Blaschke products and let A(V B ) = [lim {A(JD)k), Bn] be the corresponding ---t k ....... oo
inductive limit of disc algebras. Then (i) A(VB) is a uniform algebra on the compact set VB = lim {Jijk+l, Bd. +k ....... oo
(ii) The maximal ideal space of A(VB) is VB. (iii) A(VB ) is a Dirichlet algebra. (iv) A(VB ) is a maximal algebra.
T. TONEV AND S. GRIGORYAN
312
(v) The Shilov boundary of A('DB) is a group isomorphic to GA, and its dual 00
group is isometric to the group rA
~
U (l/mk)Z c
Q, where mk
k=O
=
THEOREM 14 [21]. Let G be a solenoidal group, i.e. G is a compact abelian group with dual group G isomorphic to a subgroup r of JR.. The G-disc algebra Ar+ is a Blaschke inductive limit of disc algebras if and only if r is isomorphic to a subgroup ofQ. THEOREM 15 [21]. Let B = {Bdk"=1 be a sequence of finite Blaschke products on ~, each with at most one singular points zak ) and such that Bk(Zak+ 1») = zak ). Then the algebra A('DB ) is isometrically isomorphic to the algebra A(rA)+ with A = {ddk"=1' where dk = ordBk.
In particular, if every Blaschke product Bk in the above theorem is a Mobius transformation, then the algebra A('DB) is isometrically isomorphic to the disc algebra A z = A(lI'). 2. Inductive limits of algebras on subsets of G-discs. Let lDJ[I",1J = {z E C : r ~ JzJ ~ I}, and blDJ[r,1J = {z E C : JzJ = r or JzJ = I} is the topological boundary of lDJ[r,1J. Denote by A(lDJ[r,1J) the uniform algebra of continuous functions on the set ~[r,1J that are analytic in its interior. Note that A(lDJ[r,1J) = R(lDJ[r,1J), the algebra of continuous rational functions on lDJ[r,1J. By a well known result of Bishop, the Shilov boundary of A(lDJ[I·,1J) is blDJ[r,lJ, and the restriction of A(lDJ[r,1J) on blDJ[r,1J is a maximal algebra with codim (Re (A(lDJ[r,1 J)JbllJ)lr.!J)) = 1. These results can be extended to the case of analytic r+-flllctions on solenoid groups (e.g. S. Grigoryan [19]). Let G be a solenoidal group, and its dual group is denoted as r c JR.. Let lDJ~,1J = [r,l] x G, 0 < r < 1 be the [r,I]-annulus in the G-disc ~G, [I" 1J [I" 1J [r 1J and let A(lDJa' ) = R(lDJa' ) be the G-annulus algebra on lDJa' ,generated by the functions a E r. Let A = {d k } k"=1 be a sequence of natural numbers and T~+1 (z) = zd k , and let r be a fixed number, 0 < r ~ 1. For every kEN consider the sets
xa,
where mk = I17=1 dl, mo = 1, and E1 = iij[r,1 J• There arises an inverse sequence
of compact subsets of iij. Consider the conjugate composition inductive sequence
where the embedings
iZ+ 1 : A(Ek)
-+
A(Ek+d are the conjugates of zd k , namely,
(iZ+1 0 J)(z) = J(zd k ). Let GA denote the compact abelian group whose dual group
ANALYTIC FUNCTIONS ON COMPACT GROUPS
rA
=
GA
313
is the subgroup of Q generated by A. The algebra [lim {A(Ek)' iZ+1}] ---+
k-+oo
is isomorphic to A([J)~,11). THEOREM 16 [21]. Let Fn+1 = B:;;l(Fn), Fl = [J)~,11. If the Blaschke products Bn do not have singular points on the sets Fn for any n E N, then D~,11 ~ [J)~,11, and the algebra A(D~,11) = [lim {A(Fn), B~}] is isometrically isomorphic to the ---+
n-+oo
G-annulus algebra A([J)~,11). Below we summarize some of the basic properties of the algebra A(D~,11) (see [21]). (a) The maximal ideal space of A(D~,11) is homeomorphic to the set [J)~,11. (b) The Shilov boundary of A(D~,11) is the set b[J)~,11 = {r, I} x G. (c) A(D~,11) is a maximal algebra on its Shilov boundary. (d) co dim (Re(A(D~,11)lb]IJ)[r"I)) = 1. G
Let B = {B I , B 2, ... , B n , ... } be a sequence of finite Blaschke products on ii} and let 0 < r < 1. Let D n+ l = B:;;l (Dn), Dl = [J)[O,r1 = {z E [J) : Izl ::; r}. Consider Tn\[O,r1
llJI
~
D2
~
D.3
~
D.........fu. 4
-n[O,r1
.•. VB
of subsets of [J). The inductive limit A(D~,r1) = [lim {A(Dn), B~}] is a uniform ---+
n-+oo
algebra on its maximal ideal space ~ {Dn, Bn-tlDn} = D~,r1
c DB.
