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A with multiplication defined by (A ® a)(fj, ® b) = \n 0 (Xb + pa + ab)
V A, fj, G C; o, b 6 A .
(16)
Clearly the map a >—> 0 © a embeds A isomorphically into this algebra. Furthermore 1 © 0 is an identity element. W e can partially alleviate the
1- I . i l
Norms and Semi-norms on Algebras
19
awkwardness o f this construction by the following device, which was already used by Segal [1941], 1.1.11 D e fin itio n Let A be an algebra. If A is not unital, denote the algebra described above by .A1. If A is unital, denote the algebra A itself by A 1. We will call A 1 the unitization of A . The advantages of this convention on the meaning o f A 1 will begin to become apparent when we discuss the spectrum. From now on we will always consider A iis h subset of A 1 under the (imbedding a m 0 $ a. Wc will also suppress the direct sum notation ® and will usually denote the multiplicative identity of A 1 by 1 whether or not A is unital. Furthermore, for any A G C, the element A1 of A 1 will be denoted simply by A. Even when calculating in an algebra A , which may not be unital, it is frequently convenient to use notation which must be interpreted in A 1. For instance, the expression a(X + b )(p + c) represents an element in A . We use it since it is easier to understand than Aiia + Aac. + fj.ab + abc. Similarly we will routinely denote the smallest ideal o f an algebra A containing an element a o f A by A l a A l . We remark that if X is an ideal o f A , then it is also an ideal of A 1 since A 11 A 1 C 1 holds. If (A ,cr) is a nonunital semi-normed algebra, there are several ways of extending the semi-norm cr to a semi-norm cr1 on A 1. Since different extensions are useful under different circumstances, we make no permanent choice. One method of extending cr which always works, but is frequently unnatural, is to define cr^A + o) = )A| + cr(o)
V A + o g A 1.
(17)
W G A * .
The weakest topology on X relative to which each element of A * is continuous will be called the weak topology or the A * -topology. The weakest topology on X * relative to which each element of n (X ) is continuous will be called the weak*-topology or the A -topology. Note that the Banach space dual A * is always a Banach space, whether or not A is complete. The natural map k is always an isometric linear injection. It is also easy to see that the weak and weak* topologies are locally convex. The Algebra o f Linear Maps The linear space (under pointwise linear operations) of all linear maps of one linear space A into another y (normed or not) will be denoted by C (X , 3>). If y equals A , we simplify this to £ ( A ). Then £ ( A ) is an algebra when multiplication is composition of maps. Unless the contrary is specif ically stated, all subalgebras of C (X ) will be given these same algebraic operations. We state the next definition formally because o f its importance in what follows. 1.1.16 D e fin itio n For any normed linear spaces A and y the subspace of £ (A , y ) consisting of continuous linear maps will be denoted by B (A , 30 and considered as a normed linear space under the operator norm ||T|| = sup{||Tx|| : x € X i}
V T G B ( A ’).
(19)
Whenever the linear spaces are obvious from context, we will simply call elements o f B ( X , y ) operators. Topological statements about B (X , y ) always refer to the norm topol ogy unless another topology is specifically mentioned. The strong operator topology on B (X , y ) is the topology o f pointwise convergence with the norm topology on y . The weak operator topology on B ( X , y ) is the topology of pointwise convergence when y carries its weak topology. When y equals A , we write B(X) for B ( X , y ) and consider it as a normed algebra under composition as multiplication. Many readers will already know that B( A ) is a Banach algebra if A is a Banach space, and that B(X) is a topological algebra under both the strong operator topology and the weak operator topology. In any case, these statements are easily verified. Observe that B( A , y) is complete if y is complete (whether or not X is complete).
Norms and Semi-norms 011 Algebras
1.1.17
23
The Ideal o f Finite-rank Operators 1.1.17 Definition Let W , X , y and Z be normed linear spaces. For elements y & y and u> G X *, we define an operator y ® u> € B (X , y ) by (y ® w )(z ) = u>(z)y
'i z G X .
( 20 )
We denote the linear span of all such operators by B f (X , ^ ) and call oper ators in this space finite-rank operators. We denote B jr(X , X ) by B p (X ). The tensor product notation (® ) will be explained in Section 1.10. For arbitrary S G B (W , * ) , T G B (y , Z ) , y G y , u> G * * , z G Z and r G y it is easy to check the formulas (z ® r)(j/g> u>)
=
r(y )z ® u >
T(yg>u>)
=
T(y)g>u>
(y ® u ))S
=
yg> S *(w ).
(21)
where z ® r belongs to B f W , Z ) , z ® u> and T (y ) ® w belong to B f ( X , Z ) and y ® 5 ” (ta;) belongs to B f (W , > ’)• Also the dual map o f y® w G B f ( X , y ) is given by (y ® u ) m= u ® /c(y) e BF ( y * , x * ) . Now suppose IV, and Z fill equal X in the above formulas. Then these formulas show that B f ( X ) is an ideal of B (X ). We prove more. 1.1.18 Proposition Let X be a normed linear space. The set o f operators in B (X ) with finite-dimensional range is the set n B f (X ) -
: n G N; i j G X\ u)j G X * }
(22)
j= i of finite-rank operators. It is an ideal of B {X ) which is included in every non-zero ideal. Proof Clearly every operator in B f ( X ) has finite-dimensional range. Con versely, if T G B (X ) has finite-dimensional range, let { x i , x 2, ■. ■,x n} be a basis for its range. Use the Hahn Banach theorem to extend linear func . ■. ,wn} o f X * satisfying tionals on this range space to elements U j(x k) = j) 3= 1
is the desired expansion. Suppose J is a non-zero ideal of B (X ). Then there is a non-zero operator T e l and a vector z G X satisfying T z ^ 0. By the Hahn-Banach theorem
24
1 : Normed Algebras and Examples
1.2.1
there is a continuous linear functional r 6 X ' satisfying t ( T z ) = 1. Hence for any x G X and any u> £ X * we can write x ® u> as x ® u> — (x ® t ) ( T z ® u>) = (x ® t ) T ( z ® w) which proves that x ® w belongs to the ideal I .
□
This proposition explains why we call B f ( X ) the ideal o f finite-rank operators. Note that the equations in (21) show that 8 f ( X , y ) is a type o f generalized ideal in the complex of spaces B ( X , y ) as X and y vary. The ideal B x ( X ) of compact operators will be defined and discussed in §1.7.7.
1.2
D o u ble Centralizers and Extensions
A double centralizer (sometimes called a double multiplier) of an algebra A is a pair of maps of A into itself. The set of double centralizers of A forms a unital algebra T>(A) under natural operations. We will begin by introducing the regular representation which defines a homomorphism of the original algebra onto an ideal in T>(A). The mapping is surjective if and only if A is unital and, in this case, it is actually an isomorphism. In the situations o f most interest, this homomorphism is injective and one often thinks of the original algebra A as a subalgebra of the double centralizer algebra 'D (A ). In many circumstances this embedding o f A into T>(A) is a more natural way o f adding an identity to a nonunital algebra. (A fter Proposition 1.2.6, we will mention three classes o f examples which are given in more detail later in this chapter.) Moreover, T>(A) is an example o f an extension of the algebra A : that is, there is a short exact sequence { 0} -* A -
V (A ) -
V (A )/ A -
{ 0 }.
It turns out that this short exact sequence is universal for the algebra A in the sense that ai] short exact sequences {0 } -
A -
B -
C-
{0 }
can be constructed from this one and a naturally defined homomorphism o f C into T>(A)/A- When we think o f B in such a short exact sequence as being constructed from the “simpler” algebras A and C, we call it an extension. Thus, this is a satisfying way of constructing and classifying extensions o f A . We explore all these ideas in this section starting with the regular representation. The Regular Representations 1.2.1 Definition Let A be an algebra. For each o g A , let L a and R a be the linear maps in C (A ) defined by L 0( 6) = ab
and
Ra{b) = ba
V b 6 A.
( 1)
1.2.1
Double Centralizers and Extensions
25
W e call the map L (i.e., a ►-» L a) the left regular representation o f A and the map R (i.e., a h-> R a) the right regular representation o f A . The kernel A l a = {a € A : ab = 0 V b € -4} of L is called the left annihilator o f A and the kernel A h a = {a G A : ba = 0 V 6 G A } of R is called the right annihilator o f A . The intersection A a o f the left and right annihilators o f A is called the annihilator o f A ■ The extended left and right regular representations are the maps L 1 and R 1 of A into L ( A 1) defined by L\(Xb) v R^(X + b)
=
Ao + ab
=
Xa + ba
. V a e A , X + b e A 1.
