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A defined by r(w) = (i 0 w)(V) is a non-degenerate representation of M, and V is the generator of T. Proof. By 2.6.4, we have:
(10 W)(1 0 o)(V (& 1)(10 o)(1 0 W*) = (V 0 1)(10 o)(V 0 1)(1 (9 o) which can also be written:
(V(D 1)(1®o)(1®W*)(1®c)(V*®1) = (10 o)(10 W*)(V®1)(1(9 o) (*)
2.6 The Fundamental Operator W Is Unitary
73
Let us consider the representation of M. in A 0 £(H) defined by: µ(w) = V(1 0 A(w))V* Therefore, for all w in M* 1, we have, using 2.6.1 (i):
µ(w) _ (i 0 i 0 w)((V 0 1)(10 vW*v)(V* ® 1)) _ (i 0 i 0 w)((1 0 o)(1 0 W*)(V 0 1)(1 ® a)) For all f 2l
by (*)
A* 1, Q2 in ,C(H)* 1, and w in M*, we have:
(,z(w),11 (9 12) _ (11 ®12 0 w)((1(9 o)(V* ®1)(1® W)(1® o)) _ (1®®w 0 02)((V* ®1)(1 ® W)) (1® ®w)((i 0 i 0 12)((V* ®1)(1® W))) _ (11 ® w)(V*(1 ® (i ®12)(W))) = w((11 0 i)(V*(10 (i 0 Q2)(W)))
= w((11 0 i)(V*)(i (9 12)(W)) Therefore, by 1.5.1 (ii), we have:
K((11 0 i)(V*)(i 0 02)(W)) _ ((11 0 i)(V*)(i 0 02)(W))* which can be written as follows:
K((11 0 i)(V))K((i ®12)(W*)) = (11 0 i)(V*)(i ®12)(W) since 11 and 12 are positive. If we apply 1.5.1 (ii) again, we get:
K(11 0 i)(V)(i ®12)(W) = (11 ® i)(V*)(i ®12)(W) or:
(i 0 12)((x(11 0 i)(V) ®1)W) = (i ®12)(((11® i)(V*) ®1)W) which, by linearity, will still holds for all 12 in £(H)*, and therefore implies:
(K(11® i)(V) ® 1)W = ((11 ® i)(V*) ®1)W As W is unitary, we have:
K(11 ® i)(V) _ (11 0 i)(V*)
74
2. Kac Algebras
and by 1.5.1 (i) and (ii), r is a representation and we have, for all w in M*: (i ®w o sc)(V) = (i ®w)(V*)
As V is unitary, r is non-degenerate (1.5.3).
2.6.6 Corollaries. (i) Let u be in the intrinsic group of (M, T, ,c). Then, we have: IC(u) = u*
and the application w -+ w(u) is a one-dimensional representation of M*, which has u as generator. (ii) If u, v are two elements of the intrinsic group of (M, T, n) the Kronecker product of the two one-dimensional representations w -+ w(u) and w - w(v) is w -+ w(uv). So, the Kronecker product, restricted to the intrinsic group, is the usual product. (iii) If µ is a representation of M*, we have, for all w in M*, with the notations of 1.1.1 (ii):
(µ x u)(w) = µ(u w) (u x µ)(w) = µ(w u) . Proof. By 1.2.3, u is a unitary and satisfies F(u) = u ® u. We can then apply 2.6.5, with A = C, and (i) is proved; then (ii) is just a corollary of 1.5.5; (iii) is an application of 1.4.3, with the representations u and u.
2.6.7 Proposition. Let R be in the centre of M, such that:
1'(R)>R®R #c(R) = R
Then, we have:
(i)W(R®R)=(R®R)W (ii) F(R)(R ®1) =1'(R)(1 ®R) = R ®R.
Proof. By hypothesis, we have, using 2.6.3 (ii):
R ® R = (R ® R)F(R) = (R 0 R)W(1® R)W* Using 2.6.2, we get:
R®R=(R®R)(J®J)W*(J®J)(1®R)(9®J)W(J®J)
(*)
2.6 The Fundamental Operator W Is Unitary
75
and, using the fact that JRJ = R (because R belongs to the centre of M) and that JRJ = R (because ic(R) = R), we have:
R0R=(R0R)W*(10R)W Taking adjoints, we get:
R0R=W*(10R)W(R0R) and, as W is unitary:
W(R®R)_(10R)W(R®R) = (R (& R)W(1® R)
_ (R ®R)W
by 2.6.1 (iv) by (*)
So, (i) is proved. We have then:
1'(R)(R ®1) = W(1® R)W*(R 0 1) = W(R 0 R)W*
= R ®R
by 2.6.3 (ii) by 2.6.1 (iv) by (i)
Applying this result to (M, s1', rc, cp o ic), we get:
1'(R)(1®R)=R®R which ends the proof.
2.6.8 Corollary. Let P, Q two projections in the centre of M, such that:
1'(P) > P ®P
fc(P) = P
r(Q)?Q®Q
K(Q)=Q
P+Q >>- 1
Then, either P or Q is equal to 1. Proof. From 2.6.7 (ii), we have:
r(P)((1 - P) ®P) = 0 r(Q)(Q ®(1 - Q)) = 0 which implies, as 1 - Q < P and 1 - P < Q, by hypothesis:
r(P)((1 - P) ®(1 - Q)) = 0 r(Q)((1 - P) 0 (1 - Q)) = 0
76
2. Kac Algebras
Taking the sum of these equalities, as r(P) + I'(Q) > 1, we get:
(1-P)0 (1-Q)=0 which ends the proof.
2.7 Unicity of the Haar Weight Let (M, I', #c) be a co-involutive Hopf-von Neumann algebra and cp, 'O two faithful semi-finite normal weights on M+, satisfying both (HWi) and (HWii).
2.7.1 Lemma. Let z be in M such that r(z) = z ® 1; then z is scalar. Proof. We have, for all t in R:
r(ot (z)) _ (i 0 oT)r(z)
by 2.5.6
=(ioat)(z®1) =z®1 = r(z) As r is one-to-one, o`p(z) = z and z belongs to MP. Let x, y be in 0t P 0 1, the proof if analogous; taking A = C, one gets (ii).
78
2. Kac Algebras
2.7.4 Proposition. (i) Let x in £(H). Then, x belongs to i2t' if and only if.
(10 x)W=W(10 x) (ii) Let x be in M', such that: W*(1®x)W = 1® x then x is scalar. Proof. We have the following sequence of equivalences:
x E M q x*A(wy,a) = )t(wy,a)x* x*.(wy,a)8)
(a 1
Va, y E H = (a I A(wy,«)x*8) Va, 0,'y E H
(x/3I A(wy,a)8) = (1 I A(wy,a)x*8)
da,/3,y E H a (W(a®x/)I y®8) = (W(a0/3) I y®x*8) by 2.6.1 (iii)
a W(1®x)=(1®x)W So, (i) is proved; (ii) is then clear from (i) and 2.7.2 (ii).
2.7.5 Proposition. For all t in R, we have:
O
awon = KOIW -t c t ratoK=(QtOK®i)r aj OK = (p
(iv) The space App( p fl 9L,,oK) is dense in HW. (v) The space A,(g't
0 such that: cP2 o u(x) = acoi(Rux)
Vx E M1
Proof. From 1.2.7 we get T1(Ru) > RU ®Ru and ic1(Ru) = Ru. So, from 2.2.6, the quadruple K1Ru = (M1Ru , F1R I 1Ru , w1Ru) is a Kac algebra. Let r be the canonical surjection M1 - M1Ru . We can define a bijective IHI-morphism
v from M1Ru to M2 by v(r(x)) = u(x). Clearly, cp2 o v is a Haar weight on K1Ru and then, using 2.7.7, there is a > 0 such that, for all x in M1Ru : cP2 o v(x) = acP1Ru (x)
So, for all x in M+W2 o u(x) = c'2 o v(r(x)) = awlRu (r(x)) = acoi(Rux) 2.7.9 Corollary.Let K1 = (M1, T1, ,c1, w1) and K2 = (M2, T2, K2, w2) be two
Kac algebras, u an H-isomorphism from (Ml,Tl,k1) to (M2iT2,Ic2). Then there exists a > 0 such that w2 o u = aw1i and so the Kac algebras K1 and K2 are isomorphic in the sense of 2.2.5.
Chapter 3 Representations of a Kac Algebra; Dual Kac Algebra
In this chapter, we shall use the notations hereafter: K = (M, T, ic, cp) will be a Kac algebra, A its Fourier representation, W its fundamental operator and M the von Neumann algebra generated by A. This chapter deals with the representations of the Banach algebra M*, following Kirchberg ([79]) and de Canniere and the authors ([21]), and the construction of the dual Kac algebra, as found independently by the authors ([34]) and Vainermann and Kac ([180]).
This chapter begins with a Kirchberg's important result on Kac algebras: every non-degenerate representation of the involutive Banach algebra M* has a unitary generator (3-1-4). For the Kac algebra Ka(G) constructed with L°°(G), one recovers the well-known result that every non-degenerate representation of L1(G) is given by a unitary representation of G. As a corollary, we get that, for any non-degenerate representation p, the Kronecker product A x y is quasi-equivalent to A (3.2.2); in the group case, that means that, for every unitary representation µG of G, the tensor product AG 0 PG is quasi-equivalent to AG, which is Fell's theorem ([48]). When we choose p = A, we then get a coproduct T on k (3.2.2). A co-involution k on M is then defined, for all w in M,, by the formula: ;c(a(w)) = A(w o ic)
The triple (M, T, k) obtained is a co-involutive Hopf-von Neumann algebra. For locally compacts groups, that means (3.3.6) that the von Neumann algebra £(G) generated by the left regular representation AG has a co-involutive Hopf-von Neumann structure given by a coproduct I'8 and a co-involution rc9 such that, for all s in G: 1'e(AG(s)) = AG(S) 0 AG(S) AG(S-1) lce(AG(S)) =
By predualizing the canonical surjection from the von Neumann algebra generated by M. onto k, one obtains an isometric, multiplicative and involutive morphism from the Banach involutive algebra M. in the Fourier-Stieltjes
84
3. Representations of a Kac Algebra; Dual Kac Algebra
algebra defined in Chap. 1, the image of which will be called the Fourier algebra of the Kac algebra, and is a self-adjoint ideal of the Fourier-Stieltjes algebra. For locally compact groups, we recover the situation of the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) defined and studied by Eymard in [46]. Using the canonical Tomita-Takesaki construction, we define, starting from the left ideal I,, of M*, a left Hilbert algebra dense into the Hilbert space H., which generates the von Neumann algebra k and a faithful semi-finite normal weight cp on k (8.5.2), satisfying a Plancherel-type relation: cp(A(w)*A(w)) = Iia(w)1I2
for all w in I,,
Moreover, the modular operator d, is affiliated to the centre of M, and is, in the sense of [114], the Radon-Nikodym derivative of the weight cp with respect to the weight cpoic (3.6.7). We prove that this weight is a Haar weight (S. 7.4 ), and we have so defined a dual Kac algebra ]K = (M, I', k, 0), the fundamental
operator W of which is oW*o, and the Fourier representation A of which is given by nA. (where a* : M* -> M is obtained by predualizing A). As A is non-degenerate, A is faithful.
On C(G), the weight so constructed is equal to the Plancherel weight pp, studied by Haagerup in [58]. So, K3(G) = (L(G), I's, mss, spa) is another example of a Kac algebra (3.7.5), which is symmetric and will be studied in Chap. 4. Another essential result about Kac algebras is the following: characters
on M* (that is, elements x of M such that x # 0 and P(x) = x 0 x), are unitaries, verify rc(x) = x*, and, with the weak topology of M, form a locally compact group, called the intrinsic group of the Kac algebra (3.6.10). See also (1.2.2), (1.2.3) and (2.6.6).
3.1 The Generator of a Representation 3.1.1 Lemma. With the definition of 1.3.6, we have:
PR(K) fl% C \. (k,,)
.
Proof. Let x be in PR(K) fl 9t ,. It follows from 1.3.6 that there exists a Hilbert space 7-l, a non-degenerate representation p of M* on 7-l and a vector e in 7-l, such that, for all w in M*: (x,w) = (p(w)ele)
3.1 The Generator of a Representation
85
As the space iv fl Il is norm dense in M* by 2.4.6 (iii), the algebra µ(I4, fl 10)
is dense in µ(M*). Thanks to Kaplansky's theorem, it exists a sequence {wn}nEN of elements of II such that IIp(wn)II < 1 and that /.tµ(wn) strongly converges to 1. Let us consider the linear forms on M* defined by: Pn(w) = (x, w,°, * w * wn)
(w E M*)
We have, by hypothesis on x: !Pn(W' * W) = (x, (W * Wn)' * (W * Wn))
0
Therefore !Pn is positive definite. Moreover, since x belongs to 9q* and since, by 2.4.5, w'n * w * wn belongs to I., we get: by 2.1.6 (ii) by 2.6.1 (v) by 2.5.3
4'n(w) = (a(wn * w * Wn) I = (.(w,°a * w)a(wn) I A p(x*))
= (A(w)a(wn) I A(wn)A,(x*)) (^(w), i2a(wn),A(wn)Ap(x*))
Let us put f1 = da(wn)
We have then 4n(w) = (.1*(.fln),w),
and therefore: 45n = .*(Iln) E .1*(M*)
From 1.6.10, we get the existence of xQ in B(K)+ such that x = 7r* (, 12). Let us also consider the decomposition A* = lr*(sa)* (1.6.1 (ii)). We have:
I(x - (sA)*(J?n),lr(w))I = I(ir*(xQ) - A*(,fln),W)l
= I(x-Pn,w)I = I (X, W) - (0.,W) = I (X, W) - (x,w,°, *W * wn)I
I)
= I (,(w)e I ) = I (µ(w), (l£ -
IIi (w)II III -,(Wn)eII IIC + /L(wn)6II
by ([57] Proof 2.11) < 211y(w)II II II 116 - µ(Wn) II < 2II r(w)II 11611 IIe
- µ(wn)6'II
Therefore, thanks to Kaplansky's theorem, we get: 11-f2 - (sA)*(nn)II Jx*J for all x in L(Hw) is an involutive anti-automorphism
of L(Hp). The restriction of this mapping to k is an involutive antiautomorphism of M in the sense of 1.2.5. It shall be denoted k. Moreover, the involutions rc and k are linked by the following relations: k(A(w)) = A(w o tc) rc(A*(w)) = a*(w 0 k)
(w E M*) (w E M*)
.
