REDUCED C*-CROSSED PRODUCTS AND CONDITIONAL EXPECTATIONS

Bul]. Kyushu Inst. Tech.

(M. & N. S,) No. 29, 1982, pp. 1-8

REDUCED C*-CROSSED PRODUCTS AND CONDITIONAL EXPECTATIONS

By

Shigeru IToH

(Received Oct. 2, 1981)

1. Introduction

Anantharaman-Delaroche [1, 2] investigated conditional expectations in W*-crossed products. In [9] we studied similar results in C"-crossed products. In this paper we treat conditional expectations in reduced C"•-crossed products. In particular, if (A, G, ct) is a C*-dynamical system with A unital and A has an ct-invariant state, then the existence of a conditional expectation of C,"(G, A, ct) onto C,"(G) is proved.

2. Preliminaries

Let (A, G, ct) be a C*-dynamieal system, that is, A is a C*-algebra; G is a Iocally compact group; and ct : G--ÅrAut(A) is a strongly continuous homomorphism, where Aut(A) is the set of *-automorphisms of A. Denote by A" the dual space of A and let Åq•, •År be the duality pairing of A and A".

Let L'(G, A) be the set of all (equivalence classes of) A-valued Bochner integrable functions on G with respect to the left Haar measure dt of G. Then L'(G, A) becomes a Banach*-algebra with multiplication, involution and norm respectively given by

(Xy) (t) = S x(s)ct,(y(s-i t))ds

(x*) (t) == A(t- ')ex,(x(t- ')") llxIl , == S ll x(t) ll dt

for any x, yEL'(G, A) and tE G, where A is the modular function of G (Doplicher-Kastler- Robinson [5, gll, glll], cf. Takesaki [14], Bratteli-Robinson [3, g2.7.1], Pedersen [11, g7.6]). It is known that Li(G, A) possesses an approximate identity.

For each xEL'(G, A), define Ilxil by

llxI! =sup di(x"x)i/2=sup IIfl(x)ll ,

Åën

where di varies over the set of states of Li(G, A) and n varies over the set of (non••degenerate)

Kastler-Robinson [5, glV], cf. Dixmier [4, 2.7.1], Bratteli-Robinson [3, g2.7.1]), Let C"(G, A, ct) be the completion of L'(G, A) by this norm ll•ll, that is, C*(G, A, ct) is the enveloping C"-algebra of L'(G, A). C"(G, A, ct) is called the C"-crossed product (or the covariance algebra) of (A, G, ct) (Doplicher-Kastler-Robinson [5, gIV], cf. Takesaki [14], Bratteli-Robinson [3, g2.7.1], Pedersen [11, 7.6.5]).

Now, let p be a (non-degenerate) representation of A on a Hilbert space H. Let K=

L2(G, H) be the set of all (equivalence classes of) H-valued strongly measurable functions n on G such that j llny(t)II2dtÅq co. Then K is a Hilbert space with inner product given by

(n, 4) =j (n(t), C(t))dt (n, CG K).

For each aEA and each se G, define operators z(a) and U(s) on K by •

(n(a)ny)(t) =p(ct,-i(a))n(t)

(U(s)q) (t) =n(s-'t) (ny G K, te G) .

Then (z, U) is a (non-degenerate) covariant representation of (A, G, ct) on K, that is, z is a representation of A on K and U is a continuous unitary representation of G on K such that for every aeA, teG, U(t)z(a)U(t-') =n(ct,(a)) (cf. Takesaki [14], Takai [13], Bratteli- Robinson [3, g2.7.1], Pedersen [11, 7.7.1]). Now, define a representation ll, of Li(G, A)

on K by

ll,(x)ny =j n(x(t))U(t)ndt

for xeLi(G, A), qGK (Doplicher-Kastler-Robinson [5, gIII], cf. Takesaki [14], Bratteli- Robinson [3, g2.7.1], Pedersen [11, 7.6,4]). For any xeL'(G, A), let

llxII,=sup ll llp(x)H , p

where p varies over the se't of non-degenerate representations of A. Then ll • ll, is a C*- norm of Li(G, A) and the completion C,"(G, A, ct) of Li(G, A) by ll • II, is called the reduced C*-crossed product of (A, G, ct) (Zeller-Meier [18, 4.6 (for G discrete)], Takai [13], cf.

Landstadt [10], Pedersen [11, 7.7.4]). For any state ip of A, let (pdi, Hip, gip) be the GNS representation of A associated with ip, and let (llip, Kip) be the representation of Li(G, A) on Kip==L2(G, Hip) constructed as above from pip. Then it is not diMcult to observe that for every xeL'(G, A),

llxll,=sup llllip(x)ll , ip

where ip varies over the set of states of A (cL Takai [13]).