( 2)
Note that the left and right regular and extended regular representations are bounded if the original algebra is normed. In that case, we give them their operator norms, using the norm ||A + a|| = |A| + ||a||
V A G C; a e A
(3)
for A 1 if A is nonunital. Then it is easy to see H I = ll^ ll = P i l l
V a G A.
(4)
In particular, L 1 and R 1 axe injective. When A is noncommutative, the right regular and extended right regular representations are anti-representar tions rather than representations in the general terminology introduced later, but the present usage is well established. It is of some interest that the extended left regular representation was used by Charles Saunders Peirce [1881] to prove that every finite-dimensional algebra could be represented by matrices. See Thomas Hawkins [1972] for remarks on the context in which this proof was given. Note that the annihilator ideals satisfy ( A l a )2 = ( A r a )2 = ( A a )2 = { 0 }
(5)
and hence the annihilator ideals themselves are { 0 } if A is semiprime (Defi nition 4.4.1 below) and o fo rtio ri if A is semisimple (Definition 4.3.1 below). O f course, they vanish if A is unital or even approximately unital (Defi nition 5.1.1 below). Following Wang [1961], Ronald Larson [1971] says a Banach algebra A is without order if A l a — {0 } or A r a = {0 }. This ungainly terminology is more common in commutative algebras. Cent'alizers When the annihilator ideal is zero, there is another more natural way of embedding A into a unital algebra. The following construction was first explicitly studied by Gerhard P. Hochschild [1947]. It was introduced into analysis by Barry E. Johnson [1964a], [1964b], [1966], who was apparently
26
1: Normed Algebras and Examples
1.2.2
unaware o f Hochschild’s work and abstracted the idea from the work of James G. Wendel [1952] and Jeffery D. Weston [I960) (c/. our remarks below on Ju-kwei Wang’s [1961] study of multipliers). See also Sigurdur Helgason [1956], John Dauns [1969] and Dauns and Karl H. Hofmann [1968], [1969].
1 . 2 .2 D e fin itio n Let A be an algebra. A ( left centralizer / right centralizer ) o f A is an element ( L / R ) G C (A ) satisfying ( L(a b ) = L (a )b / R (ab) = aR(b) )
V a ,6 e A
(6)
A double centralizer of A is a pair (L , R ) where L is a left centralizer, R is a right centralizer and together they satisfy aL(b ) — R (a)b
V o, b G A .
(7)
The algebra T>(A) o f double centralizers of A is the set o f double centralizers with pointwise linear operations and with multiplication defined by ( L u R l ) ( L 2 , R 2) = ( L l L 2 , R 2R l )
( 8)
for all ( L \ , R i) and ( L 2, R 2) in V (A ). If .4 is a normed algebra, then the set V b (A ) — { ( L , R ) £ T>(A) : L , R G B(^4)} is called the algebra o f bounded double centralizers and is given the norm *
||(L, *)|| = max{||Z||, p H }
V (L , R ) g V B (A ),
(9 )
w here ||I>|| and ||i?|| are the operator norms of L and R , respectively. A ( left / rig h t) centralizer is simply a map in the commutant ( (R a ) ' / ( L a ) ' ) of the { right / l e f t ) regular representation in C (A ). As mentioned above, this definition of double centralizers is due to Hochschild [1947]. Johnson [1964a] considered any pair (L, R ) of maps in C (A ) which satisfies equation (7). W e will show in Theorem 1.2.4 below that such a pair is a double centralizer in our terminology if both the one-sided annihilator ideals are zero. Therefore, our definition agrees with Johnson’s definition in this case, which is the most frequently considered case anyway. (Note that the identities ( A l a )2 = ( A r a )2 = {0 } imply that any semisimple or even any semiprime algebra (as defined in Chapter 4 below) satisfies this condition.) W lien A l a and A r a are not both zero (particularly if their intersection A a is zero), the definition given here is more convenient. It is easy to check that V ( A ) is an algebra under the stated operations and that T>b (A ) is a subalgebra when it is defined. Clearly (/, I ) is an iden tity element for V ( A ), where I is the identity map in C (A ). Moreover the map a *-* ( L Q, Ra) is a homomorphism (called the regular homomorphism)
Double Centralizers and Extensions
:.2 ,3
27
of A into T>(A) which maps into T>b(A ) when A is normed. The kernel of the regular homomorphism is obviously the annihilator ideal A l a O A r a = A a of A . I f A is unital, then for all o G A and all ( L , R ) G T>(A) we have L (a ) = 1L (a ) = R ( l ) a = L r ( i ) ( o ) and R (a ) = R (a )l = a L ( 1) = R m ^ (a ). Note also that L ( 1 ) = 1Z/(1) = /£(1) 1 = R ( 1). Thus the regular homomor phism is surjective if and only if A is unital. If A is a normed algebra, it is simple to check that T>B (A ) is a normed algebra under the norm given in the definition. O f course the identity element o f T>b (A ) satisfies ||1 || = ||(/, / )|| = 1 . For a ll 0 , b G A and ( L , R ) G T>(A), we have L aL (b )
= = = =
R R a(b) L L a(b) R aR{b)
aL(b) R(ba) L(ab) R (b)a
= = = =
R (a )b bR (a) L (a )b bL(a )
= = = =
LR(a)b, RR(a)b, LL(a)b, R[j(a)b.
Therefore the image of the regular homomorphism is an ideal in V ( A ) and we have ( L , R ) ( L a, R a)
=
(^L(a)i R l ( u))
(^fl> R a )(L , R )
=
(^H(a)) RR(a))
V (L ,R )e V (A ); a e A .
(11)
If the annihilator ideal is zero and A is considered as embedded in T>(A) ( via the regular homomorphism), then these relations can be written as (L , R )a = L (a ), a (L , R ) = R (a )
V (L , R ) g V ( A ); a e A .
(12)
We will now state some of these results formally and add a few more. 1.2.3 P roposition Let A be an algebra. (a) T>(A) is a unital algebra. (b) The regular homomorphism o f A into T>(A) has the annihilator ideal A a as kernel and its range is an ideal. It is surjective if and only i f A is unital, in which case it is an isomorphism. (c) I f either A 2 = A or the annihilator ideal A a zero, then the fo l lowing conditions on (L , R ) G V { A ) are equivalent: (c i) (L , R ) belongs to the center o f V ( A ) . (c2) L = R, (C3 ) L and R are both left and right centralizers. (d ) I f the annihilator ideal is zero, then A is commutative if and only if T>(A) is commutative. Proof It only remains to show (c) and (d). Under the hypothesis of (d) the regular homomorphism is injective, so A is commutative if D ( A ) is conur„utative. Th e implication (C 3 ) => (c j) shows the converse. Hence it remains only to prove the equivalence o f the conditions in (c).
28
1: Normed Algebras and Examples
1.2.3
First we will show, without any restriction on A , that either (c i) or (c2) implies (C 3 ) . The second of these implications is immediate from the definitions. If (c i) holds, then any o, b G A satisfy L(ab) — L L a(b) = L aL (b ) = a L(b ) and R (ab) = R R b(a) = R b R (a ) — R(a)b. Hence ( c j implies (C3 ). Next we will show that (C3 ) implies (c i) and (c2) under either o f the hypotheses of (c). Suppose (C 3 ) holds. Then any o, b G A and any left centralizer V satisfy L L '(a b ) — L (L '(a )b ) — L '(a )L (b ) — L '(a L (b )) = L 'L (a b ). A similar calculation for right centralizers shows that (C 3 ) implies (c j) when A satisfies A 2 = A . Also any o, b G A and any double centralizer (L ', R !) satisfy the two relations b L L ‘ (a ) = L (b L '(a )) = L(R ?(b)a) = R , (b )L (a ) = b L 'L (a ) L L '(a )b = L '(a )L (b ) = L '(a L (b )) = L '(L (a )b ) = L 'L {a )b . A similar calculation for R >R shows that (c3) implies ( c i ) when A satisfies A a = {0 }. I f A a = { 0 } and (C3 ) are both true, then the identities L (a )b = L(ab) = aL (b) = R (a )b and bL(a) — R (b )a = R(ba) = b R (a) for all o, b G A imply (C2 ). I f A 2 = A and (C3 ) both hold, then the identity L (ab) = aL (b ) = i2(d)6 = R (ab) for all a, b G A implies (c2). O A linear operator T = L = R G C (A 1) satisfying condition (c) in the above theorem is an example o f a multiplier. For any algebra A , we define T € C (A ) to be a multiplier if it satisfies T (a )b = aT{b)
V a, b G A .