3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra
93
Proof. For all x in k, let us put k(x) = Jx*J. Let w be in M. We have: Ic(A(w)) = JA(w)*J = A(w o sc)
by 2.5.5 (ii)
which altogether provides the first equality and ensures, by continuity, that for all x in k, &(x) belongs to k; the involutive character of k is trivial. For all w in M*, we have: (a*(w), w o ic)
_ (A(w o ic), w)
by the first equality _ (A(w), w o Ic)
_ (a*(w o k),w) which completes the proof.
3.3.2 Theorem. The triple (M, T, k) is a co-involutive Hopf-von Neumann algebra. It will be called the dual co-involutive Hopf-von Neumann algebra of K.
Proof. Let w be in M*, w1,w2 in M. We have: (I'kA(w), wl ® w2) = (I'A(w o ic), ci l ®w2) = ((A x A)(w o 1c), w2 ®(21i)
= (A*(w2)A*(wl),w o K)
by 3.3.1 by 3.2.2 (iv) by 1.4.3
_ by 3.3.1 by 1.4.3 _ ((A x A)(w), (wl 0 w2) o (k 0 k)) _ (c(k 0 k)PA(w), (w1 ® w2)) by 3.2.2 (iv) (A*(wl o k)A*(w2 o ic), w)
Therefore we have:
PA(w) = c(k 0 k)I'A(w) and we can complete the proof by continuity. 3.3.3 Proposition. The mapping sa is an H-morphism from (W*(K), cs,rxn, s;.) to (M, T, k). If K is abelian, (M, T, Ic) is symmetric, and if K is symmetric, (M, T, Ic) is abelian. Proof. Let w be in M*. We have: 5As,\7r(w)
= csaxA7r(w) = c(A x A)(w)
= c(sa 0 sa)(lr x 7r)(w) _ (sa 0 sa)cs,rx,.7r(w)
by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.1 (ii) by 1.6.4 (iii) by 1.6.1 (ii)
94
3. Representations of a Kac Algebra; Dual Kac Algebra
and, we get:
fsa = (sa 0 sa)csax. And: ksA7r(w) = kA(w)
=A(worc) = sa7r(w 0 IC)
= salr(w)
=
by 1.6.1(ii) by3.3.1 by 1.6.1 (ii) by definition of r (1.6.6)
sAs*7r(w)
and we get:
hs'\ = sAs* At last, since A is non-degenerate, we have sa(1) = 1, by 1.6.1 (iii); as sA is one-to-one, it completes the proof, together with 3.1.5 (ii). 3.3.4 Definition and Notations. Let us denote C *(K) the C*-algebra generated by the Fourier representation A of K. By 1.6.1 (i), we may identify its dual
(C (K))* with a closed subspace BA(K) of B(K). More precisely, to each element 9 of (C (K))*, one associates the element 9 o A of B(K) (where A denotes the restriction of sA to C*(K); cf. 1.6.1 (1) and (ii)). By 1.6.2 (ii), the mapping (sa)* is an isometry from M. into B(K), the image of which is contained in BA(K). By transposing 3.3.3, ($A)* is then an isometric Banach algebra morphism, the image of which will be denoted by A(K) and called the Fourier algebra associated to K. Every element of A(K) vanishes over KersA; conversely, let 9 be in B(K), such that (x, 9) = 0 for all x in Ker sA. We can define a linear mapping w on M by writing: (SA(z),w) = (z,9) (z E W*(K)) In fact, w appears as the composition of the restriction of 9 to the reduced algebra W*(K)suppsA, with the canonical isomorphism between W*(K)suppsa and Af. Therefore w is ultraweakly continuous and belongs to M*, and we have 9 = (sA)*(w). So, an element of B(K) belongs to A(K) if and only if it vanishes over Ker sa.
3.3.5 Proposition. The mapping rcA* is a non-degenerate faithful representa-
tion of k* in M. Its generator is W and we have: (i) (ii)
(i0I')(W)=(W ®1)(10o)(W0 1)(1®v) (i ®w 0 k)(W) _ (i 0 w)(W*)
(w E M*) .
Proof. By 1.6.1 (ii) we have rc.1* = n-7r*(sa)*, it then results from 1.6.9 and 3.3.4 that it is a faithful representation. By 2.6.1 (i), for all w in M*, and all
3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra
95
w in Al, *we have: (A* (w), w ors) = (A(w o k), w) = (w ® c..)(W)
therefore: rcA*(w)
= (i ®w)(W)
and we get (i) and (ii) through a straightforward application of 1.5.1 (i) and (ii).
3.3.6 Theorem. Let G be a locally compact group, G(G) be the von Neumann algebra generated by the left regular representation AG. There exists a unique
normal morphism I'e from G(G) to G(G) 0 C(G), and a unique normal antiautomorphism ice in G(G) such that, for all s in G: rs(AG(s)) = AG(S) 0 AG(S) AG(s-1) Ks(AG(s)) =
Then, (G(G), I's, rcy) is a symmetric co-involutive Hopf-von Neumann algebra; we shall denote it II1[e(G). It is the dual co-involutive Hopf-von Neumann
algebra associated to the Kac algebra Ka(G). Moreover, the morphism sA from the envelopping W*-algebra W*(G) to G(G) such that we have, for all
s in G: sA(lrG(s)) = AG(s)
is an H-morphism from the Ernest algebra of G (cf. 1.6.8) to H,(G). Proof. The existence of rce is the only non-trivial result; applying 3.3.1, we see there exists a co-involution k on (C(G), F.), defined in 3.2.8, such that,
for all f in Ll(G): I(A(.f)) = A(f o Ka)
From 1.1.3, we have, for any s in G, f in L1(G): (f 0 rca)(s) =
f(s-1)QG(s-1)
and so, we have:
h(IGf(s)AG(s)ds) =I)G(s)= JGf(S)AG(S_1)ds rc(AG(s)) =
AG(s-1)
.
3. Representations of a Kac Algebra; Dual Kac Algebra
96
3.3.7 Theorem. Let K = (M, T, ic, cp) be a Kac algebra, K' the commutant Kac algebra. Then the dual co-involutive Hopf von Neumann algebra (M'", I" k is equal to (M, ST, h).
Proof. Using 2.5.9, we see that the Fourier representation A' generates M;
so M'" = M. By 2.4.8, the fundamental operator W' associated to K' is W' = (J ® J)W(J 0 J). The coproduct T'" is, then, using 3.2.2 (iv), such that:
(J ® J)o W*a(J 0 J)(1® x)(J 0 J)aWo,(J 0 J) _ (h (D k)1'r.(x)
by 3.3.1 and 3.2.2 (iv) by 1.2.5 applied to (M, I', k)
= cP(x)
As, by the identification of H, with HH,, the associated antilinear isomorphism J. and J., are equal, we see, by 3.3.1, that ic'" = k and the theorem is proved. 3.3.8 Proposition. Let K = (M, F, a, cp) be a Kac algebra, and 1K = (11%I, T, kc, cp)
be a Kac subalgebra in the sense of 2.2.7. Let us denote j the canonical imbedding from k into M, which is an H-morphism. There is then a canonical surjective H-morphism r from (M, T, k) to (M", T", rkc") such that r(Afw)) = A(w o j), for all w in M*, where A is the Fourier representation of K.
Proof. Let us call I the isometry from Hc, to H. defined, for all element x in
=
nMby:
A,(j(x)) (i) As j is an H-morphism, the application w -+ w o j is multiplicative and involutive from M* to (M)*. This application is surjective because, for any
a,'yin H.,we have wy,aoj=wj,.yI*a. Let x in
w in M*. We have:
Ia(w o j)A,(x) =
o j o Tc ® i)P(x))
by 2.3.5
= IAc,((w o is o j ® i)T(x)) = AW((w 0 is o
i))F(j(x)))
by (i)
= A(w)AW(j(x))
= A(w)I
(ii)
Let us put, for x in k, r(x) = I*xI. We have r(1) = 1 and r(a(w)) = (wo j), because I is an isometry.
3.4 Eymard Algebra
97
Let now Co be in (M")*. We have: (.7(A* (w)), w) _ (a*(w)),w o j)
o j),w) _ (r(a(w)), w} _ (a*r*(ca), w) then:
A*r*(w) = jL(2) As A* is injective, it can be then easily deduced that r* is involutive and multiplicative, and therefore that r is an H-morphism.
3.4 Eymard Algebra 3.4.1 Lemma. The set B,(K) is a self-adjoint part of B(K). Proof. For all x in M and w in M*, we have:
(x*,w) = (x,w° o r.)-
by 1.2.5
and, by 3.3.1: IIA(w° o 1011=
IIA(w)II
By using 1.6.2(i) we then see that it*(Ba(K)) is stable by involution, and, thanks to 1.3.4, we get the result. 3.4.2 Proposition. (i) Any norm-one positive element of BA(K) is the limit, for the v(B(K), C*(K)) topology, of norm-one positive elements of A(K). (ii) The space BA(K) is the o(B(K),C*(K))-closure of A(K). (iii) The space BA(K) (resp. 7r*(BA(K))) is composed of the elements of the form (sµ)*(.(2) (resp. µ*(.Q)), where it is a representation of M* weakly contained in A, in the sense of [47], and Q an element of (Ap)*.
Proof. Any element of BA(K) vanishes on KerA (cf. 1.6.1(ii)). By ([25], 3.4.2(i)) any norm-one positive element of B,\(K) is thus the limit, for the o(B(K), C*(K)) topology, of elements of the form (s,\)* (fl), where .fl is normone positive in M*; at last 3.3.4 allows to deduce (i). It results from (i), by linearity that BA(K) is contained in the closure, for the a(B(K), C*(K)) topology, of A(K); on the other hand, since BA(K) is the annihilator of Ker A, it is o(B(K), C*(K)) closed, which completes the proof of (ii).
98
3. Representations of a Kac Algebra; Dual Kac Algebra
Let t be a representation of M* weakly contained in A and Si in (Aµ)*. By ([25], 3.4.4), (s,)*(.(l) is the o(B(K),C*(K)) limit of elements of the
form (SA)*(w) where w belongs to k+*, i.e. of positive elements of A(K). By using (ii), we can conclude that (sµ)*(,(l) belongs to BA(1K); by linearity this conclusion still holds for any Si. To prove the converse, let us note that those elements of the form (sµ)*(.(l)
with u weakly contained in A and Si is in (A,)* compose a vector space. Indeed if µ (resp. p') is weakly contained in A and Si (resp. Sly) belongs to (A,,)* (resp. (A,,)*), it is easy to check that: (sp)*(Q) + (8µ,)*(fl') = (Sµ®µ')*(S ® (2')
and that p ® µ' is weakly contained in A. By linearity, it is therefore enough to consider x in BA(K)+. Let y be the positive linear form on C *(K) such that x = y o A (cf. 3.3.4). The Gelfand-Naimark-Segal construction allows to associate to y a triple (f, l;, µ). Then y o A is a representation of M* and we have:
(,u(A(w))
((µ o
_ (y, A(w)) _ (y o A, ir(w))
_ (x, ir(w)) _ (7r*(x),w) Therefore, we have: 7r*(x) = (µ o'A)*(SO
and: x = (sµoa)*(S£) since it is clear that p o A is weakly contained in A, it completes the proof.
3.4.3 Lemma. The product of B(K) is o(B(K), C*(K)) separately continuous on the bounded parts.
Proof. Let 9i be a bounded family of elements of B(K) converging to an element 9 in B(K). For all 9' in B(K) and w in M*, we have: (ir(w), 9i * 9') _ (7r*(9i * 9'), w)
_ _ (i*(9i),w . r* (0')) _ (7r (w 7r* (0')), 9i)
which converges to: (7r(w . i*(9')), 9) = (ir(w), 9 * 9')
thanks to the same computation.
by 1.6.9
3.4 Eymard Algebra
99
Since the II9i * 9' I I are bounded by 110'11 sup II9i I I which is finite, by the
density of 7r(M*) in C*(K), we get that 9. * B' converges to 9 * 9' for a(B(1K), C*(K)). The left multiplication is dealt with in the same way.
3.4.4 Theorem. The sets A(K) and BA(K) and are norm-closed self-adjoint ideals of B(K). Specifically, B,\(K) is an involutive Banach algebra which we shall call the Eymard algebra associated to K.
Proof. Let ca be in k,, 9 in B(K) and w in M*. We have: by 1.6.9 and 1.6.1 (ii) = ((A x 7)* (w 0 9), w) by 1.4.3 _ ((A x 7r)(w), w 0 9) _ (y ra(w), 9 0 (Z') by 3.2.2 (ii)
(7r* (0 * (sA)*(w)),w) = (A*((w)ir*(9),w)
_ (A(w), (9 ®w) 0 7a) w) _ (A*((9 ®w) 0
From what we obtain that: 9 * (sA)*(w) = (sA)*((e ®(Z) 0''ir)
By 3.3.4, it follows that A(K) is a left ideal of B(K). As A(K) is self-adjoint and norm-closed, the first part of the theorem is secured. Now, let 9 be in BA(1K)+1. By 3.4.2 (i), 9 can be a(B(K), C*(K)) approximated by norm-one positive elements of A(K). Applying the first part of this proof, as well as 3.4.3 and 3.4.2 (ii), we find that, for all 9c in B(K), 9' * 9 belongs to BA(1K). By linearity, we can conclude that B,\(K) is a B(K)-left ideal; since, by 3.4.1 and 1.6.1 (i), B,\(K) is norm-closed and self-adjoint, the proof is completed. 3.4.5 Proposition. The restriction to BA(1K) of the Fourier-Stieltjes representation of B(K) is the transposed of the mapping kA from M* to C1(1K) (once BA(1K) is identified with the dual of C *(K)).