k-+oo
PROPOSITION 14 [21]. LetB = {B l ,B2,B3, ... } be a sequenceoffinite Blaschke products on ii} and let 0 < r < 1. Suppose that the set Dn does not contain singular points of B n- l for every n E N. Then (i) There is a compact set Y such that D~,r1 = ~ {D n+ l , BnIDn+l} n-+oo
M A(D~.rl) is homeomorphic to the Cartesian product [J)[O,r1 x Y. (ii) A(D~,r1) is isometrically isomorphic to an algebra of functions f(x,y) E C([J)[O,r1 x Y), such that f(· ,y) E A([J)[O,r1) for every y E Y.
(iii) A(D~,r1) 1]IJ)[o.rl x {y} ~ A([J)[O,r]) for every y E Y. The proof makes use of the fact that every finite Blaschke product of order n generates an n-sheeted covering over any simply connected domain V c [J) free of singular points of B. Proposition 14 implies that the one-point Gleason parts of the algebra A(D~,r]) are the points of the Shilov boundary bD~,r1 ~ 'll'r x Y. PROPOSITION 15 [21]. Let B = {B I , B 2, B3,"'} be a sequence offinite Blaschke products on ii}, and let 0 < r < 1. Suppose that (a) For every n E N the points of the set :F = (Bl 0 B2 0 ' " 0 Bn_d-l(O) are the only singular points for B n- l in Dn (b) All points in (a) have one and the same order dn - l > 1.
314
T. TONEV AND S. GRIGORYAN
Then (i) There is a compact Y such that
V~·r] = ~ {Ir»n+1' BnID,,+!} = M A(V~.rl) k-+oo
is homeomorphic to the Cartesian product Ir»~:] x Y, where A = {dd~1 is the sequence of the orders of B k . (ii) The algebra A(V~,r]) on V~,r] is isometrically isomorphic to an algebra of functions f(x, y) E C(Ir»~:] x Y), such that f( . ,y) E A(Ir»~:]) for every yEY. (iii) A(V~,r])IIIli[(l.rlx{y}
= A(Ir»~:]) for every y
E Y.
The set Y in Propositions 14 and 15 is homeomorphic to the set {{yn}~=I' Yn E (Bl 0 B2 0 ' " 0 Bn_t}-I(O)}. Proposition 15 implies that there are no single-point Gleason parts of the algebra A(V~,r]) within the set M A(V~.rl) \ bV~,r] U {w} x Y, where w is the origin of the G A-disc ll}a/l' As an immediate consequence we obtain that A(V~,r]) is isomorphic to a Gdisc algebra if and only if the set Y consists of one point. In particular, in the above setting the algebra A(V~'1']) is isomorphic to a G-disc algebra if and only if every Blaschke product Bn has a single singular point z6n ) in D~) such that B n (Z6 n») = z6n+1) for all n hig enough. 3. Gleason parts of inductive limits of disc algebras on G-discs. The celebrated theorem by Wermer [36] asserts that in every non-single-point Gleason part of the maximal ideal space of a Dirichlet algebra can be embedded an analytic disc. Therefore it is of particular interest to locate single-point Gleason parts of an algebra, and especially those of them that do not belong to the Shilov boundary. While every point in the Shilov boundary is a separate Gleason part (e.g. Gamelin [14]), the opposite is not always true, i.e. there are single-point Gleason parts outside the Shilov boundary. For instance, if G is a solenoid group with a dense in lR. dual group, then the origin w = ({O} x G)/( {O} x G) E Ir»a of the G-disc Ir»a is a single-point Gleason part for the G-disc algebra Ar+. Of course w (j. 8 Ar+ = G.