I f the ( left / right ) annihilator ideal of A is zero, then the calculation ( cT(ab) = T (c)a b = cT (a )b / T (a b )c = ab T (c) = a T (b )c ) for all c G A shows that T is a ( left / right ) centralizer. Hence Proposition 1.2.3(c) shows that when the annihilator ideal is zero, the set o f multipliers is simply the center of the double centralizer algebra. Multipliers were defined by Helgason [1956] and first systematically studied on commutative Banach algebras (with annihilator ideal { 0 } ) by Ju-kwei Wang in his thesis [1961]. They are still most often considered in this context. The standard reference for their theory is Ronald Larsen [1971]. However, we are aware o f a number of papers which purport to use multipliers on noncommutative algebras but apply properties which hold only in the commutative case. Hence we shall not use the term “multiplier” again. The second statement in the next theorem is our first example o f au tomatic continuity, in that a purely algebraic condition on a map implies that it is continuous. Both results are due to Johnson [1964a). A related result is given in Proposition 5.2.6.
1.2.4
Double Centralizers and Extensions
29
1.2.4 T h eorem Let A be an algebra satisfying A l a — A r a = {0 }. (a) I f L and R are arbitrary (not necessarily linear) maps o f A into A satisfying the double centralizer condition (7) aL(b) = R (a )b
V o, b G A ,
then (L , R ) is a double centralizer. (b) I f A is a Banach algebra and (L , R ) is a double centralizer, then L and R are bounded linear maps, so T>b {A ) equals T>(A). In this case, 2?(.4) = V b ( A ) is a Banach algebra under its norm and the regular homo morphism is contractive.
Proof (a): For any o, b, c G A and any A, n € C, we get c L (\ a + fib) = R (c )(X a + fib) — X R (c)a + fiR (c)b = c(AL (a ) + i*L (b )), and cL(ab) = R (c)ab — cL(a)b. Since c G A is arbitrary and the right annihilator of A is zero, we conclude that L is a left centralizer. Similarly, R is a right centralizer. (b): Suppose A is a Banach algebra, {a n} „ eN is a sequence in A converg ing to zero and the sequence { L ( o „ ) } ngn also converges to some element b in A . Then any c G A satisfies cb =
lim cL (a n) =
n —*oo
'
'
lim R (c )a n = 0.
n —*oo
Hence, A r a = { 0 } implies 6 = 0. Thus L is closed and, by the closed graph thecrem, it is continuous. Similarly R is continuous. It is easy to check that T>B( A ) is a Banach algebra since the limit in B (A ) of a sequence o f left or right centralizers has the same form. Clearly the regular homomorphism is contractive. □ If A is an algebra satisfying A a = { 0 } , then V (A ) is a unital algebra which contains an ideal isomorphic to A under the regular homomorphism. If A is a Banach algebra satisfying A l a — A r a — { 0 }, the last theo rem shows that V (A ) — 1 > b (A ) is a unital Banach algebra which contains an ideal continuously isomorphic to A under the regular homomorphism. Hence, under suitable restrictions on the annihilator ideals, the construc tion of V ( A ) provides another way to embed an algebra in a unital algebra and a Banach algebra in a unital Banach algebra. (Examples are mentioned after Proposition 1.2.6.) Note that A equals T>(A) if and only if A is uni tal. For technical reasons, it is sometimes desirable to replace D { A ) by the poss:bly smaller unital subalgebra V ( A ) z + A . For a discussion of this, see Dauns [1969] and Dauns and Hofmann [1968], [1969]. Left and Right Idealizers 1.2.5 Definition Let A be a subalgebra of an algebra B. The left idealizer of A in B is the set {6 G B : bA C A ) . The right idealizer is defined similarly. The idealizer Ba of A in B is the intersection o f the left and right idealizer.
30
1: Normed Algebras and Examples
1 .2.6
Clearly the ( left / right ) idealizer is the largest subalgebra of B in which .4 is a ( left / right ) ideal. Similarly the idealizer is the largest subalgebra in which A is an ideal. If b belongs to the left idealizer of A in B , then L b maps A into A and its restriction to A is a left centralizer. The same remarks hold for the right idealizer and right regular representation. Hfnre if 6 belongs to the idealissor of A ih B, then (Lb\A, /£b|»4) is a double centralizer. This is the most natural way in which double centralizers arise. See the proof of the following result. 1.2.6 P r o p o s itio n Let A be a subalgebra o f an algebra B and let B a be the idealizer o f A in B. Then there is a natural homomorphism 0 o f B a into T>(A) defined by: 0{b) = ( L b\A, Rb\A)
V b e B A.
(13)
The homomorphism 0 extends the regular homomorphism o f A into V ( A ) . It is the only such homomorphism i f the annihilator o f A is {0 }. The kernel o f 0 is {6 € B : bA = Ab = { 0 } } . I f A is an ideal in B, then 0 maps B into V ( A ). I f B is a normed algebra, then 6 is a contractive map into V b (A ). P roof The description o f the kernel o f 0 and the last sentence are obvious. Thus we only need to check the uniqueness assertion. Let 0: Ba - * 1)(A) be any homomorphism which extends the regular homomorphism. Let a € A and b € B ji be arbitrary and denote 0(b) by ( L , R ). Then formula (10) gives (Lab,Rab) = 0(ab) = 0(a)0(b) = ( L a, R a) ( L , R ) = ( L R{a), R R(a))Since A satisfies A a — {0 }, this implies R(a) = ab. A similar argument gives L(a) = ba. Hence we conclude that 0(b) = ( L , R ) = (Lb\A, Rb\A) (where we use the left and right regular representations of B ), proving the uniqueness. □ Examples o f Double Centralizer Algebras Let fl be a locally compact topological space. In §1.5.1 we will more for mally introduce the following two important commutative Banach algebras of bounded continuous complex-valued functions on fl. Both are Banach algebras under pointwise multiplication and the supremum norm: Il/lloc = sup{|/(w)| : w e n } . Let C (f i) be the algebra o f all bounded continuous functions and let C'o(fJ) be the algebra o f continuoiis functions vanishing at infinity. Clearly Cq(£1) is a closed ideal in C (Q ). Hence by Proposition 1.2.6 there is a natural homomorphism of C (Q ) into Z?(C'o(n)). In §1.5.1 we show that this is an isometric algebra isomorphism onto 'D (C0(Q )). Similarly, in §1.7.1 we will show that the double centralizer algebra of the ideal B k ( X ) of compact operators on any Banach space X can be
2.7
Double Centralizers and Extensions
31
identified with the algebra B (X ) o f all bounded linear operators on X . In fact, we show that this result extends to a general class of ideals in B (X ). Finally in §1.9.13 we will show that for any locally compact group G the double centralizer algebra of L 1 (G ) can be identified with the measure algebra M (G ). We have already noted that the use of this identification by Wendel [1952] provided an important stimulus to the study of double centralizer algebras in analysis. Automorphisms The proof of the next result is straightforward. When derivations axe introduced in Chapter 6 , a similar result will be proved there. 1.2.7 P r o p o s itio n Let A be an algebra in which the annihilator ideal is zero and let a be an automorphism o f A ■ Then we may define an automor phism a of T>(A) by a (L , R ) = (a o L o a - 1 , a o R o a T 1) which extends a when A is viewed as a subalgebra o f V ( A ) . Then a —►a de fines an isomorphism o f A u t(A ) onto the subgroup o f A ut(T > (A )) consisting o f those automorphisms fo r which A is invariant as a subset o f V ( A ) ■ I f A is normed, we may replace the word “automorphisms”, by “homeomorphic automorphisms”, everywhere. I f a is an isometry, so is a. Topological and Geometric Categories o f Normed Algebras There are two categories of normed algebras (or normed linear spaces) in common use. Since confusion between them sometimes occurs, it is worthwhile to point out the distinction clearly here. We will only describe the case of normed algebras, since the case of normed linear spaces is exactly similar. In the more common and important category, the morphisms are con tinuous algebra homornorphisms. That is, 0 (2 1) — ► 0