Let C*(G) be the group C"-algebra of G and C,*(G) be the reduced group C"-algebra

Reduced C"-Crossed Products and Conditional Expectations 3

of G (cf. Dixmier [4, 13.9.1], Pedersen [11, 7.1,5, 7.2.1). Then C*(G)=C"(G, C, ct,) and C,"(G)=C,ee(G, C, cto), where C is the comPlex numbers and cto: G-ÅrAut(C) is the trivial homomorphism. If G is amenable (cf. Greenleaf [7], Pedersen [11, g7.3]), then for any C"-dynamical system (A, G, ct), C,"(G, A, ct)=:C"(G, A, ct) (Zeller-Meier [18, 5.1 (for G discrete)], Takai [13, Proposition 2.2], cf. Pedersen [11, 7.7.7]).

3. Somelemmas

Let (A, G, ct) be a C"-dynamical system. Denote by K(G) the set of complex-valued continuous functions on G with compact support. If H js a Hilbert space, then for any feK(G), 4EH, define f46L2(G, H) by (f4)(t) ==f(t)4 (teG). For each state ip of A, let

(pip, Hip, 4ip) be the GNS representation of A associated with ip and let (llip, KÅë) be the representation of L'(G, A) induced by pip, where KÅë=L2(G, Hip) (cf. Preliminaries). In the sequel, we call (llip, Kip) the representation of L'(G, A) associated with ip and we always keep these notations in mind.

The proof of the following lemma is obtained by a direct calculation (cf. Takai [13, p. 27]).

LEMMA 3.1. Let ip be a state of A and (flÅë, Kip) be the representation of Li(G, A) associated with ip. Then for any xGL'(G, A),f, geK(G),

(udi(x) (f4di), g,4di)

= jSf(t"s)gum/s) Åq cts-i(x(t)), ip År dsdt.

PRoposmoN 3.2 (Landstad [10, Lemma 3.1], cf. Pedersen [11, 7.7.9). Let B be a C"- subalgebra ofA such that for every teG, ct,(B)cB, Then C,"(G, B, ct) is a C"-subalgebra Of Cr"(G, A, ct)•

PRooF. For each yGL'(G, B), let Ilyil, be the norm of y in C,"(G, A, ct) and llyll;

be the norm of y in C.*(G, B, ct). It suffices to show that llyll;=llyll,. The inequality llyil;År. ilyll, is obvious from definition. Let ut be any state of B. Let (pe, Hut, 4w) be the GNS representation of A associated with W and (lldi, Ke) be the representation of L'(G, B) associated with W. Take a state ip ofA extending V. Denote by (pip, Hip, 4ip) and (I7ip, Kip) the GNS representation of A and the representation of L'(G, A) respectively associated with ip. By Pedersen [11, 7.7,2], the set {f4ut:feK(G)} is cyclic for (I7ut, K"), that is, the linear subspace spanned by {IIut(.v)(f4,): yeL'(G, B),fGK(G)} is dense in Kdi. For any nut == Z llut(yi) (fi4ut) (yieL'(G, B), fi eK(G)), by Lemma 3.1 we have

i

(nut(y)nut, lle(y)nQ

= 2. (nut(y,"•y"yyi) (fi4ut),fj4ut)

t,J

= ,].E),, jSfi(t-is)fj(s)Åq ct,-i((y,*• y*yy,) (t)), ip Årdsdt

- 2. (i7di(y,*• y*yyi) (fi4Åë), fj4ip)

l,J

=(nip(y)nip, llÅë(y)ndi) ,

where qdi= \. Uip(yi)(f,4ip). By the same way as above, we obtain (ne,nv)=(ndi,nip).

Hence [IIIIw(y)IIÅqÅ~III7ip(y)ll. This implies that ilyll;Åq.,IIyll,. Therefore llyll;= llyll'

,•

Now, let B be a C*-subalgebra of A and suppose that there exists a conditional ex- pectationP ofA onto B such that for any t6G, ct,P==Pct,. Then for each tEG, ct,(B)cB, (Note that a conditional expectation P is an onto, bounded linear and idempotent mapping of norm one (cf. Umegaki [17], Tomiyama [16], Takesaki [15, III.3.4], Itoh [9]).) Define a linear mapping 9: Li(G, A).L'(G, B) by 2(x)(t) =P(x(t)) (xeL'(G, A), teG).

LEMMA 3.3. Let B, P and e be as above. Then for any positive linear form Y' on Li(G, B) and any xeLi(G, A),

Y(9(x)"9(x)) xÅq Y'(9(x"x)) ,

In particular, if (ll, K) is a representation of Li(G, B), then for every xeLi(G, A), nEK, (fl(9(x)"9(x))n, n) ÅqÅ~ (ll(e(x"x))n, n) •

PRooF. We may assume that !l' isastate of Li(G, B). Let ut:G-ÅrB" be the norm continuous positive definite function corresponding to Y' (Pedersen [11, 7.6.7, 7.6.8]).

Define ip: G.A" by Åqa, ip(t)År =ÅqP(a), ut(t)År (aeA, tGG). Then di is a norm continuous positive definite function (Itoh [9, ProofofProposition 3.1]). To this ip, there corresponds a state Åë of L'(G, A) (Pedersen [11, 7,6.7, 7.6.8]). By Itoh [9, Proof of Theor.em 3.2] we have

V(e(x)*9(x)) Å~Åq Åë(x"x) .