Given a sequence of Blaschke products B = {Bn}~=1 on ll}, consider the Blaschke inductive limit algebra A(V B ) = [lim {A(ll}k) , Bn] on the compact ---> k-+oo
set VB = lim {ll}k, Bk-d. Recall that the Shilov boundary of A(VB ) is the group f--
k-+oo
TB = lim {1l'k' Bk-d. Let Br be the set of all Blaschke products on ll} whose zeros f-k-+oo
are inside the disc Ir»[O,r] ones at O.
= {Izl
~ r}, and let B~
c Br
be the set of the vanishing
PROPOSITION 16 [21]. Let B be a finite Blaschke product with B(O) = O. Consider the sequence B = {B, B, ... }. If the Blaschke inductive limit algebra A(VB) = [lim {A(Ir»k) , Bd]' Ir»k = Ir», Bk = B is isometrically isomorphic to a ---> k-+oo
G-disc algebra, then necessarily B(z) = cz", where c E Ir»,
lei =
1, and n E N.
ANALYTIC FUNCTIONS ON COMPACT GROUPS
315
THEOREM 17 (Grigoryan, Tonev [21]). Let B be a finite Blaschke product on IDJ. The Blaschke inductive limit algebra A(D B ) is isometrically isomorphic to a G-di8c algebra if and only if B(z) is conjugate to a power z'" of z, i.e. if and only if there is an mEN and a Mobius transformation 7 : IDJ --t IDJ such that (7- 1 OB 0 7)(Z) = zm.
m
THEOREM 18 [21J. Suppose that Bn E and ordBn > 1 for every Then tllere is only one single-point Gleason part in the set DB \ TB.
17,
E N.
In particular, if B E Sr, B(O) =f. 0, and Bk(z) = zd k BCk, dk > 1 then there is only one single-point Gleason part in the set DB \ TB. The proof of Theorem 18 involves a thorough study of one-point Gleason parts of the algebras involved. 4. Inductive limits of Hoo-spaces on G-discs. Let I = {iZ+ 1}k'=1 be a sequence of homomorphisms iZ+l : HOO(IDJ) --t HOO(IDJ). Consider the inductive sequence HOO(lDJd -iL H OO (1DJ 2) 2L HOO(IDJ3) -.fL ... of algebras HOO(lDJ k ) ~ HOO(IDJ). Every adjoint mapping (iZ+l)* : Mk f-- Mk+l maps the maximal ideal space Mk+l of HOO(IDJk+d into the maximal ideal space Mk of HOO(lDJ k ). The inverse limit
Ml
1;2)" ~1_
M2
1;3)" ~2_
M3
1;4)* ~3_
M4
1;5)" ~4_
••• f - -
DI
is the maximal ideal space of the inductive limit algebra
HOO(DI) = [lim {HOO(lDJk),i~+I}J. ---+ k-+oo Recall that the open unit disc IDJ is a dense subset of every Mk. In general, the mappings (i~+l)* are not obliged to map IDJ k+1 onto itself. The most interesting situations, though, are when they do. Here we suppose that the mappings (iZ+l)* are inner non-constant functions on IDJ. For instance, algebras of type H oo (D I ) are the algebras [lim {H OO (lDJ k), (zdk)*}dkEAJ = HOO(D A) c H OO (IDJ CA ), and also ---+ k-+oo the algebras of type HOO(DB) = [~{HOO(IDJd, Bk}], where B = {Bdk'=1 is k-+oo a sequence of finite Blaschke products Bk : IDJ --t IDJ. Note that HOO(DB) is a commutative Banach algebra of functions on DB. Let A = {dd~1 be the sequence of orders of Blaschke products {Bd~1 from the mentioned above example, and let rA C Q be the group generated by l/mk, k = 0,1,2, ... , where mk = I1~=1 d/, mo = 1. THEOREM 19 [21J. Let B = {B k }k'=1 be a sequence of finite Blaschke products on ~, each with at most one singular points zbk ), and such that Bk(zbk+ 1») = zbk ). Then the algebra HOO(D B ) is isometrically isomolphic to the algebra HOO(DA) for A = {ddk'=1 with dk = ordBk. For instance, if the Blaschke products Bk are of type Bk(Z) = zd k 'Pdz), where 'Pk are Mobius transformations and dk > 1, then the algebra HOO(D B ) is isometrically isomorphic to the algebra HOO(D A), where A = {1/dd~I' If every Blaschke
316
T. TONEV AND S. GRIGORYAN
product Bk in Theorem 19 is a Mobius transformation, then the algebra HOO(DB) is isometrically isomorphic to the algebra H oo . Indeed, the last theorem implies that HOO(D B ) ~ HOO(DA) with A = {I, I, ... }. Therefore rA = Z and GA = T. Let iP = {'P1, 'P2, ... ,'Pk, ... } be a sequence of non-constant inner functions on ][)l. Consider the inverse sequence ][)l1 +---'£l. ][)l2 where][)lk
~][)l.