also commutes. Define 0'-.B" —►B' by 0 '(b ") — (0 "(b "), ip "{b ")), so that it is a continuous homomorphism which makes the analogue o f diagram (17) commute. The argument preceding this theorem shows that 6' is a homeomorphic isomorphism, so that the two extensions are equivalent. O Recall that after Proposition 1.1.3 we defined CHom(^4, B ) to be the set o f continuous homornorphisms from a normed algebra A to another B. 1.2.11 Theorem Let A and C be Banach algebras with the first satisfy ing A r a — A l a — {0 }. Let 0— * A -^ > 'D (A )-—->'D (A )/A — >0 be the short exact sequence in which p is the regular homomorphism and a is the nat ural map. The construction o f Theorem 1.2.10 applied to this short exact sequence gives a bijection o f CH om (C,T>(A)/A ) onto E x t(C ,A ). Further more, the extension corresponding to t 6 C H o m (C ,V (A )/A ) is a semidi rect product if and only if there exists a map to € CHom(C, 'D (A )) satisfying
1: Normed Algebras and Examples
36
1.2.11
r = a o u> and this extension is a direct product i f and only i f t is the zero map. P roo f The remarks following Theorem 1.2.4 show that T>(A) = V g ( A ) is a Banach algebra. For any r e CHom(C, V (A )/ A ), the construction of Theorem 1.2.10 gives the following commutative diagram. 0
— *
A
-*->
V (A )
-Z-*
V (A )/ A
—♦
0 ( 22 )
]r
0
—
—
c
0
where the lower line is an extension of A by C, which is unique up to equivalence. This establishes a map o f CHom(C, V (A )/ A ) into Ext(C, A ). We must show that the map is both injective and surjective. We will prove injectivity first. Suppose that 0— >A-^-+B— *C— >0 and 0 — ►A B' C — ►0 are two extensions o f A by C constructed from r:C -> V ( A )/ A and r ': C - * V (A )/ A , respectively. Let 6:B - * T>(A) and O': B ' —►V { A ) be the maps associated with this construction, which satisfy Oot p — p — 9 'o t p '. I f these extensions are equivalent, there is a homeomorphic isomorphism 8" : B —* B’ which makes the following diagram commute. A JL* & c — > 0
|7A
|e" JfU
B'
(
|'c C
— ► 0
We have 0 o ip — p = &' o ip' = (9' a ff" ) o ip. Therefore, the uniqueness assertion o f Proposition 1.2.6 implies 6 = 6' o 6" . Since any element of C can be written as ip(b) for some b e B, the calculation r ( t />(£>))
=
a o 6 (b) = a o 9' o &"(b)
=
t ' 04 , ' o0"(b) = r'(iP(b))
VbeB
proves r = r'. Hence the map is injective. Now suppose an extension 0
—*
A
B
C
— .
0
of A by C is given. Proposition 1.2.6 shows that there is a continuous homo morphism ff:B —> T>(A) satisfying p — 8 cup. Define r € CH om (C, T>(A)/A) by r(\p(b)) = 0 0 6 (b) for all ip(b) € C. First we check that r is well de fined. If V'(b') = ip(b), then there is some a e A satisfying b' = b 4- 1} is any finite set of Banach algebras, the Cartesian product x ag_4^4 a can be made into a Banach algebra by coordinatewise algebraic operations and the P -n o rm : ll(aQ)Qe 4 ll = max{||a„|| : a e A }
V (aa ) aeA € x aeAA a .
It is again clear that this construction with the obvious definition o f pro jections gives a product in either category of Banach algebras. In the topological category of Banach algebras this construction satis fies another universal mapping property, so long as {.4 ° : a € A } is finite. Let { A u : a € A ) be a collection of objects in a category. An object A together with maps (called injections) r]a'.A a —> A is called a coproduct of { A a : Ot G A } if, whenever another object A ! and another collection of maps T]'u : A a —* A ! for each a € A is given, there exists a unique map i p : A —> A ' satisfying q'a = ip ° »/Q for each a e A. This definition deter mines a coproduct up to isomorphism. It is again easy to check that the construction described above, together with the obvious injection maps, defines a coproduct in the topological category o f Banach algebras. To get a coproduct in the geometric category, we merely need to change the norm, so long as { A a : a € A } is finite. The norm we need in the -norm: li (aa)a€.4 II l = ^ l!aa|| a£A
^ ( aa)a£A £
•
When we come to infinite families { A a : a G A } of algebras, normed algebras or Banach algebras, the construction of the product and coprod uct diverges more decisively, and becomes dependent on whether algebras, normed algebras or Banach algebras are involved. Throughout this work we always interpret the Cartesian product x aeJ4«Aa of a family o f sets as the collection of all functions a: A —►Uae,4.4 a satis fying a (a ) e A a for each a € A. An element o f the Cartesian product is called a cross section when considered as a function. The direct product fIQg,4.4 a o f a family { A a : a € A } of algebras is sim ply the Cartesian product * a e .4A a made into an algebra by coordinatewise algebraic operations. 1.3.1 D e fin itio n Let {.4 ° : a 6 A ) be a family of normed algebras or Banach algebras. The t°°-direct product H^eAA a of { A a : a € A } is the subset o f the Cartesian product x n f AA a consisting o f all cross sections a such that l l ai L = sup{||a(a)|| : a € A } (1) is finite. The subset is made into a normed algebra or Banach algebra by coordinatewise algebraic operations and the f°°-norm defined by equation (I)-
Sums, Products and Limits
i.3.2
39
It is easy to see that the above definitions subsume those given earlier and define products in all o f the relevant categories when the projections are defined in the obvious way. Direct Sums ( Let { A a : a € A ) be a collection of algebras or normed algebras. The direct sum ® Qg 4 ^4a o f { A a : a € A ) is the subset o f the Cartesian product x o 6 i4^ ° consisting of all cross sections which are zero except at a finite nu’Tiber of elements of A. (Th e finite set of a with o (a ) non-zero varies with a e ©c>e,4-4 “ .) This set is made into an algebra by coordinatewise algebraic operations. I f the summands A a are normed algebras, then the direct sum is a normed algebra with the t l -norm defined by N i t = £ ||a(a)||. aeA The direct sum just described is frequently called the algebraic direct sum. In order to define the direct sum in the categories o f Banach algebras, arid for many other purposes in this work, we need the notion o f an un ordered sum o f numbers. Let A be an arbitrary index set and for each a e A , let aa be a complex number. The unordered sum fla of this arbitrary set of numbers is defined as follows. Let T be the collection o f all finite subsets o f A. When T is ordered by inclusion, it is a directed set. For each F € T define ap — Y I c z f a«- Then {ap-Jp-gjr is a net. If this net has a limit, then this limit is the unordered sum. If the net has no limit, then ilie sum does not exist. If it should happen that each number aa (a € A ) is non-negative, then it is easy to see that s u p {a f : F g F ] is the limit o f the net (and hence the unordered sum) if and only if it is finite. Hence for non-negative series we define s u p {a f : F e !F} to be the unordered sum, whether or not it is finite. 1.3.2
D e fin itio n
Let {.4 “ : a € .4} be a collection o f Banach algebras.
The Banach algebra direct sum A a o f {.4 Q : a e A } is the subset of the Cartesian product x aeAA a consisting o f all the cross sections a such that the unordered sum
INK = £
M a )ll
(2)
is finite. This set is made into a Banach algebra by coordinatewise linear operations and the ^'-norm defined by equation ( 2 ). The definitions o f direct sums o f algebras, normed algebras, and Banach algebras just given subsume the previous definitions and give coproducts in each of the relevant categories with respect to the obvious injections. Suppose {.4 ° : a e A } is a collection of Banach algebras. Then its Banach
40
1: Normed Algebras and Examples
1.3.2
algebra direct sum ® ^ g/4.4Q is just the completion of its normed algebraic direct sum © Qe,4-4 a . Internal Direct Sums The direct sums we have just discussed are sometimes called external direct sums. There are corresponding notions o f internal direct sums. Let A be an algebra, let A be an index set, and let { l a : a € A } be a family of ideals o f A . Then the sum J^aeA this family is the set o f finite sums o f elements from Uae/( I n. The algebra A is calk'd the internal direct sum o f the family { I Q : a e A } if A is the sum o f the family and satisfies
1/5 = { 0 }
r n
V a € A.