Proof. Let 9 be in (C (1K))* and w in M*. Then by 3.3.4, 9 o A belongs to BA(K) and we have: (/cir*(9 o ), w) _ (?r(w o /c), 9 0 (A(w o cc), 9)
_ (kA (w))), 9)
which completes the proof.
by 3.3.1
100
3. Representations of a Kac Algebra; Dual Kac Algebra
3.4.6 Eymard's Theorem ([46]). Let G be a locally compact group, and f an element of B(G). The following assertions are equivalent: (i) There exists a Hilbert space H, vectors C, q in H, and a unitary representation µG of G on H, weakly contained in the left regular representation, such that, for all t in G: f(t) _ (pG(t) 177) (ii) We haven
r sup{
l JG
If (t)h(t)dtl, h E L'(G), IIA(h)II < 1} < +oo J
The space of such functions is noted BA(G); it is a closed ideal of the FourierStieltjes algebra of G, and will be called the Eymard algebra of G. Moreover,
if f is in BA(G), its B(G)-norm is equal to:
sup{J If(t)h(t)dtl, h E L1(G), IIA(h)II G
:1}
The space BA(G) can be identified with the dual of C *(G) (the C*-algebra generated by the left regular representation of L'(G)), the duality being given, i f f(t) = (pG(t) 171), and h in C (G), by: (f, h) = (p(h)i 17])
where p denotes again the associated representation of C *(G) (recall that p is weakly contained in A). Moreover, for every w in £(G)*, the set:
A(G) = Is -+ (A(s-1),w)
(s E G)}
is a norm-closed ideal of BA(G). It will be called the Fourier algebra of G, and its dual is £(G). Proof. Using 3.4.2 (iii), we see that property (i) characterizes 7r*(BA(Ka(G))); by 1.6.3 (iii), it is the same for property (ii). So, we get BA(G) = 7r*(BA(Ka(G))) = /Ca7r*(BA(Ka(G))) and all other properties of BA(G) come then from 1.6.3 (iii) and 3.4.4. Let us now consider na7r*(A(Ka(G))). Using definition 3.3.4, it is the set of all elements which may be written, for all S2 in £(G)*: tCa7r*(sA)*(Q) = KaA*(Q)
In 1.3.10, we have seen that A*(J2) is the function s -i (AG(s), Si); by then IcaA*(Q) is the functions -' (AG(s-1),.f2).
3.5 Construction of the Dual Weight
101
So, we get A(G) = Kalr*(A(Ka(G))), and all properties of A(G) come from 3.4.4.
3.5 Construction of the Dual Weight In that paragraph, we consider the set '8 = a(I. fl I,,). 3.5.1 Proposition. Let w, w' be in I, fl I. The formulas: a(w)Ta(w) = a(w * w)
(i)
a(w)l = a(w°)
(ii)
allow us to equip B with a structure of left Hilbert algebra, dense in H.. Let us denote by fr the left multiplication of B. We have, for all w in I,, fl I"P:
(iii)
fr(a(w)) = A(w)
and the von Neumann algebra generated by fr(93) is equal to M.
Proof. (a) We have seen in 2.4.6 (iii) that iv fl I,,, is an involutive subalgebra of M,,. As a is a bijection from i. fl I,', to 93, we see that 93, equipped with T and d is an involutive algebra. (b) Let be in H V, orthogonal to '.B; by 2.4.5, we have, for all w1, w2 in IV:
0 = (a(w1 * w2) I ) _ (A(wl)*a(w2) I C)
by 2.4.6 (ii)
_ (a(w2)IX(wi)e) because of the density of a(I() in H. (2.1.7 (ii)), it implies A(w1)C = 0 for all wi in II; because of the density of I, in M* (2.1.7 (ii)), it implies, for all w in M*, A(w)C = 0, which, in turn, because of A being non-degenerate (2.6.3 (i)) implies C = 0. Therefore 93 is dense in H,. (c) For all wi fixed in I, fl P. the mapping a(w) -+ a(wi)Ta(w) is continuous from B to B. In fact, we have:
a(wi)ta(w) = a(wi * w) = A(wi)a(w)
by definition by 2.6.1 (v)
(d) For all wi, w2, w3 in I, fl I,,, we have: (a(wi)Ta(w2) I a(w3)) = (A(wi)a(w2) I a(w3))
= (a(w2)IA(wi)a(w3)) = (a(w2) I a(wi)Ta(w3)) = (a(w2) I a(wi)1Ta(w3))
by (c) by 2.5.3
by (c)
by definition
3. Representations of a Kac Algebra; Dual Kac Algebra
102
(e) Let
in H, , orthogonal to BTB. We have, for all wl, w2 in I,p fl I,,,: 0 = (a(wl)Ta(w2) 10 = (A(w1)a(w2) 10 = (a(w2)Ia(wl)e)
by (c) by 2.5.3
By (b) it implies A(wl)e = 0, since ag. fl i.1) is dense in H by (b); by continuity, it implies A(w)e = 0, for all w in M*; therefore because of A being non-degenerate, it implies e = 0; so, BTB is dense in H,p. (f) Let w be in I. fl I and x in T. fl OT<po,c. We have: (A,p(x) I a(w)0) = (A,,(x) I a(w°))
= (x*, w°)= (k(x),w) = (a(w) I A,p(tc(x*)))
by definition by 2.1.6 (ii) by 1.2.5 by 2.1.6 (ii)
because fl,poro = ,c(92,). Therefore, the mapping i has an adjoint, the restriction of which to A,p(9t,p fl'J2,po,c) is the mapping A,p(x) - A,p(,c(x*)). By 2.7.5 (iv), this adjoint mapping is densely defined, therefore i is closable. Following ([158], def. 5.1; cf. 2.1.1 (iii)), if we remark that (iii) has been
proved in (c), and that, thanks to 2.4.6 (iii), fr(B) generates M, we have completed the proof. 3.5.2 Definitions. We shall denote B' the right Hilbert algebra associated to B, V' the achieved left Hilbert algebra, and B° the maximal modular subalgebra of B" (cf. 2.1.1 (iii)).
We shall still note t and d (resp. b) the product and the involution on B" (resp. B'). We shall note E and F, the closures of D and b, with respective domains denoted to Di and Z*. In particular, we have, for all x in YL,, fl gtpo,c: FA,p(x) = Aw(n(x*))
If in Hp is left bounded with respect to f8, we shall still note r(e) the "left multiplication" by e ([14], def. 2.1). 3.5.3 Definitions. We shall note 0 the faithful, semi-finite normal weight on M canonically associated to B ([14], th. 2.11), and call 0 the dual weight associated to K. For all w in I,p fl I,', by 3.5.1 (c), A(w) belongs to fr(B) and therefore to 01,P fl 9 it. Moreover, for wl and w2 in I,p fl Io, we have: c (A(w2)*A(wl)) = (a(wl) I a(w2))
3.5 Construction of the Dual Weight
103
To the weight cp we associate the Hilbert space HO and the canonical oneto-one mapping AO : 910 - HO. We shall note 21 the left Hilbert algebra associated to 0, i.e. fl'7l ), which is isomorphic to B" (2.1.1 (iii)). More precisely, the mapping which, to every 6 in B", associates the vector:
fE = can be uniquely prolonged into a unitary operator from HW to H, still denoted by F. It will be called the Fourier-Plancherel mapping and will allow us to identify HO and H., and, through this identification of H., we have B" = 2l. Using the definition 2.1.6 (ii), we shall note a instead of a o.
3.5.4 Proposition. For all w in I., a(w) is left-bounded with respect to B, and we have:
(i) *(a(w)) = A(w) (ii) a(w) = Ao(,\(w)) (iii) for all a, y in 21', yTab is left-bounded with respect to B and we have:
fr(yTal) = A(wy,a) . Proof. Let wi be in I., w2 in i. fl I,,, l: in V. We have: A(w2)*'(e)a(wl) _ 7 (4)a(w2)a(w1) = fr'(e)a(w2 * wl)
by 2.6.1(v)
= fr(a(w2 * wl)) because w2 * wi belongs to Iw fl I._ A(w2 * w1)e by 3.5.1(c) _ A(w2)A(wl)e
As 1 is in the closure of \(I. fl 1) = $(B), we have: fr'(C)a(wl) = \(wl)C
which yields (i), and (ii) immediately, then (iii) follows by applying 2.1.7 (i).
3.5.5 Remark. Thanks to 2.1.1 (iv), we know that M is in a standard position in H; so, by 1.1.1 (iii), every element of M* can be written ,flap ( M for some vectors a, /3 in Hey. This element shall be written wa p.
3.5.6 Corollary. (i) The algebra £(G) is in a standard position in L2(G). (ii) The predual L(G)* is equal to the set {,fl f g I £(G), f, g E L2(G)}.
104
3. Representations of a Kac Algebra; Dual Kac Algebra
(iii) For all f in L2(G), let us put f (s) = f (s-1)- for all s in G. The set of all If * g, f, g E L2(G)} is the Fourier algebra A(G) defined in 3.4.6.
Proof. The assertions (i) and (ii) are just applications of 3.5.5 to Ka(G). In 3.4.6, A(G) has been defined as the set of functions s -- (.XG(5-1),w), for all w in £(G)*. But we have: (AG(s-1),Cof,g)
=
ig) =
JGf(st)g(t)dt =
JG.f(t)g(s-lt)dt
= JG f(t)(ts)dt _ (f * 9)(S) And so, (ii) implies (iii).
3.6 Connection Relations and Consequences 3.6.1 Proposition (Connection relations). The operations T and t are linked by the following relations: (i) For any a, -y in 2V and 8, b in Hip, we have:
(W(a 0 0) I7 0 6) = (a I k(7Ta1)b) (ii) For any a, 7 in W and /3, b in '.B', we have: (W(a (& l3) I7 0 b) = (/3Tbb 17Tab)
(iii) The set'.B'T'B' is included in App( /3, b in B', we have:
p) and, for any a,7 in Hip and
(W(a 0 0) I7 0 6) = (n(/3Tb)a I7) Proof. Combining 2.6.1 (iii) and 3.5.4 (iii), we get (i). Therefore, with the hypothesis of (ii), we have: (W (a 0,8) 17 0 6) = (/3 I ir(b)(7Tab)) = ( (b)*Q 17Tab)
_ ( (5 which is (ii).
)/3I7Tab) _ (/3Tbb I7Tab)
3.6 Connection Relations and Consequences
105
It can also be written as follows: (W(c, 0 Q) I7 (9 6) = (QTbb 17r I W)-f)
= (ir (a)(/T8) I7))
It follows that:
III (a)(/Tbb)II < sup{I(W(a 0 0) I7 0 )I, 7 E 2t',
11-t11:5 1}:5 IIali 11#11 IIbII
Thus, /3Tbb is left-bounded with respect to 2t (cf. 2.1.1 (iii)), and we can write: (W (a ®Q) 17 (9 b) = (7r(/3Tbb)a 17)
which, by continuity, still holds for any a, 7 in H.. This completes the proof.
3.6.2 Lemma. (i) The set B'TB' is included in A ,(gi, fl givoc). More precisely, for /3, b in B', we have: ir(t63
) = *(ws,R)*
and:
(ii) The space A,(91 fl
is a core for F.
Proof. Let /3, 6 be in B', a, 7 in HV. We have: (A*(wb,Q)*a 17) _ (a*(2'5,A),w7,a) _ (l3 I a(w7,a)b)
=(W(a0P)I7®b) (7r(/3Tbb)a I7)
by 2.6.1 (iii) by 3.6.1 (iii)
from what follows the first equality. For w in M*, we have, then:
(K(ir(atbb)),w) _ (,(a*(wd,Q)*),w) _ (A*(wS,A),w )
_ (a(w),wQ 6)
by 1.2.5
106
3. Representations of a Kac Algebra; Dual Kac Algebra
Therefore, we have:
x(i(8Tab)*) = A*(wa,b)* = 7r(dTa) by the first equality; as 91wo,c =
the proof of (i) is completed. As, by ([158], p. 17), 'B't B' is a core for F, (ii) is immediate.
3.6.3 Lemma. Let x be, in M. (i) For any a in Db, xa belongs to Db, and we have:
Fxa = rc(x)*Fa (ii) For any Q in Vi, x/3 belongs to Vi, and we have: Sx/3 = rc(x)*SQ
.
Proof. Let a be in A,P(91,p fl 91worc), w in I. fl I,*, x in M. We have: (xa I a(w)#) _ (xa I a(w°))
by 3.5.1 by 2.1.6 (ii)
_ (nA,1(xa),w)
_ w)
by 3.5.2(i)
_ (A,1(K(x*)F"a)*,w) _ (a(w) I ic(x*)F"a)
by 2.1.6 (ii)
Therefore xa belongs to Db and:
Fxa = tc(x*)Fa As A,p(Ol,p fl,pO1C) is a core for P by 3.6.2 (ii), we have proved (i).