~
][)l3
~
][)l4
~
...
Denote by Dq, its inverse limit. The inductive limit lim {HZ", 'PkH" ---t
k-+oo
of the adjoint composition inductive sequence
Hf" 1.4 H2'
~
H'3
~
... 10 'Pk,
of algebras HZ" = HOO(][)lk) ~ HOO(][)l), where 'PkU) = is a subalgebra of BC(Dq,), the algebra of bounded continuous functions on the set Dq,. The closure HOO(Dq,) of lim {H OO , 'Pk} in BC(Dq,) is a commutative Banach subalgebra of ---t
k-+oo
HOO(][)le). Its elements are referred to as iP-hyper-analytic lunctions on Dq,. Recall that according to the classical corona theorem for the space HOO on the unit circle (Carleson [8]), given h, ... , /k, functions in Hoo with L~=lllil :::: a > 0 on ][)l, there exist functions gl, ... ,gk in Hoo such that L~=lligj = 1 on ][)l; If IIIi 1100 :::; 1, then 9j can be chosen to satisfy the estimates II 9j II :::; C (k, a) for some constant C(k, a) > O. Next theorem is the corona problem for the algebra HOO(Dq,). THEOREM 20 (Grigoryan, Tonev [21]). If h, 12, ... , In, IIIi II :::; 1, are iP-hyperanalytic functions on Dq, for which Ih(x)1 + ... + I/n(x)1 :::: 8 > 0 for each x E Dq" then there is a constant K(n,8) and iP-hyper-analytic functions gl, ... , gn on Dq, with Ilgj II :::; K(n, 8), such that the equality h (x )gl (x) + ... + In (x )gn (x) = 1 holds for every point x in the set Dq,. In the case when iP = {Z2, z3, ... , zn+1 ... } the corresponding set Dq, coincides with the open big disc ][)le over the compact abelian group G = ij, and the algebra HOO(Dq,) coincides with the set He of hyper analytic functions. In this case Theorem 20 reduces to the corona theorem for the algebra He of hyper-analytic functions on G with estimates (cf. Tonev [32]). 5. Hoo-spaces on solenoidal groups. Let G be a solenoidal abelian group, i.e. r = G c R Let HOO(][)le) be the algebra of bounded functions in the open G-disc ][)le that can be approximated on compact subsets of][)le by functions 1, I E Ar+. For every I E HOO(][)le), the limits
f*(g) = lim I(r)(g), where I(r)(g) = r-+1
f(r. g)
exist for almost all 9 E G, and f* E Hoo (G, a). The space of functions f*, I E HOO(][)le) we denote again by HOO(][)le). The algebra HOO(][)le) we interpret as a subspace of the set of functions in LOO(G,a) that are boundary values of continuous functions on ][)le, equipped with the norm 11/1100 = lim sup I/(r)(g)l. Denote
r-+1 gEe
ANALYTIC FUNCTIONS ON COMPACT GROUPS
317
by 'HOC(lD>a) the weak*-closure of Ar+ in LOC(G,a) (cf. Gamelin [14]). Clearly HOC(lD>a) is a closed subalgebra of 'HOC (lD>a). Let I be a directed set. We consider a family {rdiEI of subgroups of r indexed by I, such that ril C r i2 whenever il -< i2. Let r = limri , and H~(lD>a) denotes --+ iEI the space of functions f E Hoc (lD>a) with sp (I) C ri . The family {Hr: (lD>a) hEI of subalgebras in HOC(lD>a) is ordered by inclusion. Denote by Hr(lD>a) the closure of the set U H~(lD>a) = limH~(lD>a) with respect to the norm II . lIoc. Hr(lD>a) iEI ~ is the set of I-hyper-analytic functions on lD>a. In a similar