aeA\{a] When these conditions hold, we write A = ® QgA%a- It is clear that (external) algebraic direct sum ® a 6 ,4.4 “ is the internal direct sum of ideals = {a € (Bae A A a : a(/3) = 0 for all 0 ^ a }. Conversely, if A is internal direct sum o f { ! “ : a € A }, then A is naturally isomorphic to external direct sum ® ae^ Z a . I f A is a nonned algebra, we say it is the internal direct sum o f family { ! “ : a € A } if each 1 “ is a closed ideal, A = A^a ^ d r n
^
= {0 }
V a € A,
the the the the the
(3 )
/3€A\{a} where the bar denotes closure. Again it is easy to check that the external direct sum © a e^^4Q of a family {^4“ : a € A } of normed algebras is the internal direct sum o f {Z “ : a € A } , where each T a is defined by I a = {a € ® a eA-A0 : a ((i) = 0 for all 0 ^ a }. Suppose A is the internal direct sum o f { I Q : q e A }. The definition of a coproduct guarantees that there is a map (in either category) of the external direct sum onto A . The map is an algebra isomorphism but not necessarily an isomorphism in either category o f normed algebras. In either category of Banach algebras we say that A is the internal direct sum o f ( J “ : a € A ] if each T a is a closed ideal, is dense in A and Xa fl $^/3e 4 \{a} ^ 's zero for each a 6 A. Clearly an external direct sum is an internal direct sum as before. If A is the internal direct sum o f ( I “ : a e A }, then the definition of a coproduct shows that there is a map (in either category) of the external direct sum ©ag^X 0 onto a dense subset o f A (which includes Y I c ^ a J Q), but this map need not be a homeomorphism nor onto A . Subdirect Products Finally, we discuss what are usually called subdirect sums o f algebras and nonned algebras, but what are more correctly called subdirect prod
:.3.3
Sums, Products and Limits
41
ucts. These arise naturally in many contexts as will be shown in Chapter 4 end elsewhere. It should be noted at the outset that representation o f an al gebra as a subdirect product furnishes relatively little information about its structure unless additional information, such as that discussed in Chapter 6 , is available. Subdirect products were first considered by H. Priifer [1925] for commutative rings and by Gottfried Kothe [1930] for noncommutative rngs. For further information and history, consult Neal H. McCoy [1947]. The normed case is an obvious extension of the non-normed case. Sub direct products are simply algebras, normed algebras or Banach algebras together with a suitable embedding as a subalgebra, or normed subalgebra o f a direct product. Direct sums are examples o f subdirect products when provided with the obvious embedding into the direct product. 1.3.3 D e fin itio n Let A be an index set and let : a € .4} be a family of algebras. A subdirect product o f { A a : a € .4} is a subalgebra A of Ha€AA a such that for each a € A, na: A —* A a is surjective. The subdirect product in either category o f normed algebras or Banach algebras is simply a normed or complete normed subalgebra A o f I I Q6 ^^4a defined in the appropriate category with the same restriction that each tt0 :.4 —►A a should be surjective. Let A be a subdirect product in any o f these categories and, for each fi € A, define X/} to be {a € A Q x aeA A a : a(f3) = 0 }. Clearly, Xa is an iteal which is closed in the nonned cases. Furthermore, each na can be identified with the natural map A —►A/Xa — A a. Finally, n ae^ I a = { 0 } holds. Conversely, if {Xa : a € A } is any collection o f ideals (closed in the nonned case) with n a6AI a — { 0 }, then A is isomorphic to a subdirect product o f {A/Xa : a € >1}. This is the way in which subdirect products most commonly arise. A ll o f the foregoing discussion o f direct products, direct sums, internal d rect sums and subdirect products applies to linear spaces, normed linear spaces and Banach spaces. Simply dropping all references to multiplicative structure, (and thus replacing ideals by linear subspaces) makes this clear. Later we will need to deal with direct sums in the category of Hilbert spaces. The discussion o f that case is postponed until needed. Inductive or Direct Limits We begin by defining inductive limits which can be considered as a massive extension o f direct sums. W e will then give one example which is important in the definition o f K-theory. The example is often called the a jebra of matrices of arbitrary size. 1 3.4
D e fin itio n
Let (j4, < ) be a directed set, and for each a € A let
/ a be an algebra. For each a, (3 £ A with a < /?, let ) = w (a) for any o 6 A and u> € A ‘ .) We shall denote the two products by juxtaposition and by a dot, respectively, (i.e., we use f g and f • g for the two products o f / and g in In most cases the two products do not agree and the double dual algebra with either product is too large, too complicated and too badly behaved to be useful without some simplification. However, there are special cases (notably the C*-algebras which will be considered in the second volume o f this work) in which this algebra is easy to comprehend and exploit. In other cases, quotients of the algebra modulo various natural ideals are easier to exploit. In all cases that have been studied in detail, the structure is intimately connected with that o f the original algebra and has interesting, if sometimes complicated, relationships to significant mathematical objects. In this section we explore identities and one-sided identities in the Arens products, connections with double centralizers, the case in which the two products agree and a few simple examples. Tw o representations which also stem from Arens’ very general construction are defined. Although the two Arens products are usually called the first and the
1.4.1
Arens Multiplication
47
setfohd Arens product, they stand on a completely equal footing since their definition simply depends on choosing left or right first. The first product is coa:inuous in the weak* topology in its first (left) variable for any choice of the second (right) variable in A '* ; it is also continuous in the same topology in its second variable for any choice of the first variable in the canonical image of A in A " ■ (That is, f } g is continuous in / e .4** for any g e ' A ” , and g h-> k ( a)g is continuous in g for any a e A .) The second product enjoys the reflected continuity properties. (That is, g >—* / • g is continuous in g e A ** for any / € A **, and / > - * / • /c(a) is continuous in / for any a € >1.) Basic properties o f the weak* topology guarantee that any element / € A ** is the limit of a net in k ( A ) within the ball o f radius ||/|| (cf. Dunford and Schwartz [1958] V.4.2). These facts allow the calculation o f both products, but the reader should be warned that this is a subject on which an unusual number of false results have been published, and several of the errors stem from attempting to use continuity results in proofs. The entirely algebraic original construction is less subject to erroneous interpretation. We now give that definition, which guarantees the existence o f products with the continuity properties just described. Definitions and Basic Properties 1.4.1 Definition Let A be a normed algebra, let k : A —> A ** be the nat ural injection and let L and R be the left and right regular representations of A. For any a e A and w € A *, define elements uia and 0w o f A * by
811(1
Wa = ( £ a ) * M
aU = (R a )*(w ).
(1)
For any ui 6 A ' and / € A **, define elements fui and uij o f A * by fu>(a) - /(w0)
and
w/(a) = / (aw)
( 2)
for all a € A . Finally for any f , g € A * ' and w € A *, define elements f g and / •g o f A ** by fg (u >) = / ( 9w)
and
/ • g(w) - g (u f ).