Let a be in Db, /3 in Vi and x in M. We have: (xf I F"a) = (Q I x*F"a) = (/31 F"ic(x)a) = (ic(x)a I S"Q) = (a I ,c(x*)S73)
which completes the proof.
by (i)
3.6 Connection Relations and Consequences
107
3.6.4 Proposition. The modular operator A = z , is affiliated to M'. Proof. Let a be in Zo and x in M. We have:
xLa = xE a = Frc(x*)Sa by 3.6.3(i), because S'a belongs to Db = FSxa by 3.6.3 (ii)
= dxa as Bo is a core for a, we have xLi C dx, which completes the proof. 3.6.5 Corollary. For all t in R, we have:
Tat =(i®at)I'. Proof. For all x in fl, and t in R, we have:
I'vt (x) = aW*(ot (x) 01)Wa = aW*(Li'tx0-:t ®1)WQ = aw*(ast ®1)(x
by 3.2.2 (iv)
®1)(a-:t
(& 1)Wa
Now, by 2.6.1 (iv), W belongs to M 0 M and by 3.6.4, A't belongs to M', therefore, we have: 01)w(a-'t ®1)o ra (x) = a(Li't (& 1)W*(x = (1 (DLi't)aW*(x 0 1)Wa(1 ®L-it)
= (1
®ast)I'(x)(1
03-it)
by 3.2.2 (iv)
= (i ®at )r(x) which completes the proof.
3.6.6 Corollary. For any x in. M, we have:
(i) (ii)
rc(x*) = JxJ
(J ®J)W(J ®J) = W*
.
Proof. Let a be in 'Zio and x in M. We have:
xfa = xL1/2Sa = al/2x,Sa
= Li1/2,Sic(x*)a = JKc(x*)a
by density, we get (i). By (i) and 2.6.2, we get immediately (ii).
by 3.6.4 by 3.6.3 (ii)
108
3. Representations of a Kac Algebra; Dual Kac Algebra
3.6.7 Theorem. The modular operator 3 is affiliated to the centre of M; moreover, it is the Radon-Nikodym derivative of the weight cp with respect to the weight cp o r., in the sense of [114] (cf. 2.1.1 (v)).
Proof. By 2.7.6 (i), we have a` or. = ate for all t in R. The theorem 5.4 of [114] gives then the existence of a unique positive self-adjoint operator h, affiliated to the centre of M, such that cp o r. = cp(h.) (cf. 2.1.1 (v)). Let x be in'7t 0 such that: cr
(R.X(w)*A(w))
= allRa(w)ll2
by 3.5.2 (ii)
120
3. Representations of a Kac Algebra; Dual Kac Algebra
So, we have IIPa(w)112 = aIIRa(w)112 for all w in I.; so we get P = R (and so
P belongs to the centre of k), a = 1, c3^ = Op, which ends the proof of (ii). Now, E ® E is a faithful normal conditional expectation from M ® M to M ® M such that, for all X in AW®W((E ® E)(X)) = (P ® P)A,®,(X) So, for x, y in 91., we shall have:
A'P®'v((E 0 E)(r(y)(x ®1))) = (P ® P)A,p®,(I'(y)(x ®1)) = (P ® P)W(Aw(x) ® A,(y)) = (P ® 1)W(1® P)(Aw(x) ® A,(y)) because P belongs to Z(M) = (P ® 1)W(A,(x) ® A,(Ey)) = (P ®1)A,a®sv(I'(Ey)(x ®1))
As E ® i is also a faithful normal conditional expectation from M 0 M to M 0 M such that, for all X in %1.O.:
(9 i)(X)) = (P ®1)A ,®(X) We have then:
Aw®,((E ® E)(F(y)(x ®1)) = A,®w((E 0 i)(I'(Ey)(x 0 1))) and, therefore:
(E ® E)(F(y)(x ®1)) = (E 0 i)(.P(Ey)(x ®1)) By continuity, we get, for all y in 9'l.:
(E ® E)(P(y)) = (E 0 i)(F(Ey)) = F(Ey) and, by continuity again, we have:
(E®E)F=FE Let now x be in M; we have:
rc(Ex)P = J(Ex)JP
by 3.6.6 (i)
= J(Ex)PJ = JPxPJ = PJxJP
because P belongs to Z(M)
= Pic(X)P = Erc(X)P
because P belongs to Z(M) by 3.6.6(i)
3.7 The Dual Kac Algebra
121
We have then rc(Ex) = Erc(x), which ends the proof of (i) and of the proposition.
3.7.10 Proposition. Let K = (M, T, ic, cp) be a Kac algebra, R be a projection
of the centre of M such that r(R) > R ® R, tc(R) = R and KR be the reduced Kac algebra in the sense of 2.2.6. There is a canonical one-to-one H-morphism j from (KR)" to K which identifies (KR)" with a Kac subalgebra of K (cf. 2.2.7). More precisely, if r denotes the reduction x -+ xR of M on MR, we shall have, for all w in (MR)*: j(ARM) = .X(w o r) where AR denotes the Fourier representation of KR. Proof. Let us call I the projection R, considered as an element of £(Hv, We have, then:
I*I = R II* = 1H(PR
(x E 9p)
IAcp(x) = AVR(r(x))
r(x) = IxI* wy,q o r = WI*a,I* y
(x E M) (a, y E H(PR)
Moreover, if w is in I,pR, it is easy to check that w o r is in I,p and that:
a(w o r) = I*a(w) The reduction r is an 1-morphism, thus the mapping w - wor from (MR)* to M. is multiplicative and involutive; therefore, the set {.,(w o r), w E (MR)* } is an involutive subalgebra of M; let us call N its weak closure. Let 6 in Hip such that (/3 xS) = 0 for all x in N and S in HV. We have, for all a, y in HWR, all 6 in Hip: 0 = (Q 1 A(wI*y>I*a)S)
=(W(I*a0/3)1 I*y®S) =((I®1)W(I*a0/3)1-y®S) which implies, for all a in H,R:
(I0 1)W(I*a®/3)=0 or:
(R 0 1)W(I*a ®/3) = 0
by 2.6.1 (iii)
122
3. Representations of a Kac Algebra; Dual Kac Algebra
as R is in the centre of M, W belongs to M 0 M and RI* = I*, it implies: W(I*a ®/3) = 0 As W is unitary, it gives I*a ®/3 = 0, for all a in HER, which implies /3 = 0. Then, N is a non-degenerate algebra on Hg,; it is a von Neumann subalgebra w in (MR)*. We have: of if. Let y be in IA(w o r)Ap(y) = IAp((w o r o is ®( i)r(y)) = A,, ((Lo o /C ® 2)(r 0 r)r(y))
by 2.3.5
= AWR((w o n 0 r)r(ry)) = AR(w)AWR(r(y)) = AR(w)IAW(y)
therefore:
IA(w o r) = AR(w)I and:
RA(w o r) = I*)tR(w)I
By passing to the adjoints this equality yields that R belongs to N'. So R belongs to the centre of M, and, for all x,11 in 21y, we have:
(R& =
RC E 2[W
RC'
ReTRii = R(ETn)
Now, let z be in k such that zR = 0. For all , i in 21'' we shall have: zA(wM R,?) = zi(ReTR77b)
= zR*(6Trjb)
by 3.5.4 (iii)
by the above remarks
which implies, by continuity, for all 6, 77 in HH:
zA(wNR,)=0 or, also, for all y, a in HER :
z.\(wl.y I.a) = 0 that is, for all w in (MR)*:
zA(wor)=0 and zN = 0, which ensures z = 0 by the above results on N. So, the reduction N - NR is an isomorphism. Let us call 3° the inverse isomorphism, and for x
3.7 The Dual Kac Algebra
123
in (MR)", let us put j(x) = Z(I*xI). It is clearly a one-to-one homomorphism from (MR)" to k such that j(1) = 1. Moreover, we have, for all w in (MR)*:
j(AR(w)) = s(I*AR(w)I) = (RA(w o r)) = A(w o r) The range of j is therefore equal to N. Now, let w be in M. We have: (ra*(w),w) =
o r) = (A(w o r), w) = (jAR(w),w) = (AR*j*((Z1),w)
and then rA* = \R* j*. From what it is straightforward to prove that j* is involutive and multiplicative and therefore that j is an 1-morphism. For all t in R, we have: vf(A(w o r)) = .X(w o r o Lost) by 3.7.7 (iii) = A(w o Lr(a;t) o r) which belongs to N
therefore N is at -invariant. Let w in IVR. We have: c (i(AR(w)*AR(w)) = O(A(w o r)*A(w o r)) = 11a(w o r)112
= III*a(w)112 = Ila(w)112
_ IPR(AR(w)*.XR(w))
Therefore j(.1R(IWR) C
which implies that cp I N is a semi-finite weight.
Finally, we see that N is a Kac subalgebra of k, j is an l-isomorphism from (MR)" to N and, by 2.7.9 and the above calculation c o j = OR, which completes the proof.
Chapter 4 Duality Theorems for Kac Algebras and Locally Compact Groups
In that chapter, we obtain a duality theorem for Kac algebras, namely that the bidual Kac algebra is isomorphic to the original Kac algebra (4.1.1). From that, we can successively deduce that the Fourier representation A is faithful,
and that M* is semi-simple (4-1.3). We also see that the dual Kac algebra of the Kac algebra K8(G) constructed in Chap. 3 is the Kac algebra Ka(G) constructed in Chap. 2 (4-1.2). These results were found, independently, by the authors in [36], and Vainermann and Kac in [180]. Moreover, we obtain that the relative position of the von Neumann algebras M and M is such that (4.1.5):
MnM=MnM'=M'nk=M'nk' This result, from [136], leads, in the case of Ka(G), to Heisenberg's theorem (4.1.6). The crucial link with duality of locally compact groups is given by Take-
saki's theorem ([157]), which states that every symmetric Kac algebra K is isomorphic to the symmetric Kac algebra constructed from the intrinsic group of K (4.2.5). By duality, we get that every abelian Kac algebra K is isomorphic to the abelian Kac algebra constructed from the intrinsic group of K. Applied to a standard Borel group with a left-invariant measure (4.2.6), we get A. Weil's theorem [197].
Applied to K3(G), Takesaki's theorem leads immediately to Eymard's duality theorem ([46]), which states that G is the spectrum of the Fourier algebra A(G) (4.3.8), and, which, in turn, contains, in the commutative case, Pontrjagin's duality theorem (4.3.8). Eymard's duality theorem allows us to give a precise description of all the objects constructed from the symmetric Kac algebra K3(G); in particular, the Fourier algebra of K3(G) is L1(G), and its Fourier-Stieltjes algebra is M'(G) (4.4.1), which leads, in the commutative case, to Bochner's theorem (4-4-3). We then, after [38], characterize all the morphisms which realize the quasi-
equivalence of A with A x p, for all non-degenerate representations 1t, as
4.1 Duality of Kac Algebras
125
proved in Chap. 3 (4-5.6). When p is of dimension 1, we then get another characterization of the intrinsic group of a Kac algebra (4-5.8), due to De Canniere ([18]), which leads to Wendel's duality theorem ([199]) for locally compacts groups (4-5-9). To each couple of non-degenerate representations of M* and fl, respectively, we functorially associate a unitary operator belonging to the tensor product of the von Neumann algebras generated, as it was done in [40]; it is called the Heisenberg's pairing operator (4.6.2). For one-dimensional representations, i.e. for the intrinsic groups, we get a bicharacter in a situation similar to Heisenberg's commutation relation (4.6.7). This Heisenberg's pairing operator allows us to construct the extension of any non-degenerate representation of M* to a representation of the Fourier-Stieltjes algebra B(fc), just the same way non-degenerate representations of L1(G) are extended to M1(G) (4.6.8). This will be essential to define the arrows of the category of Kac algebras in Chap. 5. Chapter 4 ends with a Tatsuuma type theorem about Kac algebras (4.7.2), which gives, as corollaries, Ernest's duality theorem ([44]) and Tatsuuma's duality theorem ([168]) on locally compact groups.
4.1 Duality of Kac Algebras 4.1.1 Theorem. Let K = (M, T, rc, cp) be a Kac algebra. The bidual Kac algebra
K"" is isomorphic to K (equal if we identify Hp and Hcp), and the Fourier representations A and A are linked by: A=rcoA* A=koa* .
Proof. The von Neumann algebra M"" is, by definition, generated by the Fourier representation A which is equal to rcA* (cf. 3.7.3), up to the isomorphism between H. and HO; we thus have:
M""CM The fundamental operator W associated to K is equal to oW*o (cf. 3.7.3); so 2.7.4 (i) applied to K gives that, for any x in £(H), x belongs to M" if and only if (x 0 1)W = W(x 0 1). So, by 2.7.6 (v), we get M""' = M' and then M = M" Similarly, for any x in M"" = M, we have: F^^(x) = oW*o(1 0 x)oT%Vc by 3.2.2 (iv) applied to K = W(1 0 x)W* by 3.7.3
= I'(x) and:
by 2.6.3 (ii)
4. Duality Theorems for Kac Algebras and Locally Compact Groups
126
is"-(x) = Jx*J
by 3.3.1 applied to k
= K(x)
by 3.6.6 (i)
By 2.7.7 the two Haar weights cp and cp"" are proportional. Let /3, b be in 21',
such that /Tbb # 0. We have:
by 3.7.3 by 3.6.2(i)
= cp(7r(/3Tbb)*7r(/3Tb))
_ Ilap0 6)jj2
by 2.1.7 (i) applied to cp
by 3.5.2 (ii) applied to K Therefore cp"" = cp and the theorem is proved, using 3.7.3 applied to K and K.
4.1.2 Corollary. Let G be a locally compact group, and K9(G) the symmetric Kac algebra associated by 3.7.5. The dual Kac algebra K.,(G)" is equal to Ka(G) (when L2(G) and H(p, are identified).