way we define the algebra 'Hf(lD>a) as the II . Iloc-closure of the inductive limit ~ 'H~(lD>a), where iEI 'H~(lD>a) = {f E 'HOC(lD>a) : sp (I) c rd· THEOREM 21 (Grigoryan, Tonev [22]). Let G be a solenoidal group such that its dual group r = G is the inductive limit of a family {rdiEI of subgroups of r, i.e. r = lim r • Let Hr='• (lD>a) and 'Hr', (lD>a) be the spaces offunctions in Hoc (lD>a) --+ i iEI [resp. in 'HOC(lD>a)] with spectra in ri , i E I. Then the following statements are equivalent. (a) HOO(lD>a) = Hr(lD>a) and 'HOC(lD>a) = 'Hf(lD>a). (b) HOC(lD>a) = U Hr:(lD>a) and 'HOC(lD>a) = U 'H~(lD>a). iEI iEI (c) Every countable subgroup ro in r is contained in some group from the family {rdiEI. Example 4. Let r = Q be the group of rational numbers with the discrete topology. Assume that {rdiEI is an inductive system of subgroups of Q such that Q = lim n. The last theorem implies that if Q itself is not one of the groups in the --+ iEI family {rdiEI, then Hr(lD>a) =I- H""(lD>a). In the case when all subgroups ri , i E I are isomorphic to Z, the algebra Hr(lD>a) coincides with the algebra of hyper-analytic functions (e.g. [34]). As seen above, in this case the space Hr (lD>a) differs from HOC (lD>a). The properties of subalgebras of HOC (lD>a) on general compact groups G are less known. In particular it is not known if they possess a corona, and their maximal ideal spaces and Shilov boundaries lack a good description. Example 5. Let r = R and let A C R+ be a basis in R over the field Q of rational numbers. Then R = ~ r(-y,n), where (-y,n)EJ
Given an (-y, n) E J, consider the set
T. TONEV AND S. GRIGORYAN
318
The closure HJ'(lJJJc) of the set
U
H0',n)(lJJJc) under the
II . lloo-norm, i.e.
the
(-y,n)EJ
inductive limit algebra
lim H(oo-y,n )(lJJJc) is a subalgebra of HOC (lJJJc). There arises _ (-y.n)EJ
the question of whether or not the algebra HJ'(lJJJ c ) coincides with HOO(lJJJ c ). THEOREM 22 (Grigoryan, Tonev [22]). The set HJ(lJJJ c ) =
lim H(oo-y,n ) (lJJJ c ) _ (-y,n)EJ
is a proper closed subalgebra of H OO (lJJJ c ).
c HOO(lJJJ c ) is easy (e.g. [12]). Assume that H0',n) (lJJJc). By the previous theorem, the countable
Proof. The inclusion HJ'(lJJJc)
HOO(lJJJ c ) = HJ(lJJJc) =
~ (-y,n)EJ
subgroup Q c IR is a group in the family {rh,n) h-y,n)EJ' which is impossible since r(-y,n) is isomorphic to -Z} for some kEN.
The algebra HOO(lJJJ c ) is isometrically isomorphic to the algebra HfPr+(IR)(IR) C
HOO (1R) of boundary values of almost periodic r+-functions on IR that are analytic in the upper half plane. Similarly, the algebra HJ'(lJJJ c ) is isomorphic to a subalgebra HJ'(IR) of HZ,r+ (R) (1R). As the last theorem shows, these algebras are different. Algebras of type HI (lJJJ c ) were introduced in connection with the corona problem for algebras of analytic r-functions (Tonev [32]). R. Curto, P. Muhly and J. Xia [12J have introduced similar algebras of this type in connection with their study of Wiener-Hopf operators with almost periodic symbols.