(3)
These two products in A ** are called the first and second Arens product, respectively. When the two products coincide, the algebra A is said to be Arens regular. We gather some elementary properties o f these two products. 1.4.2 Theorem
The elements uia and auj belong to A " and satisfy: u>a{b) = u)(ob)
l k . l l < IM I ||a||
au{b) = u>(ba)
(4)
||„w|| < IM I ||a||
(5)
48
1: Normed Algebras and Examples (u}a) b = u }ab
( au ) b = a(ub)
a(bV) ~ abU
1.4.2 (6)
fo r all ui e A ' and a,b 6 A - The elements fU) and uif belong to A * and satisfy H H I < |M|||/|| IK|| < M\\f\\ (7) (/w)o = f ( u a) k(o)W —
o(w/) = (o w)/
( 8)
wii(a) = wa
(9)
/or all a € A , u) € A * and f e .4*’ . Finally the two products satisfy
l\fg\\ < l l / l l W faw
=
11/ - d l < l l / l l IM I
/ (s w )
* (a )/ = « ( a ) • / = (£ « )* * (/ )
w / 9
=
( a ’/ ) s
/ « (a ) = / • « ( a ) = (/**)**(/ )
/c(a)/c(6) = /c(a) • /t(6) = k ( ab)
do) ( H
)
( 12 ) (13)
fo r all a,b € A , u € A * , f , g € A " . Each product makes A '* into a Banach algebra, and k is an injective homomorphism from A to A " with respect to either Arens product. The two products agree whenever one o f the factors is in k (A ). Furthermore, the map ( L . A '* - B(-4*) / R - A " - B ( A ') ) defined by ( Z , (w ) = fui j Rf{u>) = u>f ) is a { homomorphism / anti-homomorphism ) with respect to the ( first / second ) Arens product. The following maps are continuous in the weak* topology (I.e., the A * topoLogy) on A ' * : f
f9
g>~>n(a)g
9 - + f -9
( 14)
f >-> f ■n(a )
(15)
fo r all a € A and f , g € A ‘'*. I f ip. A —> B is a continuous ( homomorphism / anti-homomorphism ), then ifi, , :A " * —►B ** is a continuous { homomorphism / anti-homomorph ism ) with respect to ( either Arens product / the opposite /Ireus products ) on A ** and B *m. Proo f A ll these results are easy to obtain from step by step calculations starting with the three step definition. □ We will show in Proposition 2.5.3(e) that n (A ) is a spectral subalgebra o f A * mwith respect to either Arens product and in 4.1.15 that various kinds o f ideals in A and A " are closely related. In Proposition 8.3.7 and Theorem 8.7.14 we will give more details on the relationship between minimal ideals in A and A " . Fereidoun Ghahramani, Anthony To-Ming Lau [1988] and Ghahramani, Lau and Viktor Losert [1990] show that, for locally compact groups G and
1-4,2
Arens Multiplication
49
H , any isometric isomorphism
= we or, equivalently, that e(u>a) — u>(a) = e (Qw) for all cu € A * and a € A - There is no reason to expect t h a a mixed identity is unique. In fact, any e! € A ** such that e and
50
1: Normed Algebras and Examples
1.4 3
e' agree: on ( L ^ ) ' { A ' ) + is also a mixed identity. Anticipating the terminology and results of Chapter 5, we note that A " has a mixed identity if and only if A has a bounded two-sided approximate identity. In this situation we say that A is approximately unital. Similar results involving one-sided approximate identities hold for a right identity for the first Arens product or a left identity for the second product individually even if they are not mixed identities. If A " has a ( left / right ) identity for either product, then equations (16) and (4) show that the annihilator ideal ( A b a / A l a ) ) is zero. The situation on uniqueness of mixed identities changes if A " has a (two-sided) identity element for either one of the Arens products. In that cam: any mixed identity must equal this unique (two-sided) identity since ( e — eej — e' / e ~ v!c e‘ ) if a is an identity and e' is a ( right / left ) identity. When A ** is unital or when it just has a mixed identity, there is another way to consider the Arens product which is helpful in some situations. In slightly less generality, it was first published by Duncan and Hosseiniun [1979]. Recall that Theorem 1.4.2 already showed the existence o f a ( ho momorphism L : A ** —* B ( A ') / anti-homomorphism R: A " —> B (A * ) ) defined by ( Lf(u>) = fui / Rf(u>) = ujf ). In fact we have already derived in equations ( 1 ) and ( 8 ), but not yet stated explicitly, that the range of ( L / R ) is included in the commutant { ( ( L a ) * ) ' / ( ( R a ) 'Y )• When A ’ * has a one- or two-sided identity we get the following.
1.4.3 Proposition Let A be a nonned algebra. Then ( I / R ) has the whole commutant ( ( ( L a ) * ) ' / ( ( R a ) * ) ' ) o* its range i f and only i f A ** has a ( right / left ) identity fo r the ( first / second ) Arens product. Further more, ( L / R ) is a homeomorphic { isomorphism / anti-isomorphism ) of A " with the ( first / second ) Arens product onto ( ( ( L a ) ' ) ' / ( ( R a ) * ) ' ) i f and only i f there is a two-sided identity fo r the ( first / second ) Arens product. Finally, ( L / R ) is an isometry i f and only i f the two-sided identity fo r the ( first / second ) Arens product has norm one. P roo f We work only with L since the case o f R is similar. Suppose L has all o f the commutant as its range. Then there is some e G A** satisfying L e = I € B (A ’ ). This means eui = oj, from which it follows that e is a right identity for the first Arens product. If L is an isomorphism, then e must be a two-sided identity and if L is an isometry, ||e|| must be one. Conversely, suppose e g A " is a right identity for the first product. Let S be arbitrary in the commutant ( ( L a )* )'- Define f s G ^4** by /s(w ) = e(S(u>) ) for all u) € A * ■ Then equation (16) shows fs(o)_= e(S(uia) ) = e (S o (L a)^(uj)) = e ((L a)* o S (u i)) = e(S (u})a) = S (u i)(a ), so L maps f s onto S. Thus L is surjective. Now suppose e is a two-sided identity for the first product. Since /uj is just L f ( u ) , equation (17) shows that L is injective.
1A 4
Arens Multiplication
51
Finally if e is a two-sided identity element of norm one, then the map we have just constructed from the commutant to A " is contractive. Since L is obviously always contractive, we conclude that it is an isometry. □ This result shows that when A * ’ is unital for one o f the Arens products its Arens multiplication may be defined in terms of the multiplication in ( ( L a Y ) ' or ({R-a Y Y 38 a subset of B (A * ) induced by the appropriate map L or R. The existence o f an identity in >1** can be described in terms of the existence of a particularly strong kind of approximate identity in A , but we will not discuss this here since approximate identities are not introduced until Chapter 5. It is of considerable interest that if either / is a right idealizer of k (A ) in A ** or if is left idealizer of k(-4) in .4**, then L j and R g commute in B (A * ). It is easy to see that these two operators commute exactly when the o ix ed associative law g • (h f ) = (g ■h ) f holds for all h € A **. Hence they commute for all /, g € A "* if A is Arens regular or k(.A) is a one-sided ideal in A ** with respect to one, and hence both, Arens products. Relationship to the Double Centralizer Algebra It should not be surprising that there is an intimate connection between the Arens products on A ** and the double centralizer algebra D (A ). We will define various maps between subalgebras o f >1**, T>(A) and V ( A * r ) under minimal hypotheses on A beginning with the following two easy propositions. Some o f these ideas are related to Laszlo Mate [1967] and Pak Xen Wong [1985]. 1.4X- Proposition Let ( L , R ) be a double centralizer o f a Banach algebra A in which the annihilator ( ideal A a is / ideals A l a and A b a are ) { 0 }. Then (18) is an isometric injective isomorphism o f { T>b { A ) / T>(A) = T>b { A ) ) into V b ( A " ) when A * ' is endowed with either Arens multiplication. Note that L ** and R** leave k ( A ) invariant as a subset o f A * * . P roo f Under the second alternative hypothesis, all double centralizers are bounded by Theorem 1.2.4. (In that case the same theorem shows that it is enough to consider the third and sixth lines below.) One easily checks the following results for sill a € A , ui G A " , and g € A **: L'{u>)a R*(u>)a £ *K )
9J?*( w) i?*( 9u,)
-
wL(o)
= = = = =
Ufi(a) L W fi“ (s)W L "(g )U
aL *(u j) aR * ( w) L(a)W
= = = =
L'(aU>) fl(a)W R*(aU>)
R (tJ)» £ * (w» )
=
R'(u>g)
=
52
1: Normed Algebras and Examples
1.4.5
These equations show that ( L ” / R " / (L **, R " ) ) is a bounded ( left / right / double ) centralizer whenever ( L / R / (L , R ) ) has the same property, no matter which of the Arens products we use. Since ( S o T )* * equals and ||S|| equals ||S**||, the maps L >—> L ** and R >—> R ** are homornorphisms and isometries. (This proof could also be based on looking at L " and R " as extensions by continuity of L and R , respectively.) □ 1.4.5 C o r o lla r y I f A " has a mixed identity e and ( L , R ) is a double centralizer o f A , then they satisfy the following identities: R “ ( f ) = f R » ( e ) = / L **(e ) and L ** (/ ) = L * * (e )- / = i?*‘ ( e ) - /
{
fo r all f € A " . Thus the map (L , R ) h-. ( L m\ R “ ) o f V { A ) into V ( A ” ) has range in (the image o f ) A ** fo r one of the Arens products i f and only i f there is a two-sided identity e € A** with respect to this product. In that case, ( L , R ) L ° '( e ) = R mm(e ) is a homeomorphic isomorphism o f T>(A) = T>b ( A ) into A * * , which is an isometry i f and only i f the identity has norm one. Proof As noted already, the existence of a mixed identity ensures that A l a and A r a axe zero so the stronger form of the last proposition is available. Equations (19) now follow by applying the basic properties of double centralizers to ( L " , R " ) and the equations f e ~ f or e • / = /, respectively. In particular, (/ **,/ **) is a norm one, two-sided identity for V ( A **) in whichever Arens product is being used. It can be identified with e in A " if and only if e is an identity for A ** in that Arens product. (Recall that, if any algebra like A " has an identity o f norm strictly greater than one, then the embedding o f A ** into T>(A**) is an isomorphism but not an isometry.) □ When A ' m is not unital, the situation becomes more complicated, and in particular it may be necessary to consider quotient algebras. We begin to study this situation by noting that the idealizer A ^ of k(.A) in A ** does not depend on which Arens product we use, since the two products agree when one factor is in k (A ): A'X = { } e >1** : M A ) = f ■k {A ) c k (A ) and n ( A ) f = k(.A) • / C k(.A )}. Proposition 1.2.3 describes the obvious homomorphism of this set into V ( k ( A ) ) , and we may consider this as a map into T>(A). Technically we can define this as / _► 9 ( f ) = ( L f . R f ) where (21) L/ = K~l o L f O K
and
R f = k ~* o R j o k
Arens Multiplication
1A3
53
with ( L f / R f ) being the ( left / right ) representation of / in A '* with either of its Arens products. Now it is easy to check
( L f y = R f e ((R Ay y
and
( R f y = If e((LA yy
( 22 )
in t e r m s of the notation used in Theorem 1.4.2 and Proposition 1.4.3. In the next proposition, we examine the second conjugates o f the actions L f and R f. For any subset S of A *, we use S x to denote the subset of A ’ * cons;sting of those functionals which vanish on 5. 1.4.8 P r o p o s itio n The subset ( ( L a ) * ( A * ) x / (R A y ( A * ) x ) is an ideal in A ** with, respect to the ( first / second) Arens product which is the kernel o f the maps { L and R / R and L ) . In fact, any element f from this ideal satisfies ( A * * f = 0 / / ■A ” = 0 ). (23) The subset A ^ = (R A y ( A * ) x n ( L A y ( A * ) x = {( i ? ^ ) * M * ) + ( ^ ) * M * ) } x is an ideal in A™ and is the kernel o f the homomorphism 8 : A % —> 'D (A ), defined above as 0 (f) = (L f , R f )
V / G ^ *.