Proof. It is a combination of 4.1.1 and 3.7.5. 4.1.3 Corollary. Let K = (M, T, n, cp) be a Kac algebra, ik = (M, I', k, 0), its dual Kac algebra. Then:
(i) The modular operator A is affiliated to the centre of k, and is the Radon-Nikodym derivative of cp with respect to c o k.
(ii) The Fourier representation ..\ is injective and, therefore, M* is semisimple.
(iii) For all t in R, L't belongs to the intrinsic group of K. (iv) Let x in 010 fl 910.p and 77 in 2to. Then the element (i ® (z,7)I'(x) belongs to 'J'l, fl 0110ok and we have:
0 wn)P(x)) =
(v) Let A be a von Neumann algebra, tb 'a faithful semi-finite normal weight on A+, and let X in 0'l,p®o fl %tV,®Ook and rl in 2to. Then the element
(i 0 i
0 I')(X) belongs to 9ip®O fland we have: ®i ®wn)(i ®I')(X )) _ (1 ® JA(wo_14,)J)A+G®cv(X )
4.1 Duality of Kac Algebras
127
Proof. As, by 4.1.1, K"" = K, the first assertion results of 3.6.7 applied to K. By 3.3.5, 0 is faithful, therefore so is A, thanks to 3.7.3. Applying this result to K and using 4.1.1, we get the second assertion. The third assertion is 3.7.7 (ii) applied to K = K" By applying 3.7.7 (v) to K, we get (iv). Let us assume 1177 11 = 1. As i ® i ®w,, is a conditional expectation, we get:
(1G ® k)(((i ®i ®wn)(i ®r)(X*)((i ®i ® wn)((i ®r)(X))))
Vy,r.
4.7.3 Corollary (Ernest's Theorem [44]). Let G be a locally compact group. The mapping aG : G --* W*(G) implements an isomorphism and a homeomorphism from G onto G(W*(G)). Proof. By 4.7.2, sA implements an isomorphism and a homeomorphism from G(W*(G)) onto G(1Ks(G)) and, by Eymard's theorem (4.3.2), AG implements an isomorphism and a homeomorphism from G onto G(Ks(G)). The results come from the formula AG = sA7rG.
4.7.4 Corollary (Tatsuuma's Theorem [168]). Let G be a locally compact group. Let x in W*(G). The two following assertions are equivalent: (i) there exists some s in G such that x = 7rG(s) (ii) for every continuous unitary representations y, v of G, we have: (a) sµ®v(x) = sµ(x) ®s,(x) (b)
sar,(x) # 0.
Proof. By 4.7.2 and 4.7.1, the second assertion is equivalent to x belonging to G(W*(G)), the corollary then results from 4.7.3.
Chapter 5 The Category of Kac Algebras
In what follows, K1 = (M1, T1, ic1, V1) and K2 = (M2, r'2, !2,'P2) are two Kac algebras, K1 = (M1, r'1, k1, 01) and 12 = (M2, r'2, k2, 1P2) their duals. In this chapter, we put on the class of Kac algebras a structure of category, by defining convenient morphisms (called K-morphims), as it was done by the authors in [40]. The definition is not straightforward and involves the von Neumann algebras generated by the preduals of dual Kac algebras (5.1.1). Thanks to the Heisenberg's pairing operator, it is then a routine to define a dual K-morphism, and we have now a duality functor in the category of Kac algebras (5.1.8). Moreover, the full subcategory of symmetric Kac algebras, will be equivalent, thanks to Takesaki's theorem as seen in Chap. 4, to the category of locally compact groups (5.1.4). Let now u be an H-morphism from (MI, fl, tcl) to (M2, T2, K2); we can define a K-morphism it from K1 to K2, called the extension of u (5.2.3), and give a characterization of those K-morphisms which are extensions (5.2.4). If the dual K-morphism u" is an extension too, the H-morphism u will be called strict (5.8.1), and we have then got another class of arrows, stable by duality (5.3.2), which is the category introduced by the authors in [36]. Strict IEI-morphisms are characterized (5.3-4); in both categories, isomorphisms are the same (5.6.8). The end of the chapter is devoted to other results about isomorphisms, due to de Canniere and the authors ([22]). Let us suppose now that the preduals (Mi), and (M2),k are isomorphic Banach algebras (nothing is assumed about the involutions); then the Kac algebra K1 is isomorphic either to K2 or to K2 (5.5.5), and the dual Kac algebra K1 is isomorphic either to K2 or to (K2)S (5.5.9). In the case of two locally compact groups G1 and G2, we recover Wendel's theorem ([198]) about isomorphisms between the Banach algebras L1(G1) and L'(G2) (which implies isomorphisms between G1 and G2) (5.1.11), and Walter's theorem ([194]) about isomorphisms between the Fourier algebras A(G1) and A(G2) (which implies isomorphisms between G1 and either G2 or G2P1') (5.5.12). Moreover, from that result, we can deduce the unicity of the co-involution of a Kac algebra (5.5.7).
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5. The Category of Kac Algebras
As a corollary, similar results occur for isomorphisms of the FourierStieltjes algebras B(K1) and B(K2) (5.6.6), and, in the case of two locally compact groups, we recover Johnson's theorem ([65]) about isomorphisms between the Banach algebras M1(G1) and M1(G2) (5.6.9), and Walter's theorem ([194]) about isomorphisms between the Fourier-Stieltjes algebras B(G1) and B(G2). Thus, these four different results, which used different types of proof, are there shown, as M. Walter guessed (see the introduction of [194]), to be actually the same property.
5.1 Kac Algebra Morphisms 5.1.1 Definitions. We shall call K-morphism from K1 to K2 an H-morphism a from W*(K1) to W*(K2). By transposition, we get an involutive Banach algebras a* from B(1K2) to B(K1). The class of the Kac algebras, equipped with these morphisms, thereby becomes a category. It shall be denoted by k. 5.1.2 Theorem. With the above notations and those of 4.6.8, there is a unique normal morphism & from W*(K2) to W*(Ki) such that: &a2 = *1a*
or also, and equivalently, such that: (a ®i)(Vlri,*i) = (i ®a)(V,r2ifr2) Moreover, & is a K-morphism from ]K2 to 1K1. By iterating the process, we find:
a=a
We shall say that & is the dual morphism of a. Proof. Let w2 in M2*. By 4.6.8, we have: 1 ®a*(sA2)*(w2))(VVrl,fq)
®w2)(i 0 As a(1) = 1 and sA2 (1) = 1, the operator (i 0 of W*(K1) 0 M2i and by 1.5.3, it implies that Thus, if we put: a = sfrla.(8X2).
0 a)(V,.1,fr1)
)(i 0 a)(V,ri,fri) is a unitary 'kla*(sa2)* is non-degenerate.
5.1 Kac Algebra Morphisms
163
we have: &(1) = 1
In another way, & is defined as to make the following diagram commute: al
, B(K2)
M2*
B(K1)
) W*(K1)
W2
W*(K2) We shall have: afr2(SA2)*(w2) = &'7r2(w2)
= irla*(3j2)*(w2)
by 4.6.8 by definition of &
Therefore &-r2 and *la* coincide over A(1[(2); as A(k2) is a an ideal of B(1K2) and as the restriction of aia* to A(1[K2) is non-degenerate, we easily get:
afr2 = fla* By definition of irl and *2, this equality can be equivalently written as follows, for all B in B(1[K2):
a(i 0 0)(V,r2,*2)
®B o a)(Vr1,*1)
which is also equivalent to: (& 0 i)(V7r2,fr2) = (i ®a)(V1,.1 )
or, thanks to 4.6.4, to: (z ®&)(Vr2,lr2) = (a 0 i)(Vr1,,1)
(*)
Let /3 be a normal morphism from W*(K2) to W* (Ki), such that /3*2 = frla*. For all w2 in M2*, we have: 13r2(w2) = Q7r2(SA2)*(w2)
= Fria*(SA2)*(w2) = alr2(SA2 )*(W2) = &12(W2)
therefore /3 = &, which yields the unicity of &.
by 4.6.8
by assumption by definition by 4.6.8
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5. The Category of Kac Algebras
Let &* be the transposed of & and 01 in B(K1). We have: 7r1(&-(e1)) _ (2 0 &*B1)(V*2,7r2)
(i ®B1)(a air1(B1)
by 4.6.8
by (*) by 4.6.8
Therefore, we get: 7r1&* = a*r1
and, as, by 4.6.10 (ii), 7^r2 is faithful, we obtain that, as ail, &* is an involutive algebra morphism.
By transposing, recalling that &(1) = 1, we get the fact that & is a morphism of Kac algebras. Finally, it is clear, by (*), that a"" = a, which completes the proof.
5.1.3 Theorem. The correspondance which associates to any Kac algebra its
dual Kac algebra (as defined in 3.7.4), and to any morphism the dual Kmorphism (as defined in 5.1.2), is a duality functor of k into itself. It shall be denoted by D.
Proof. Let K1i K2 and K3 be three Kac algebras, a be a morphism from K1 to K2 and 8 be a morphism from K2 to K3. Let us consider the morphism Pa from K1 to K3. By using 5.1.2 repeatedly, we get:
(/3a 0
Therefore, we have (la)" completes the proof.
(Q ®i)(i 0 (90i)M2,112 ) which, because of the already known results,
5.1.4 Theorem. Let G1 and G2 be two locally compact groups and m be a continuous morphism from G1 to G2. Then: (i) There exists a unique K-morphism, denoted by Ka(m), from Ka(G2) to Ka(Gl) (i.e., here, an IRI-morphism from Ml(G2)* to Ml(G1)*), the transposed of which (it is an involutive Banach algebras morphism) is the mapping from M1(Gl) to M1(G2) that sends every measure of M'(Gl) on its image by m. (ii) There exists a unique K-morphism, denoted by K3(m), from Ks(Gl) to Ke(G2) (i.e. an ]El-morphism from W*(Gi) to W*(G2)) such that, for all g in G1: Ks(m)7rl(g) = 7r2(m(g))
5.1 Kac Algebra Morphisms
165
where 7r1 and ire stand respectively for the universal representations of G1 and G2. (iii) Let us denote 1Ca (resp. A.8) the full sub-category of K made up of the abelian (resp. symmetric) Kac algebras. The mapping which associates to a locally compact group G the Kac algebra Ka(G) (resp. K8(G)), and to a continuous morphism of groups the morphism Ka(m) (resp. K8(m)) as above defined, is a duality (resp. an equivalence) functor between the category of locally compact groups equipped with the continuous morphisms and the category )Ca (resp. 1C8); it shall be denoted by lea (resp. K8). (iv) We have: K8 = D o Ka .
Proof. The mapping g -+ 7r2(m(g)) is a continuous representation of G1 in W*(G2); thus there exists a normal morphism, denoted by ][he(m), from W*(Gl) to W*(G2) such that, for all g in Gl: K8(m)iri(g) = 12(m(g))
In particular, we have: K8(m)(1) = 1
by using 1.6.8, we immediately check that K3(m) is an H-morphism; the unicity is trivial, which completes the proof of (ii). The transposed of the dual morphism, i.e. (K3(m)")*, is an involutive Banach algebra morphism from M1(G1) to M1(G2) such that, for all y in M1(Gl), we have: 7r2((K8(m)^)*(u)) _ K8(m)(*1(,u))
by 5.1.2
2(m(g))d)
= JG1
From where we immediately get that (K8(m)")*(p) is the image measure m(µ); starting from this equality, by transposing and dualizing, the unicity in (ii) implies the unicity of this morphism Ka(m), which yields (i). Let /3 be a morphism from K3(Gl) to K8(G2). It is clear that ,Q will map the intrinsic group of W*(Gl) in the intrinsic group of W*(G2). As, by 4.7.3, these groups are isomorphic both algebraically and topologically to G1 and G2 respectively, there is a continuous morphism m from G1 to G2 such that, for every g in G1, we have: 7r1(g) = ir2(m(g))
which is nothing but to say that:
Q=Ks(m)
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5. The Category of Kac Algebras
As each symmetric Kac algebra is of the form K.(G) (4.2.5 (i)), it follows that Ks is an equivalence between the category of locally compact groups and 1Cs. The ends of (iii) and (iv) are straightforward.
5.1.5 Remark. With the above notations, and identifying B(K (Gi)) to B(Gi), for i = 1,2, it is easily checked that for all f in B(G2), we have:
Ks(m)*(f) = f o m .
5.2 IEI-Morphisms of Kac Algebras 5.2.1 Lemma. Let K1 and K2 be two Kac algebras, S be an 1-morphism from (M1, j'1,,1) to (M2, F2, Kc2) and y be a non-degenerate representation of Ml*. Then p o 5* is a non-degenerate representation of M2* on Hµ, the generator of which is equal to (i 0 8)(Uµ). Proof. Let W2 in M2*. By definition of the generator Uµ, we have: A 0 8*(w2) = (i 0 25)(UU) = (i (9 w2)(z 0 8)(U,)
As we have 8(1) = 1, the operator (i®8)(Uµ) is a unitary of A.OM2i by 1.5.2, it is the generator of the representation p o b*, which is then non-degenerate. 5.2.2 Theorem. Let K1 and K2 be two Kac algebras, 8 be an H-morphism from
(Ml,Tli,cl) to (M2,I'2,K2), S* be the involutive Banach algebras morphism from M2* to Ml* obtained by transposing b, W*(S*) be the homomorphism from W*(K2) to W*(K1) obtained by applying the functor W* to 5* (i.e. such that W*(S*)7r2 = 7r1S*). Then, we have: (i) W*(5*) is a K-morphism from R2 to K1;
(ii) the involutive algebra morphism W*(S*)*, from B(K1) to B(K2), obtained by transposing W*(S*), is the unique Banach space morphism which makes the following diagram commute:
W(6.).
B(K1) B(K2) lc17r1.
rc2a2.
I
Jll.