6. Bourgain algebras and inductive limits of algebras. Bourgain algebras of a Banach space were introduced in 1987 by J. Cima and R. Timoney [9J as a natural extension of a construction due to J. Bourgain [5J. The concept of tightness of an algebra was introduced by Cole and Gamelin [lOJ. Let A c B be two commutative Banach algebras, and 11" A : B ----t B I A is the natural projection. The mapping Sf: A ----t (f A+A)IA c BIA; Sf: 9 ~ 1I"A(fg) is called the Hankel type operator. DEFINITION 7. An element fEB is said to be (a) a Bourgain element, (b) a wc-element, (c) a c-element for A, if the Hankel type operator Sf : A ----t BIA is correspondingly (a) completely continuous, (b) weakly compact, (c) compact. The Bourgain algebra of A relative to B is the space A~ of all Bourgain elements for A in B [9J. PROPOSITION 17 [35J. If the range Sf(A) = 1I"A(f A) of the Hankel type operator Sf for an fEB is finite dimensional then f E A~ In particular, (As)f(C) = C(G) if As is a maximal algebra and xS \ X is a finite set for a character X E O\S. Indeed, X E (As)f(C) by the above proposition. Since X ~ S, then X ~ As, and consequently (As)f(C) = C(G) by the maximality of A.
ANALYTIC FUNCTIONS ON COMPACT GROUPS
319
Example 6. Let H be a finite group, G = (H E& Zr and S So' H E& Z+. Then (As)f(G) = C(G).
r- r
r
Note that if = G and XS\ X is finite for every X E then every character X E G has finitely many predecessors in r. As it follows from Proposition 17, (Ar)f(G) = C(G), and therefore the corresponding big disc algebra Ar possesses the Dunford-Pettis property. THEOREM 23 (Yale, Tonev) [35]. Let G =,BJR be the Bohr compactification of lR. The Bourgain algebra (AIR+ )f(G) of the big disc algebra AIR+ is isomorphic to AIR+' Proof. Clearly, JR is a subset of (AIR+ )f(G). Since, as one can easily see, the
seqnence of real valued functions 'Pn(x) = as n
--+
00 for every x
E
11 +2ei ';i 12n
converges pointwisely to
1
2n . JR, then the real valued functIOns 'l/Jn(g) = 11 + X2-a-(g)1
converge pointwisely to 1 as n --+ 00 for every 9 E G. Suppose that X3 E (AIR+ )f(G). Consider the sequence ~n(g) = 'l/Jn(g) -1, where 'l/Jn is as before. {X1~n}n is a weakly null sequence in AR+ since it converges pointwisely to 0 on G. By the Bourgain algebra property there are functions hn E AIR+ such that IIX3X1~n - hnll < l/n for every n, where II . II is the sup norm on G. By integrating over Ker(x~), if necessary, we can assume that hn
then (X',pn)(g)
1 • = qn(X") for some polynomIal qn'
~ (x«g))" ( 1 + ;~(g))
2n the j-th Cesaro mean
af" =
n
= max I(X1~n)(g) gE G = ~Eas IPn (x-a- (g)) ( z)
Fa, j
~
= (1 + z)2n + 2n(1 + z)2n-1 = (2n +
= max I(X 1'I/Jn)(g) gE G
- X1(g) - (x-a- (g)) 3n qn P (z)
~~(g))" ~ Pn(X" (g)).