(24)
The vwo Arens products agree on A*2 /^"n an(^ the map induced by 6 is an •isomorphism onto V ( A ) . In particular, A ^ is zero i f A ** has an identity fo r e ther Arens product. Wnen the map 8: A*£ —> T>(A) is followed by the embedding (L , R ) (L * * ,R * * ), the image o f f G A a is a double centralizer in either Arens product which agrees as a pair o f maps with ( L j \ R ^ ) where the super script numerals indicate the regular representation with respect to the second and first Arens multiplication, respectively. Proof We consider only the second case (for variety). For / G (R a )* ( A * ) 'L, we htve f ( auj) = 0 for all a G A and w G A *; this implies w/ = 0 from whicl the identification as kernels and equation (23) follow directly. (Note that L / R ) can be defined on the appropriate one-sided idealizer set which includes the annihilator set { (R a ) * ( A * ) x / ( L a ) * ( A * ) ± ).) Since R is an anti-homomorphism with respect to the second Arens product defined on all o f A , the set is a (two-sided) ideal even though (23) only shows that it is a right ideal. We can show directly that g ■/ is in this ideal for any € ^ **. We see g ■f { au>) = / ( ( „ « ) , ) = / ( „ K ) ) = 0 if / G ( R a W ^ . Suppose / € A **, g G A “ and gic(a) = n(b). Then f 9M
= / (6w) = p «(a )(w / ) = / •g (auj).
The case o f is similar so the two products do agree on the quotient. Now consider the last statement of the proposition. We again prove the
54
1: Normed Algebras and Examples
1.4.6
second case. For / € A * * , g € A*^ and ui € A *, we get ( R g ) * '(f ) (u j ) = f ( ( R t r ( u > ) ) = f ( g w ) = /ry(w ) = / # > ( / ) M .
Note that this proof has established more than stated in the proposition in several cases. □
( L ,R ) V {A )
V (A ") ( 4 n), 4 n))
(25)
The last three propositions can best be visualized in terms o f the com mutative diagram above, which represents the case in which the nth Arens product has an identity e either for n = 1 or 2. (A ll the maps are defined and interesting without this restriction, but the diagram is no longer com mutative in general.) Each map is a homeomorphic isomorphism onto its range and is isometric if e has norm one. Thus we see that if at least one o f the Arens products has an identity, then a substantial part of A " behaves very well indeed: just like the double centralizer algebra. Consider approximately unital algebras for a moment. If such an algebra is Arens regular, then it has only one mixed identity which is in fact a (two-sided) identity for both Arens products. Hence, in the context o f approximately unital normed algebras, A ** being unital for one of the Arens products is a weak version of A being Arens regular. Arena Semxregularxty A still weaker version of Arens regularity has been introduced by Michael Grosser [1984] and studied there and subsequently by the same author [1987] and by others. An approximately unital Banach algebra A is said to be semiregular if it satisfies R '* (e ) = L " { e ) for all mixed identities e. (The idea behind the definition is that an algebra is semiregular if and only if the two embeddings o f double centralizers, based on using their left or their right centralizers, agree for any mixed identity.) Corollary 1.4.5 shows this
1 . 4.7
Arens Multiplication
55
is true whenever A ** has an identity for at least one o f the Arens products. The above remarks show that Arens regular algebras are semiregular. The latter result and the easy fact that commutative approximately unital Bantxl algebras are semiregular were established in the first cited paper. The case of commutative algebras shows that semiregularity is strictly weaker than the possession of an identity for at least one Arens product. :Losert and Harald Rindler [1984] showed that, for a locally compact group G, the algebra L l (G ) is semiregular if and only if it is either discrete or commutative. This contrasts with the case of regularity where L l ( G ) is regular if and only if G is finite, a result proved by Young [1973a] after Civin and Yood [1961] handled the abelian case. Similarly, the algebra B a ( X ) oi approximable operators (i.e., those uniformly approximable by finiterank operators (Definition 1.1.17 above)) on a Banach space X is regular if and only if X is reflexive (Young [1976]). Much weaker conditions are knovm for semiregularity (Grosser [1987]). We shall treat these results in a preliminary way later in this section and in more detail in Section 1.7 after introducing more o f the relevant concepts. However, two special cases completely solved in the last reference are particularly interesting. Consider the algebra B a (C (Q .)) o f operators on the Banach space o f sill continuous complex-valued functions on a compact Hausdorff space Jl. This algebra o f operators is semiregular if and only if $1 is scattered, where scattered signifies that every nonempty subset contains an isolated point. Let /i be a finite measure and let L l (fi) be the usual space of (equivalence classes) of absolutely /i-integrable functions. Then B a {L 1(h )) is semiregular only when L l (fi) is finite-dimensional (in which case, of course, the algebra is Arens regular). Before discussing Arens regularity, we introduce three simple examples. Examples of Arens Multiplication fo r Commutative Algebras For a commutative algebra A , Definition 1.4.1 immediately implies aui = iva and fui = ujf, hence f g ~ - g - f (note reversal of factors) for all a & A , u) € A ' and j , g £ A **. Prom this it easily follows th^t a commutative algebra is Arens regular if and only if one, hence both, Arens products are commutative. In the following examples, this observation will be useful. For a fuller discussion of this and many other results on the Arens products for general commutative Banach algebras, see Theorem 3.1.12. 1.4.7 A re n s M u ltip lic a tio n in (co)** ^ £°° This is perhaps the simplest example in a nonreflexive space and shows that the Arens multiplication is often the natural multiplication on the double dual. (O f course, in a Banach algebra which is reflexive as a Banach space so that A and A ** are canonically isomorphic under k , both the Arens products agree with the original multiplication.) As usual, let co be the collection o f all sequences converging to zero with elementwise algebra operations and the supremum
56
1.4.7
1: Normed Algebras and Examples
norm:
I M L = sup{|an |
:n£ N}.
We identify t 1 (the space o f all sequences for which the norm, X.'