M1
5
1 .l
M2
where iri stands for the universal representation of Mi*, and Ki1ri for the Fourier-Stieltjes representation of B(Ki) (i = 1, 2);
5.2 H-Morphisms of Kac Algebras
167
(iii) The mapping W*(S*)" is the unique morphism from K1 to K2 which makes the following diagram commute: W*(b*)"
W*(Ki)
W*(K2) ea2 S
M1
M2
Proof. By definition, W*(S*) is a normal morphism from W* (K2) to W*(K1) such that: W*(6*)7r2 = 7r1S*
As, by 5.2.1, 7rl S* is non-degenerate, we get:
W*(S*)(1) = 1
and by transposing the above equality, we find: 7r2*W*(S*)* = S7rl*
and, by hypothesis: /c27r2*W*(S*)* = 6Kj7r1*
As, by 1.6.9, ,C27r2* is faithful, we get the unicity of W*(S*)*; we may show, in
the same way, that W*(S*)* is an involutive Banach algebras morphism; by transposing, we get that W*(S*) is a Kac algebra morphism, which completes the proof of (i) and (ii). Let wl in Ml*. We have: by 4.6.8
8A
)*(w1)
by 5.1.2
= /27r2*W*(S*)*(sA1)*(wl) by 4.6.10(iii) by (ii) = SIc17r1*(sA1)*(w1) = by 4.6.9(i) = SsA1*1(W1)
Therefore, we have: sa2W*(S*)" =
Conversely, let a be a K-morphism from K1 to K2 such that: sae a
SsA1
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5. The Category of Kac Algebras
by transposing, it comes: a*(3a2)* = (s0*6* _
And thus, a* and (W*(b*)")* coincide over A(II52); for A(1[52) being an ideal of B(1[52) and for the restriction of (W*(6*)")* being non-degenerate, we get the unicity and therefore (iii).
5.2.3 Definition. With the above notations, we shall say that the K-morphism
W*(b*)^ from K1 to K2 is the extension of the H-morphism 6; it will be denoted by S. The dual K-morphism W*(6*) from 1[52 to K1 will be called the coextension of 6. The 1111-morphism 6 being given, the K-morphisms b and 6" are respectively
characterized by the equalities: saz6
6sai
3°7r2 = 7r16*
5.2.4 Proposition. Let a be a K-morphism from K1 to K2. The following assertions are equivalent: (i) The representation of (M1)* in M2 defined by is quasi-equivalent to a sub-representation of the Fourier representation A1. (ii) There exists an H-morphism 6 from (Ml, F1, ic1) to (M2, F2, n2) such that a is the extension of 6. Then the H-morphism 6 is unique.
Proof. Let us assume (i). It means that there is a normal morphism 6 from Ml to M2 such that b(1) = 1 and:
N ail = 6A1 = 6sA1 i1 which gives:
s-az a=bs-ai Thanks to 3.3.3 it can easily be checked that 6 is an 1111-morphism; with 5.2.2 (ii) we then get (ii). Let us assume (ii). By 5.2.2 (iii) and 5.2.3, it implies:
ss-z a=6Al therefore:
sA2afr1 = bbl
which is nothing but (i).
5.2 H-Morphisms of Kac Algebras
169
5.2.5 Proposition. Let a be a K-morphism from K1 to K2. The following assertions are equivalent: (i) The representation of M2* in M1 defined by kii1*a*(sA2 )* is quasiequivalent to a sub-representation of the Fourier representation a2. (ii) There exists an H-morphism 6 from (M2, P2, k2) to (M1, f1, k1) such that a is the co-extension of 6. Then, the H-morphism b is unique.
Proof. Let us assume (i). It means that there is a normal morphism 6 from M2 to M1, such that 6(1) = 1, and that we have: 6A2 = k1*1*a*(sA2 )*
by 4.6.10 (iii)
Sj1ir'a*(sA2)*
by 5.1.2 by 4.6.8
s &7r2(SA2)* 3A1 &7r2
Thanks to 5.2.4, we then see that b is an H-morphism from (M2, 42, k2) to (M1, r1, k1) such that & = 6, which yields (ii). Let us assume (ii). By 5.2.3 and 5.2.2 (ii), we have: kl'kl*a* = 6k2i2* and therefore: 6k2r2*(sA2) = 6A2
by 4.6.9 (i) applied to K2
which completes the proof.
5.2.6 Proposition. Let G1 and G2 be two locally compact groups, u be a continuous morphism from G1 to G2 and Ka(u) and K3(u) be the morphisms defined in 5.1.4 (i) and (ii). Then, we have: (a) The following assertions are equivalent: (i) Ka(u) is an extension. (ii) K8(u) is a co-extension.
(iii) There exists an H-morphism Ha(u) : Ha(G2) -i Ha(G1) such that, for all f in L°°(G2): Ha(u)(f) = f o u
(iv) The image of the left Haar measure on G1 by u is absolutely (v)
continuous with respect to the left Haar measure on G2. The morphism u is strict, and has an open range.
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5. The Category of Kac Algebras
(b) The following assertions are equivalent: (i) K8(u) is an extension. (ii) Ka(u) is a co-extension.
(iii) There exists an H-morphism 18(u) : 18(G1) -+ H8(G2) such that, for all s in G2: Hs(u)(AG,(s)) = AG2(u(s))
(iv) The representation AGZ o u of G1 is quasi sub-equivalent to the representation AG1 (v) The morphism u is strict, and has a compact kernel. Proof. It is clear that (a)(i) and (ii) (resp. (b)(i) and (ii)) are equivalent. Let us assume (a)(i). From 5.2.5, it follows that there is an H-morphism 8 from L°°(G2) to L°O(G1) such that 8(f) = f o u for all f in B(G2). This equality can be extended, by norm continuity, to all continuous bounded functions f on G2, and then, by ultraweak continuity, to all f in L°O(G2). So, we have (a)(iii). Let us suppose now (a)(iii) and call cpj the Haar weight on LO°(Gi) (for i = 1, 2). It is immediate that W1 o H,, (u) is a semi-finite, normal trace on L°O(G2); therefore, there exists a positive element g, affiliated to L°D(G2) such that V 1(f o u) = c02(fg) for all f in LO°(G2), which implies (a)(iv). Let us suppose now (a)(iv); it is then clear that the application f -, f o u defined from L°°(G2) to L°°(G1) is a normal morphism; it is easy to check that it is an 1-morphism, whose extension is Ka(u). So we have proved that (a)(i)-a(iv) are equivalent. Let us assume these properties. Coming back to the H-morphism Ha(u), we see that this morphism may be decomposed into a reduction (which, by 4.3.6 (ii), is the restriction L' (G2) - L' (G'), where G' is an open subgroup of G2), and an 1-isomorphism from L°°(G') to a sub co-involutive Hopf-von Neumann algebra of L°°(G1). By 4.5.10(i), there is a normal subgroup H of G1 such that this sub von Neumann algebra of L°°(G1) is isomorphic to L°O(G1/H), and, using 4.3.5, we get that G' is isomorphic to G1/H. It is easy to see that we have just got the canonical decomposition of u, which is then strict and with an open range, which is (a)(v). Let us now assume (a)(v). Let w be in £(G2)*, f , g in 1C(G1). Then, we have (where, for sl in G1, s1 means its class in Gl/Keru): Cw o u(s1)f(s1)9(s1)ds1 1G1
=
Cw,
r G1
AGZ(u(s1)*)f(s1)9(s1)ds1\)
/
d. 1 J AG2(u(t)*)f(t)(t)dt} = Cw J l/Keru lKeru
5.2 1 Morphisms of Kac Algebras
_ (w,
JrG1 /Ker u AG2
_ (w,J
171
(u(sl)*)d31 LKeru)
AG2(s2)*h(s2)ds2) G2
by changing the variable u(sl) = s2 and defining the function h by: I
h(u(si)) =
f
f(t)g(t)dt
1 Ker u
h(s2) = 0 if s2 does not belong to u(Gi)
It is then clear that h is continuous on a compact, and is null outside it. So fG2 AG2(s2) h(s2)ds2 belongs to the definition ideal of the Haar weight on G(G2) (3.6.11). By 3.1.3, the representation w - Gw o u of G(G2)* is quasiequivalent to a sub-representation of the Fourier representation of Ke(G2), that is of the Gelfand representation of G(G2)*. So we get (a)(i), by 5.2.5. The equivalence of (b)(i), (b)(iii) and (b)(iv) is a corollary of 5.2.4. Let R = support IH[8(u); from 1.2.7 and 4.5.10 (iii), we see that Ks(Gl)R is isomorphic to K8(G1/K), where K is a compact normal subgroup of G1, and, from 4.3.6(i), we see that H8(u)(G(Gi)) is generated by all AG2(s), where s runs into a closed subgroup of G2. It is then easy to see that K = Ker u, and that G' = Imu; as ][1e(G1/K) is isomorphic to 1113(G'), we see, by 4.3.5, that u is strict, which is (v). Let us now assume (b)(v). Let f be in L1(Gi), g, h continuous functions on G2, with compact supports. Then, we have:
JG2 (Ll AG231))f(s)si) g(32)h(s2)ds2 l (I 9(u(sll)s2)h(s2)ds2J f(sl)dsl 1G1
G2
= f f(s1)k(s1)ds G1
with: k(si) = JG2 9( u(sl )52)h(s2)ds2
It is clear that k is continuous with compact support; so k belongs to the definition ideal of the Haar weight on L°°(Gi). So, using 3.1.3, we see that the representation f -> f G, AG2(u(si)) f(sl)dsl of L1(G1) is quasi-equivalent to a subrepresentation of the Fourier representation of Ka(G1), that is the left regular representation AG1. So we get (b)(iv).
172
5. The Category of Kac Algebras
5.3 Strict ]EI-Morphisms 5.3.1 Theorem. Let K1 and K2 be two Kac algebras, u be an 11-morphism from (Ml, l'1i icl) to (M2, T2, K2) and u the extension of u. The following assertions are equivalent: (i) The morphism i is the co-extension of an ]H[-morphism u from the Hopf-von Neumann algebra (M2iT2, k2) to (M1,P1,K1); (ii) There exists a von Neumann algebra morphism v from M2 to Ml such that, for every w in M2*, we have: v(A2(w)) = .1(w o u)
Then, the morphisms v and u are equal. Proof. By definition, the mapping u : W*(kl) --+ W*(]k2) satifies:
3ou=uos-Al "2 therefore, we have: (sA1 )* 0 u,k = (u)* 0
and, also: lc1ir1*(sAl)*u*
_ Alu*
by 4.6.9(i)
the theorem is then a direct consequence of 5.2.5. 5.3.2 Definition. Every H-morphism verifying the conditions of 5.3.1 will be
said to be strict. Given a pair of Kac algebras, the extension operation is clearly putting strict ]f1-morphisms, and K-morphisms which are both an extension and a co-extension, into a bijective correspondance. Equipped with the class of strict 1-morphisms, the Kac algebras form then a category with
a duality, the dual of a strict H-morphism u being then defined as the ][morphism u which verify, for all w in (M2)*: u(A2(w)) = .Xl(w o u)
We have already met some strict ID[-morphisms, namely the H-isomorphisms
(3.7.6), the reduction of a Kac algebra (3.7.10) and the injection of a Kac sub-algebra into a Kac algebra (3.3.8). 5.3.3 Proposition. Let G1 and G2 be two locally compact groups and m be a continuous morphism from G1 to G2. The following assertions are equivalent:
5.3 Strict H-Morphisms
173
(i) The morphism Ka(m) is both an extension and a co-extension. (ii) The morphism K3(m) is both an extension and a co-extension. (iii) The morphism m is strict, has an open range and a compact kernel. If they are satisfied, it is possible to define the following two strict Hmorphisms:
Ha(m) : L°°(G2) - L°O(GI) such that Ha(m) f = f o m for all f in L°O(G2) He(m) : C(Gl) -- £(G2) such that Hs(m)AG,(s) = AG2(m(s))
for all sinG1. Proof. It is a straightforward consequence of 5.2.6 (a) and (b) and 5.3.2. 5.3.4 Theorem. Let K1 and K2 be two Kac algebras and u be an H-morphism from (M1, TI,'c1) to (M2, T2, K2). The following two assertions are equivalent: (i) The H-morphism u is a strict H-morphism. (ii) The subalgebra u(Mi) is Qt 2 -invariant and the restriction of 'P2 to u(Mi) is semi-finite.
Proof. Let us assume (ii). Let Ru be the support of u. The morphism u can be decomposed into three components, u = i o a o r, where i the injection of u(Mi) in M2, a is an H-isomorphism of MIR, onto u(MI) and r is the reduction Ml -- MIRu. We know that MIRu and u(MI) can be equipped with Kac algebra structures and that r, i and a are strict H-morphisms; therefore, by composition, so is u, which brings (i). Conversely, let us assume that u is strict. With the same definitions as above, let us decompose u into the product j o r where j = i o a, i.e. j is the H-morphism from M1Ru to M2 such that j(xRu) = u(x) for all x in Ml. The dual strict H-morphism r" is injective and allows the identification of (M1Ru )^ and r((M1Ru)"); it is defined, by 5.3.2, by rA1R = .t1r*, where AIR is the Fourier representation of M1Ru . We have: r"A1Rj* = .Xir*7* = Alu* = u)2
(*)
by definition of the dual strict H-morphism u. As r" is an injective homomorphism, we get, for every w2 in M2*: IIAlRi*(w2)II = IIua2(w2)II
Let wl in MI*. Thanks to Kaplansky's theorem, we have: IIu*(u'1)II = sup{I(u*(wl),A2(w2))I, w2 E M2., IIA2(w2)II C 1} = sup{ I(uA2(w2),u'1)I, W2 E M2*, IIa2(w2)II < 1}
(**)
174
5. The Category of Kac Algebras
= sup{I(ra1Ri*(w2),w1)I, w2 E M2*, IIa2(w2)II 5 1} by (*) = sup{I(A1Rj*(w2),T*(w1))I, w2 E M2*, I1A2(w2)II : 1} 0
because, by 5.4.2 (ii), el is a Jordan isomorphism, which completes the proof.