= max 1(~X1~n)(g) - hn(g)1 gE G
(X3hn)(g)1
ZTO
= (1+Z)2n -2,
SO+S1+"'+S, j +1 J of Pn, where Sk is the k-th
partial sum of Pn, we have 4n(2n + l)a~~(z) 1 + z)(l + z)2n-1. Now II~X1~n - hnll
(1 +
If Pn(z)
zn
X1(g) - X3(g)h n (g)1
(x~ (g)) I = ~tf IPn(Z)_zn_ z3nqn (z)l·
z3n q (z)
P Note that a" " (z) because the Cesaro mean a2n de2n - (z) = a 2nn - pends only on the first 2n terms of the Taylor series. Since the inequality max la~(z)1
zE'lr
f E A(1l') we see that
:::; max If(z)1 holds for every zE'lr
maxlaPn ( zE'lr
2n
z)
-
zn
(z)1
p (z) = maxla n zE'lr 2n
zn
-
z3n q (z) n
(z)1
T. TONEV AND S. GRIGORYAN
320
However, O"~~(z)_zn (z) = O"~~(z) (z) - zn(n + 1)/(2n + 1) and thus O"~~(z)_zn (-1)
1/2 as n --> 00 for odd n. Hence AIR+ by the maximality of AIR+'
-->
-X3 f/. (AIR+)f(O) and consequently (AIR+)f(O) =
THEOREM 24 (Tonev [33]). Let {AO" }O"EL', {BO" }O"EE be two monotone increasing families of closed subspaces of a commutative Banach algebra B such that BO" are algebras, and AO" c BO" for every 0" E E. Let A = [ U AO"] be a (linear) subO"EE
space, and let B = [ U BO"] be a subalgebra of B. Suppose that for every
0"
E E
O"EE
tllere is a bounded linear mapping r 0" : B --> BO", such that (i) rO"IB" = idB" (ii) rO"(fg) = frO"(g) for every f E BO", 9 E B (iii) sup Ilr0" II < 00. O"EE Then A~ c [ U (AO")~"]. O"EE Proof. Let fEB be a Bourgain element for A. Fix a 0" E E, and consider a weakly null sequence { f. Then Ilf-rO"n(f)II::; Ilf-fO"nll+llrO"n(fO"J-rO"..(f)II::; Ilf-fO""II+supllrO"..IlllfO"n -fll· Hence rO"n(f) --> f and, consequently, f E [ U (AO")~"]. O"EE
0"
lim rO", let Hf? = {! E HOO(J]))o) : sp (f) c rO"}. Note that H't:.. is a --+ " O"EE closed sub algebra of HOO(J]))o), and H'f:. c Hr:. if and only if rO" err. Therefore, the family {H'f:. }O"EE of subalgebras of HOO (J]))o) is ordered by the inclusion. Denote by H~ the closure of the set U H'f:. with respect to the norm II . 1100' Theorem O"EE 24 implies that if r = lim rO" and G = f, then the Bourgain elements for H roo are --+ + O"EE approximable in the L 00 - norm on G by Bourgain elements for H'f:., 0" E E. Note that H~, HOO(J]))o), and the weak* closure HOO(G, dO") of Ar+ in LOO(G, dO") are commutative Banach subalgebras of LOO(G,dO"), which are in principle different from each other, except in the case of G = 'll', when they coincide (Grigoryan [19]).
If r
=
The algebra HQ;/n = H OO 0 Xl/n = {! 0 Xl/n: f E HOO} is a closed subalgebra of HOO(J]))o). The closure HQ' of U HQ' with respect to the norm II . 1100 is the + nEN lin algebra of hyper-analytic junctions on G = f3Q (cf. Tonev [34]). By Theorem 24 the Bourgain algebra of is contained in the algebra + C(G).
Hift
Hift
ANALYTIC FUNCTIONS ON COMPACT GROUPS
321
THEOREM 25 (Tonev [33]). If the hypotheses of Theorem 24 are satisfied. then A~c c [ U (AO')~~]; A~ c [ U (AO')~O']; O'EE
(H~J~:(G) .
c [U
O'EE
(H~)~:(G)]; (HrJ~OO(G)
O'EE
In particular, the algebra H~
c [U
(H~)~OO(G)].
O'EE
+ C(,8Q)
contains the spaces (H~)~:({3Q) and
(H~ )~OO({3Q). A uniform algebra A C C(X) is said to be tight [strongly tight] if every f E C(X) is a we-element [resp. c-element] for A, i.e. if (A)~~G) = C(X) [resp. (A)f(G) = C(X)] (cf. Cole, Gamelin [10], also Saccone [30]). Theorem 25 implies that the algebra H~ is neither tight nor strongly tight.
References
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DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF MONTANA, MISSOULA, MONTANA 59812-1032 CHEBOTAREV INSTITUTE OF MATHEMATICS AND MECHANICS, KAZAN, SIA
Rus-