IN , = X > n l ’ n~l
is finite, with elementwise linear operations) with the dual o f cq as follows: OO
u i:ix —►(cq)*
is defined by
Wb{a) = ^
anbn
n=l
for all a 6 co and i 6 f Then we can identify i° ° (the space of all bounded sequences with the same supremum norm as cq and elementwise linear operations again) with the dual of t l ~ (co)* by: OO
f : t ° ° -> (£ *)' where f„ (b ) = £
anbn
n= 1
for all a G i ° ° and b G t x. W ith these identifications (which the reader can easily check are isometric linear isomorphisms), the canonical injection of cq into its double dual identified as i° ° is just the usual inclusion as spaces o f sequences. Thus we will suppress the symbol for this injection. Now the following results axe immediate:
0 (w(,) = (w (,)0 = uJab
where (ab)„ = a„bn
V d G c o ; 6 g < *;
/(w(,) = (w6)/ = uifb
where (/&)„ = /„&„
\/f £ e°°; b £
( f 9 ) n = V - g ) n = fn 9 n
V / ,flG / “ .
Thus, in this case, the two products agree and just give the elementwise multiplication with which we would certainly have endowed t°° in the first place. This example is unusual. Most algebras axe not Arens regular and gen erally the double dual is quite complicated. The fact is that Co is an example, o f a commutative C*-algebra. Commutative C*-algebras will be introduced in Chapter 3, particularly Theorem 3.2.12. Not necessarily commutative C*-algebras axe briefly introduced in §1.7.17 and will be studied at more length in the second volume of this work. There we will show that any C*-algebra is Arens regular and that its double dual has a very natural interpretation as a universal W*-algebra (or von Neumann algebra) “com pletion” , o f the original algebra. Theorem 3.2.12 shows that every unital commutative C*-algebra has the form C (ft ) for some compact space ft and vice versa. Hence it is obvious
1 . 4.8
Arens Multiplication
57
that i ' ~ £°° is C ( f l ) for what turns out to be a rather complicated uxtre aely disconnected connected space. For generalizations, see Samuel Kaplf n [1985], 1 ■ S A re n s M u ltip lic a tio n on w h en £l has P o in tw is e M u l tip lic a tio n There are two possible fairly natural multiplications on l l : point'visc and convolution multiplication. We will examine the Arens multiplic t.ion in both cases. Although both multiplications are commutative on £' only the first is Arens regular. This is particularly interesting since, as we will note below, certain Banach space properties force Arens regular ity. This example shows that no Banach space property can characterize Arent regularity. Tt make these two examples more interesting, we wish to exhibit a reasonahiy concrete interpretation of the double dual space o f £l . It turns out I lat there are two concrete representations, each o f which has some interest. But before we even describe these concrete representations of ~ (co)***, we note a special property o f any triple dual Banach space X***-. it has a canonical contractive projection 7r onto X * . Any linear sub space which is the range o f a bounded projection 7r is complemented by the closed linear space (1 —1 In this case this complementary subspace is precisely « ( A ’) X, as the reader may easily check. The construction o f this projection will be given in Section 1.7. The first concrete realization o f (£°°)* ~ ( t 1)** — (co)*** we mention is the; given by Dunford and Schwartz ([1958] IV.8.16) as the space 6a (N ) of all complex, bounded, finitely additive set functions where the duality is implemented by simply integrating the sequence in £°° by the set function in 6g (N ). In this case the two complementary subspaces defined in the last paragraph are simply: 7r (ia (N )) is the space of countably additive measures on N which can obviously be identified with i 1 by the Radon-Nikodym theorem; and ( I — 7r )( 6a (N )) is the space of finitely additive measures which vanish on all finite sets. The second representation of the dual of £°° comes from noting that f.°° is a unital, commutative C*-algebra. Hence, by what is essentially a corollary o f Theorem 3.2.12 described in §3.2.15, £°° can be identified with the Banach space C'(/3N) of all continuous complex-valued functions on a compact Hausdorff space 0N, called the Stone-Cech compactification of N. There is a continuous injection of N onto a dense subspace o f /3N, and the function in C(/3N) corresponding to / 6 t°° may be considered as just the extension of / by continuity to all of /3N. Then the Riesz representation theorem identifies ~ (C(/3N))* with the Banach space M (8 W ) of all complex regular Borel measures on /3N. This time the pair of complementary spaces consists of the subspace of measures supported
58
1: Norrned Algebras and Examples
1.4.9
on N C /?N (naturally identified with £l , again by the Radon-Nikodym theorem) and the space o f measures supported on the complementary space 3N \ N. We now consider the Arens products arising from elementwise multipli cation on t 1 following Civin and Yood [1961], Theorem 1.4.2. In describing the situation, we use several concepts which will not be introduced until later in this work, and some of the ideas come from our discussion in Chap ter 3 of the Arens multiplication in commutative algebras 3.1.12. To begin with, the situation is similar to that of the last example. We identify £°° with the dual of i 1 ^ (co)* by: uj:£°°
-> ( t l y
where
OO u>a(b) = £ anbn n= I
for all a e t°° and b e t 1. We will use { «o / ) for the natural map of { Co / t 1 ) into its double dual space. We immediately see fc(w0) = (wa)fc = uiab
where
(ab)n = anbn
for all a £ t°° and 6 € £l . However, this shows that these linear functionals are all in «o(co), from which it follows that all Arens products with at least one factor from the complementary subspace to k i ( ^ ) in the triple dual space decomposition must be zero. Call this complementary subspace 71 = Ko(co)x - Thus the direct sum decomposition (£1) '* — « i ( ^ ) © TZ is an algebra direct sum, multiplication on (£*)** is commutative for either Arens product and thus the two products agree Moreover, it is easy to see that any non-zero homomorphism of ( l (and hence of (^* ) ’ * under either interpretation o f this as a space of measures) into C is given by evaluation at a positive integer n. A ll these evaluations belong to the dual space of ( l since they are obviously all contractions. Their closed linear span is just the image of co in £°° under the embedding «o-'Cq —» £°°. Thus the subspace of elements of (^’ )** which vanish on all these homornorphisms is just TL. Hence 1Z is the Gelfand radical and the Jacobson radical of (£ ')**, and £l (or equivalently k ^ 1)) is Jacobson semisimple. Thus, with pointwise multiplication on £1, the Arens multiplication on its double dual is extremely simple but badly behaved in that it has an enormous annihilator ideal.
1.4.9 Arens Multiplication on — M(/3N) when £l has Convo lution Multiplication In this example, it will be tidier if we consider the entries in the sequences in £x to be indexed starting with zero. Obviously this makes no difference to the Banach space structure. The convolution product is then defined by n { b * d )n = Y s bn- kdk k=0
V n £ N°; b,d £ £l
Arens Multiplication
59
which is easily seen to be commutative. Let Sn € £l be the sequence which is zero except at n where it is one (so (Sn)rn = Snrn, the Kronecker delta). Then So is a multiplicative identity and in general 6n * 6m — 6n+m so that £l with this product may be interpreted as the semigroup algebra of N° with addition for its semigroup operation. This time an easy calculation shows OO b (^ a )
(^ a )b
— ^aob
w h ere
(fl O 6 )n
fln + m 6m
m=0 for ell n G N°, a G £°° and b € £l . However, the formula for the next step in the definition would not help us to see that this algebra is not Arens regular. Instead we look at a very concrete construction using property (f) o f Theorem 1.4.11 below. Consider two sequences {& „} and { d „ } in £l where bo = 6q and, for n > 1, bn = and dn = ) The algebra £l ( S ) is Arens regular. (c) There do not exist sequences {u n} and {u m} in S such that the sets {u nvm : m > n ) and {u nvm : m < n ) are disjoint. (d) The semigroup operation can be extended to the Stone -Gech com pactification (3S o f S as a discrete space. • When these conditions hold, £l (S )** with its Arens product can be natu rally identified with M(/3S) equipped with the convolution product based on the extended semigroup operation. V/e refer the reader to the original paper for a proof, although one can be based on our next result and is suggested by the last example. Arens Regularity Recall that an algebra is called Arens regular if the two Arens products agree on its double dual. This topic has attracted most of the attention devoted to Arens multiplication, at least until recently. As we will note, Ar ens regularity is uncommon. Duncan and Hosseiniun [1979] first formu lated the following comprehensive theorem this way, using ideas crystallized by Young [1976] from Pym [1965], and J. Hennefeld [1968].
60
1.4.11
1: Normed Algebras and Examples For any
uj
e A ' define
(K , - . A ^ A ' / P ^ .A -* A ' )
by
( A w( a ) = w a / P w(a) = 0w )
(26)
for all a 6 A . These maps satisfy ( K ( f ) = f u > / K ( f ) = ujf )
“ d
( A :*(/)(