5.6.2 Lemma. (i) Let K be a Kac algebra. We define a set Q by:
Q = {Q E W*(K); Q projection and Q # 1, s,rx,(Q) < Q ® Q}
5.6 Isometries of Fourier-Stieltjes Algebras
185
Then Q has a greatest element and:
max Q = 1 - supp s,\ (ii) With the constructions and notations of (i) associated to the two Kac algebras K1 and K2, we have: t1(Q1) = Q2
.
Proof. Because A # 0, it is clear that the projection 1 - supps,\ is different from 1. Moreover, we have:
(sA ® i)s,rx,r(1 - supp s,\) = s,\x,r(l - supp s,\) = (1 -supps,\) = 0
by 1.6.4 (i)
by 3.2.2(iii)
Therefore s,rx,r(1 - suppsA) belongs to Ker(s,\ ® i) and: s7rx,r(1 - supp sA) < (1 - supp sA) ® 1
(*)
By, 3.3.3, we also have:
sAs*(1 - suppsA) = ksA(1 - supp sA) = 0 Therefore:
s*(1 - supp s'\) < 1 - supp s'\ by s* being involutive, we get in fact:
s*(1 - supp s,\) = 1 - supp sa
**)
We then can write down:
s7rx7r(1 - supps,\) = saxrs*(1 - suppsa) = c(s* ® s*)s,rx,r(1 - supp sA) c(s*(1 - supp sa) ® 1)
= 10 (1 - supp s,\) Finally, by using again (*), it comes:
3,rx,r(1 - supp sa) < (1 - supp s,\) ®(1 - supp s.\) therefore 1 - supp sa belongs to Q.
by 1.6.6 by (*) by (**)
186
5. The Category of Kac Algebras
Let Q be in Q. We have: PsA(Q) = 'YAsA(Q)
= S'saxa(Q) = c(sa 0 sa)sirxa(Q) < c(sa 0 sa)(Q 0 Q) = sA(Q) 0 sa(Q)
by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.4 (iii) by hypothesis
< sa(Q) ® 1
It then results from 2.7.3 (ii) that s.\(Q) is either equal to 0 or 1. Let us assume s.\ (Q) = 1; it is equivalent to Q > suppsa, which implies:
Q+(1-suppsa)>1 and:
s7rxr(Q) + s,rx1r(1 - supp sA) > 10 1 and then:
Q®Q+(1-supps')0(1-supps,\)> 101
And:
((1 - Q)®supp s,\)(Q®Q + (1 - supp sa)®(1 - supp sa))((1 - Q)(&suppsa) > (1 - Q)®suppsa which leads to:
(1-Q)®suppsa=0 which is impossible, Q being different from 1, and supp sA different from 0. Therefore we must have s,\ (Q) = 0, which is Q < 1-supp s,\ and it completes the proof of (i). Let now Q be in Q1. As £1 is a Jordan isomorphism, Bl(Q) is a projector of W*(K2). Moreover it is not equal to 1, because 21(Q) = 1 would obviously be equivalent to Q = 1. Now, let 9 and 9' be two positive elements of B(K2). We have: (s,r2 x, vi (Q)), 0 ® 9') _ (sire xir2 (P(Q)t(1)*), 9 ®9)
= (s"z xay & Q)V(1)* ®e(1)*), e ®e')
_
(s-,"xIrz(Q(Q)),L(1)*
by 5.4.1 by 5.6.1 (i)
9®L(1)* 9')
_ (Q(Q), (L(1)* 9) * (L(1)* - 0'))
_ (Q,T((L(1)* . 9 * (L(1)* . 9'))) _ (Q, T(L(1)* 9) * T(2(1)* 9'))
by hypothesis
5.6 Isometries of Fourier-Stieltjes Algebras
187
= (s,.2 x1r2 (Q), T(2(1)* - 0) 0 T(2(1)* . e')) < (Q ® Q, T(2(1)* 0) 0 T(2(1)* 9f)) by assumption
_ (Q, T(2(1)* - 0))(Q,T(2(1)* . 01)) = (Q, (21)*(0))(Q, (6)* (0'))
by 5.6.1 (iii)
_ (21(Q), 9) (21(Q), Of )
_ V1 (Q) ®21(Q), 9 ®B1)
So, we get: sir2xirr2Vi (Q)) C 21(Q) ®21(Q)
and therefore Pi(Q) belongs to Q2.
Therefore we get 21(Ql) C Q2. As 21 is bijective, we could prove 21 1(Q2) C Q1 the same way, which completes the proof of (ii).
5.6.3 Proposition. With the above notations, we have:
(ii) (iii)
supp s'\1) = 1 - supp s'\2 2(Kersa1) = Kers,\2 T(A(K2)) = A(K1) .
Proof. Let us apply 5.6.2 to prove (i), considering that 21 preserves the order. The ideal Ker sal is generated by the projection 1 - supp sal . Let x in W*(K1). As 21 is a Jordan isomorphism (5.4.2), we have:
Li(x(1 - supp sx1) _'1(21(x)21(1 - supp s,\1) x- 21(1 - supp sa1)21(x)) = 21(x)(1 - suppsa2) by (1) As 21 is bijective, we get 21(Ker sat) = Ker x1\2 . And 2(1) being unitary and Ker 3A2 a bilateral ideal, it completes the proof of (ii). Let 0 in B(K2); by (ii), T(6) vanishes over Ker S,\2 if and only if 0 vanishes over 2(Ker s,\l) = Ker sa2, which gives the result, thanks to 3.3.4.
5.6.4 Notations. The restriction of T to A(K2) satisfies the hypothesis of 5.5.10. There is thus an element u of the intrinsic group of k2 and an ]isomorphism iP from K2 to K1 or K1 such that, for all 0 in A(K2), we have: 'P7uK21r2*(0) = iciiri (TB)
Let us determine u more accurately; the mapping (sA1)* 'T(s,\2)* is an isometric linear bijection from (M2)* to (Ml)* which shall be denoted by T.
5. The Category of Kac Algebras
188
Let P : A - M2 its transposed. By 5.5.9, we get u = 1(1); and by
transposing the relation T(s,\2)* = (sal)*T which defines T, we get that 31\2P = Psal, by definition of t and P. Then, we have: U = P(1) = sa2(P(1))
5.6.5 Lemma. With the above notations, we have, for all 9 in B(K2): k17r1*T(9)
Proof. To simplify, we shall put 7 = 7sa2 (1(1)). In 5.6.4, the above relation has
been proved for 9 in A(K2). Now let w in .1%I2*. Let us recall that (sae)*(w) is the generic element of A(K2) and that A(K2) is a bilateral ideal of B(K2) (3.4.4). By applying 5.6.4, we then find: 1fi(7(K27r2*(9 * (s,\2)*(w)))) = ic17r1*(T(9 * (sA2)*(w)))
or: (457I27f2*(e))('P71c27r2*(sa2)*(w)) _ (k17r1*T(9))(l17r1*T(sa2)*(w))
and, by using 5.6.4 again: (i17rl*T(9))(457tc27r2*(sa2)*(w))
which, by 4.6.9 (ii), can also be written: ('7ic27r2*(9))(4i7A2(w)) = (K17r1*T(9))(4P762(w))
by having 2(w) converging to 1, we complete the proof. 5.6.6 Theorem. Let K1 and K2 be two Kac algebras. We assume that there exists a multiplicative, isometric, linear, bijective mapping T from the FourierStieltjes algebra B(K2) on B(K1). Then, there exists an El-isomorphism from
K2 onto K1 or K. More precisely, if t stands for the transposed of T, we have:
(i) The operator sa2(P(1)) belongs to the intrinsic group of K2. (ii) There is an ]Hl-isomorphism 4i from K2 onto K1 or K1 (in the first case Pl is a von Neumann algebra homomorphism from W*(Ki) to W*(K2), in the second case it is an anti-homomorphism) such that, for all 9 in B(K2), we have: !P7aa2(t(1))X272*0) = ic17r1*T(9)
5.6 Isometries of Fourier-Stieltjes Algebras
189
Proof. It is enough to put 5.6.1 (i) and 5.6.5 together.
5.6.7 Corollary. Let K1 and K2 be two Kac algebras. Let W be a normal isomorphism from W*(Ki) onto W*(K2) such that: 3702 X 7r2 T1 = (W ®W) s ,.1 X7r1
(i.e. such that W respects the canonical coproduct of W*(Ki) and W*(K2)). Then, there exists an H-isomorphism if from K2 onto K1 such that, for all w in Ml*, we have: W(ir1(w)) = 7t2(w o f)
We have also:
s*2w = wsi1 .
Proof. Let us apply 5.6.6 to the transposed mapping W* = B(K2) -i B(Kl). As W is multiplicative, we are in the first case, furthermore, as ll(1) = 1, there is an H-isomorphism -P from K2 onto K1 such that, for all 0 in B(K2), we have: 4i(tC27r2*(e)) = t17r1*(T/*(e))
Because Ifiic2 = tcliP, it can also be written: !F(1r2*(9)) = 7r1*(T*(8))
Therefore, for all w in Ml*, we have: (W(ir1(w)), 8) _ (w, ir1*+,*(e))
_ (w, 1r2*(B)) _ (w o t,-7r2-(B)) _ (7r2 (w o f), B)
which gives the first result. We can see that: s*2Wir1(w)=s7r27r2(wo4i)=fr2(wo4i)=7r2(Wo0oKc2)=7r2(w0 K1 0
=rl(woK1) _ W*1(w) Wsa1 -7r1(w)
which gives the second result, by the ultraweak density of 7rl(Ml*) in W*(Ki).
190
5. The Category of Kac Algebras
5.6.8 Corollary. Let K1 and K2 be two Kac-algebras, W a K-isomorphism from K1 to K2 (i.e. an IEII-isomorphism from W*(Ki) to W*(K2)). Then there exists an IIII-isomorphism u from (Ml, F1, rc1) to (M2, I'2, K2) such that W is the extension of u. Therefore, K-isomorphisms are H-isomorphisms.
Proof. It is a particular case of 5.6.7. 5.6.9 Corollary (Johnson's Theorem [65]). Let G1 and G2 be two locally com-
pact groups. Let T a multiplicative, linear, bijective mapping from M'(Gi) to M1(G2). Then there exists: (i) a character X on G2 (ii) a bi-continuous isomorphism a from G2 to G1 such that for all measure p of M1(G1) we have: T1u = Xa-'(,U)
Proof. By 4.4.1 (ii), the algebra M1(G1) is the Fourier-Stieltjes algebra associated to the Kac algebra K,(G1). Let us recall that, by 3.6.12, the intrinsic group of Ka(Gl) is composed of the characters on G1. Therefore, by using
5.6.6, we see that there is a character X on G1 and an i-isomorphism 0 from K,(G1) to K,(G2) (because K,(G2)s = K,(G2)) such that, for all p in M'(G1), we have: AG2 (Tq) = COX,AG1(0))
(*)
We easily compute that for all p in M1(G1) we have: QX1(AGi (µ)) = AGl (X µ)
On the other hand, by 4.3.5, there is a bicontinuous isomorphism a' from G1 to G2 such that, for all s in Gl: o(AG1(s)) = AG2 (a (s))
By integrating, we find, for ally in M'(Gl): O(AG,W) = AG2(a (µ)) Going back to (*), we have: XG2 (Tp) = -P (AG, WY)) = AG2 (a'(X lU)
and therefore:
Ty=a( µ)= (,'oa-1)(aG)) We finally reach the result by writing X = X' o a-1 and a = a'-1
by (**) by (***)
5.6 Isometries of Fourier-Stieltjes Algebras
191
5.6.10 Corollary (Walter's Theorem [194]). Let G1 and G2 be two locally compact groups. Let T be a multiplicative, isometric, linear, bijective mapping
from B(Gi) to B(G2). Then there exists: (i) an elements in G1 (ii) a bicontinuous isomorphism a from G2 to Gl or to GOPP such that, for all t in G2 and f in B(Gi), we have:
(Tf)(t) =
f(s-la(t))
Proof. By 1.6.3 (iii), up to the Fourier-Stieltjes representations, we have B(Gi) = B(Ka(Gi)) (i = 1, 2). Therefore, applying 5.6.6, we get the existence of an element u in G(Ks(G1)) and an IIII-isomorphism 4 from IH[a(Gl) to IEIIa(G2) or IH[4(G2)S = Ha(G2PP), such that for all f in B(G1), we have:
Tf = iP(au(f))
(*)
By 4.3.2, there exists s in G1 such that u = AGl (s). Then, we have for all f in L°°(Gl) and almost all tin G1:
(**)
(AXG, (9)(f))(t) = As-1t)
on the other hand, by 4.3.5, it exists a bicontinuous isomorphism a from G2
to Gl or GTP such that:
!P(f) = f o a
(***)
Going back to (*), we finally find, for all tin G1 and f in B(G1), that:
(Tf)(t) = (PAGl(8)(f))(a(t))
=f which completes the proof.
(s-1a(t))
Chapter 6 Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras
Let K = (M, T, ,c, cp) be a Kac algebra, 1K = (M, P, k, c) the dual Kac algebra. We have seen that the modular operator L = 4 is the RadonNikodym derivative of the weight cp with respect to the weight cp o is (3.6.7).
So, it is just a straightforward remark to notice that cp is invariant under , if and only if cp is a trace. Moreover, the class of Kac algebras whose Haar weight is a trace invariant under #c is closed under duality (6.1.4). These Kac algebras are called "unimodular" because, for any locally compact group G, the Kac algebra &(G) is unimodular if and only if the group G is unimodular. Unimodular Kac algebras are the objects studied by Kac in 1961 ([66], [70]). We show later another analogy with the group case, namely that if cp a finite weight, then (M, T, r., cp) is a unimodular Kac algebra (6.2.1); it is called "of compact type", because 1[ (G) is of compact type if and only if G is compact. We prove then, after Kac ([67]), that every representation of the involutive Banach algebra M. is the sum of irreducible representations (this leads, for compact groups, to the Peter-Weyl theorem) and that the Fourier representation is the sum of all (equivalent classes) of irreducible representations of M. (6.2.5). With the help of Eymard's theorem, this leads to Tannaka's duality theorem for compact groups (6.2.6). If K is such that the Banach algebra M. has a unit, then K is a unimodular algebra and K is of compact type (6.3.3). So, such Kac algebras will be called "of discrete type". Moreover ([67]), the von Neumann algebra M is then the sum of finite-dimensional matrix algebras:
M = ® £(Hi)
with di = dim Hi < 00
i
and the trace cp is then given by:
(px) = =
dTr'(xi) i
where xi belongs to £(Hi), and Tri is the canonical trace on C(Hi).
6.1 Unimodular Kac Algebras
193
Moreover, we get, following Ocneanu ([109]), an existence theorem for a Haar trace in this case; let (M, F,ic) be a co-involutive Hopf-von Neumann algebra, such that M = OiL(Hi) with di = dim Hi < oo and some Hi,, equal to C; let p be the one-dimensional projector associated to Hi,,; if p gives a unity of the Banach algebra Mk, and if F(p) satisfies a certain (quite natural) condition involving K, then, there is a Haar trace p and (M, 1', ,c, cp) is a Kac algebra of discrete type (6.3.5). This result appears, then, to be, in the non-commutative case, the analog of Krein's matrix block algebras (6-4-5), and, so leads to Krein's duality theorem (6-4.6). More generally (6.5.2), we can associate to each co-involutive Hopfvon Neumann algebra a discrete type Kac algebra (or, by duality, a compact type Kac algebra); in the group case, we recover Bohr compactification of locally compact groups (6.5.4). We then also get an existence theorem of a Haar state in the compact type case (6.5.8). At last, we get, after Kac ([69]), an easy result (6.6.1) which strengthen the analogy with locally compact groups: Kac algebras which are both of compact and discrete type are finite dimensional (and vice versa). Then, following Kac and Paljutkin ([75]), we give an existence theorem for a Haar state on a finitedimensional co-involutive Hopf-von Neumann algebra (6.6-4). This last result makes the link ([110]) with the algebraic Hopf algebra theory, as exposed in [1] or [154]. For other specific results about finite-dimensional Kac algebras, we refer to [74], [75], [110], [71], [72], [4].
6.1 UnimodularKac Algebras 6.1.1 Lemma. Let (M, r, ic) be a co-involutive Hopf-von Neumann algebra, cp be a faithful, semi-finite, normal trace on M (we consider elements of M as operators on Hip). For all x in 9tiEI ciTri. By using the standard representation of k on ®iEI(Hi (9 Hi), we get that 0 is equal to
202
6. Special Cases
the restriction of the canonical trace on C(®i(Hi ® Be)). The same holds for every standard representation of M, and in particular on H. The representation \ is therefore the sum of irreducible representations w -> )\(w)pi, equivalent to representations Ai of M. on spaces Hi such that G(Hi) = Ai(M*)" If {ej}jEJ is an orthonormal basis of H, for all w in I., we can compute the Hilbert-Schmidt norm of \(w): IIA(w) 112 S = i I(A(w)Ej 16k j,kEJ
_
)12
I(Ia(w)1 ek (g dfj)I2
by 6.2.3 (iii)
j,k = IIIa(w)II2 = IIa(w)I12
= IIW112 < +00.
Therefore, A(w) is a Hilbert-Schmidt operator. By density it implies that for all w in M*, A(w) is a compact operator on H. And, for all i, piA(w) is also a compact operator on piH. Now, the isomorphism which maps (up to ci) piH onto the Hilbert-Schmidt operators on Hi (i.e. onto Hi 0 Hi) is actually mapping piA(w) on A1(w)®1; thus, the compactness of these operators imply:
di = dim Hi < 00 which completes the proof of (iii). Let now {CJ} j=1 .. di be an orthonormal basis of Hi and ei k the associated
matrix units. Let yl, Y2 in fits which are decomposed into y1 = >i yi and Y2 = Ei y2 with yl, y2 in £(Hi). We have: (wyl °
®wy2)(I'(p)) = (wy2 ®wyl o k) (A x A) (W)
= (A*(wy2)A*(wyi o k),') = W(A(2 2 ° &)A(2 1))
=
W(A(t,y2)*A(c.,yl ))
by 3.2.2 (iv) by 1.4.3
by 3.7.3 by 1.2.5 and 6.1.1 (iii)
_ (A ,(A(wyi )) 1
(A,(y1) I Acp(y2))
_ O (Ml)
_ ciri(y2iAD di
ci i
(y2y1 Cj I Cj j=1
by 6.1.1 (ii)
6.2 Compact Type Kac Algebras
203
di
ci L (y=6j I ek)(y2*ej 161) j,k=1
i
di
ci L (yi6j I 61)(Ad 16j) j,k=1
i
` Loci L ai(ej,kyi)Tri(ekjy2 di
j,k=1
i
di
L ci E (Tri ®Tri)(e9,ky1 (& ek,jy2) j,k=1
i
di
(c ®0
So, fr(Ap(p*(Q,i)*)) belongs to £(Hi)+, and therefore there exists such that fr(e)t= fr(AW(p*(nn)*))1/2 and for any w in IV, we have:
in p=H
(A(w)l I ) _ (a(w) I r'(C)* ) _ (a(w) I AW(p*(Q,)*)) = (p(w)n 117)
As A; and µ are irreducible, we see that p is unitarily equivalent to )t;, which completes the proof of (ii).
Thanks to (ii) it is possible to put a bijection between Irr and the set I defined in 6.2.5 (i); then (iii) can be deduced from 6.2.5 (iii). For p E Irr, let We have: k be the matrix units associated to the basis
e
where Tr, is the canonical trace on £(Hµ). Therefore, by (iii), an orthonormal j), p E Irr, 1 < basis of H is made of the following elements
j, k < dµ}. For w in I,p, we then have: (a(w) I
k)*)) = (p(w)A; ,ek = (p(wX; I fk ) Tr/`(ej,kµ(w)) = dp 1S(ej,k)I(w))
through the identification of p to a component of A, by (iii) = dµ 1(a(w) I AO(ek,j)) An orthonormal basis of H is thus made of1/2
k)*), p E Irr, 1 < j, k < d,}
and, using the formula (*), we get:
(I
,ek)*)) =
which completes the proof of (iv) and of the theorem.
Ik)
6.2 Compact Type Kac Algebras
207
6.2.7 Corollary. Let G be a compact group, equipped with a normalized left Haar measure. We have: (i) The Hilbert space L2(G) is a subalgebra of L1(G) (ii) The Fourier algebra A(G) has a unit, A(G) is equal to B(G), and sate is an ]EII-isomorphism from W*(G) to K8(G).
(iii) Every representation of G can be decomposed into a direct sum of finite-dimensional irreducible representations. Moreover, every irreducible representation of G is finite-dimensional and is equivalent to a component of the left regular representation AG. So, the left regular representation AG is the sum of all (classes of) irreducible representations of G. Let us note this set Irr G.
(iv) (Peter-Weyl theorem) For all µ in IrrG, let {6Y}t<j (ff(s)Tq(s)ds)P,k. j,k1
Proof. Through the use of 6.2.2, it is the translation of 6.2.4, 6.2.5 (i) and 6.2.6 applied to Ka(G). 6.2.8 Corollary (Tannaka's Theorem [166]). Let G a compact group, IrrG the set of all (classes of) irreducibles representations of G, and, for each v in IrrG, H a Hilbert space such that v(L'(G)) = let us now choose x = ®IrrGxv an element of different from 0. Then, the two following assertions are equivalent:
(i) There exists a unique s in G such that, for all v in IrrG:
x = v(s) (ii) For any pair µ, v in Irr G, and 7r1 i ... , 7rn in Irr G, if V is a unitary in Hp ® H,,, and mk integers such that: µ(s) 0 v(s) = V* k
(e1(cmk) 0 lrk(s))) V
then: x® ®xv = V*
((icmk) ®x"k
V.
k
Proof. For any µ, v in Irr G, there exist a1 i ... , 7rn in Irr G, a unitary U,,,,, in and integers £(HI, 0 such that, for all s in G, we have: Uµ,v(l1(s) ® v(s))U,,,, = ®(1G(Cmµ,v,*k) 0 Irk(s))
208
6. Special Cases
By hypothesis, this implies: UM,,(-Tµ ®xOUµ,v =
®(1G(Cmµ,v,,,k) 0 xak )
(*)
and, for any s in G:
By 6.2.7, we have G(G) =
ra(AG(s)) = AG(s) ® AG(s) = ®(µ(s) 0 v(s)) µ,v
_ ® Uµ,, µ,v
and therefore, for any y =
(E)(1f_(CMA,v,Wk) 0 Irk(s)))
in G(G), we have:
r8(y) = ® Uµ,v (®(1G(C"'v,v,*k) ® yk)) Uµ,v µ,v
k
and then, thanks to (*): re(x) = ®(xµ µ,v
=x®x As, by hypothesis, x is not equal to 0, x belongs then to the intrinsic group of K8(G), i.e. to the set {AG(s), s E G} by 4.3.2. As AG = ®IrrGv by 6.2.7 (iii), (ii) is obvious, the result is proved. we get (i). As the implication (i)
6.3 Discrete Type Kac Algebras 6.3.1 Definition. Let K = (M, r, r., cp) be a Kac algebra. It shall be said of discrete type when the algebra M. is unital. 6.3.2 Theorem. Let G be a locally compact group. The following assertions are equivalent: (i) The group G is discrete. (ii) The Kac algebra Ka(G) is a discrete type Kac algebra. Proof. It is well known that G is discrete if and only if the algebra L'(G) has a unit (1.1.3). 6.3.3 Theorem. Let K be a Kac algebra. The following assertions are equivalent: (i) The Kac algebra K is of discrete type. (ii) The Kac algebra k is of compact type.
6.3 Discrete Type Kac Algebras
209
Proof. Let us assume (i). Because of M* being unital, so is A(M*); let e be the unit of A(M*). We have xe = x for all x in A(M*); by having x strongly converging to 1, we get 1 = e, and so 1 belongs to A(M*). And, as Is,, fl I1 is dense in M* and A norm-continuous, we get that 1 belongs to the norm closure of A(I, fl I,') and therefore to the norm closure of fl 91!. So, there is an invertible element in 910 fl ¶fl ,. Therefore the left ideal 91.- is equal to all k and the weight cp is finite, which is (ii). The converse implication has been proved in 6.2.5 (i). 6.3.4 Corollary. Let K be a discrete type Kac algebra. Then we have: (i) The Kac algebra K is unimodular. (ii) There exist Hilbert spaces Hi, with di = dim Hi < +oo, such that the algebra M is isomorphic to ®iEIC(Hi). (iii) There exists p, one-dimensional projection of the centre of M such that, for all x in M, we have:
F(x)(p®1)=pox F(x)(1 ®p) = x ® p F(p)
di 1
j,k
iEI
K(eii ,k)
AkJ
where eJ k are matrix units of .C(Hi). (iv) We have:
'p=1: diTri iEI
where Tri is the canonical trace on £(Hi), and the unit e of M. is defined, for all x in M, by: E(x) = 'p(px)
The Haar weight cp is also equal to the restriction to M of the canonical trace on C(H). (v) We have, for all i in I: di
(S?l ik'o
o
* .flak'Emi
_
E
k=1
where 6l a is the Kronecker symbol and of Hi corresponding to the matrix units eJ k.
is the orthogonal basis
Proof. The assertion (i) results from 6.2.1 and 6.1.4, (ii) from 6.2.5 (iii), (iii) from 6.2.5 (ii) and (iv), (iv) from 6.2.5 (v), and finally (v) from 6.2.5 (vi).
210
6. Special Cases
6.3.5 Theorem ([109]). Let (M, I', rs) be a co-involutive Hopf-von Neumann algebra such that M = ®IEIG(Hi), with di = dim Hi < oo. Then the following assertions are equivalent: (i) There exists a weight cp on M such that (M, T, r., cp) is a discrete type Kac algebra. (ii) There exists a one-dimensional projection p in the centre of M such that, for all x in M, we have: I'(x)(p (9 1) = p ® x I'(x)(1 0 p) = x ® p
F(p) _
di 1 E K(ej,k) ® e'kJ iEI
(*)
j,k
where the e k are matrix unit of £(Hi). Moreover, cp is then equal to EiEI diTri, where'd is the canonical trace on £(Hi). The unit of e of M is a homomorphism and satisfies e(x) = cp(xp) for all x in M. If we use the standard representation of M on H = ®iEI(Hi®Hi), then cp is equal to the restriction to M of the canonical trace on C(H).
Proof. By 6.3.4 (iii), we know that (i) implies (ii). Let us assume (ii), and let us put cp = >.iEI diTri. We have then cp(p) = 1. Let us put e(x) = cp(xp) for all x in M. Then, for all w in M*, we have: (x, e*w) _ (l(x), a ®w) _ (sv ®w)(I'(x)(p 0 1))
by assumption
_ (gyp 0 w)(p 0 x)
_ (x, w)
Therefore e*w = w for all w in M. We show that w*e = w the very same way.
Let {eJ}1<j