On the other hand, by Pedersen [11, 7.6.7] it follows that ÅqP(X*X) :S Åq(x*x)(t), ip(t)Årdt

- S ÅqP((x*x) (t)), W(t)Årdt

:j Åq9(X*x)(t), W(t)Årdt : IIT(9(.*.)).

Reduced C"-Crossed Products and Conditional Expectations 5

Thus Y'(9(x)"9(x)) Åqx Y'(9(x"x)).

LEMMA 3.4. Let B, P and 9 be as in Lemma 3.3. Then for each xeLi(G, A), y, zGL'(G, B), e(yxz)= y9(x)z.

PRooF. By definition it follows that

(Yxz) (t) = S S y(r)ct,(x(r- 's))ct.(z(s- ' t))drds.

Since P(bac) =bP(a)c for every aGA, b, ceB (Tomiyama [16], cf. Takesaki [15, III.3.4], Itoh [9]), we have

9(y)cz)(t) =P((yxz)(t))

== SS P(Y(r)ct,(x(r- 's))ct,(z(s- ' t)))drds

= j j Y(r)P(ct.(x(r- 's)))ct,(z(s- ' t))drds

= S S Y(r) ctr(9(x) (r- ' s)) ct,(z(s- ' t))drds

== (y9(x)z) (t) . Hence 9(yxz) == y9(x)z.

Let P*: B"-A" be the dual operator of P: A.B.

LEMMA 3.5. Let B, P and 9 be as in Lemma 3.3. Let ut be a state ofB and di == P'(W) with (17ut, Kut) and (nÅë, Kip) the respective representations of Li(G, B) and L'(G, A) as- sociated with W and di. Then for each xGLi(G, A),f, geK(G),

(lldi (9(x)) (f4ip), g9) == (lldi (x) (f4ip), g4ip) •

PRooF. By using Lemma 3.1 twice, we can easily obtain the result as follows.

(ll, (x) (feip), g4ip)

= jSf(t-'s)g-/s) Åq ct.-i(x(t)), ip År dsdt

= Sjf(t' is)g'(s) Åq ct,-i(P(x(t))), W År dsdt

= : SSf(t- ' s)g-/s) Åq ct,- t(9(x) (t)), ut År dsdt

= (llv (2(x)) (fev), g4v)•

Now we give a theorem concerning the existence of conditional expectations in reduced C*-crossed products.

THEoREM 4.1. Let (A, G, ct) be a C*-dynamical system and B be a C"-subalgebra of A. Suppose that there exists a conditional expectation P of A onto B such that for any tGG,ct,P=Pct,. Then there exists a conditional expectation of C,"(G,A,ct) onto

Cr*(G, B, ct)•

PRooF. Define e as in Lemma 3.3 using P. By Proposition 3.2, C.*(G, B, ct) is a C"-subalgebra of C,"(G, A, ct), We show that e may be extended to a conditional expecta- tion of C,"(G, A, ct) onto C,"(G, B, oc). Let W be any state ofB and (I7ut, Kut) be the re- presentation of Li(G, B) associated with W. Denote ip=P"(ut) and let ([Idi, Kip) be the representation of Li(G, A) associated with ip. Take any ne= Zllut(yi)(fiee), where yiGL'(G, B) andfieK(G). Then for each xELi(G, A), it holds tha't

(flvi(9(x)"9(x))n,fr, ijut) Åqx (I7ut(9(x"x))nut, n,)

by Lemma 3.3. By Lemmas 3.4 and 3.5, we have (llut(9(x"x))ije, nut)

= Z. (lldi (y,"• e(x"x)yi) (fi4ut), fjst)

t,J

-- l . (I7di (9(y,"• x*xyi)) (fi9), fj4ut)

l,J

= 2. (ll, (y,*• x*xy,) (f,4ip), f,4ip)

l,J

=(lldi (x"x)nip, nip) , where nip = 2 fldi(yi) (fi4ip) G Kdi. Thus

i

(flw(9(x)*e(x))nth, ndi) Åqx (llip(x"x)nÅë, nip) .

Moreover (nip, nyÅë) =(qth, qdi) by Lemma 3.5. Since the linear subspace generated by such ne's is dense in K" (Pedersen [11, 7.7.2]), it follows that llllw(2(x))ll Åq. illldi(x)ll. This in

turn gives the inequality ll9(x)ll,Åqx Hxil,. Hence e is extended to a conditional expectation Of Cr"(G, A, ct) OntO Cr*(G, B, ct)•

A state ip ofA is called ct-invariant if for every teG, aEA, Åqct,(a), ipÅr =Åqa, ipÅr.

The proof of the following corollary is immediately derived from Theorem 4,1 by

Qb$erving Itoh [9, PrQof of CQrollary 3.3].

Reduced C"-Crossed Products and Conditional Expectations 7

CoRoLLARy 4.2. Let (A, G, ct) be a C*-dynamical system with A unital. If A has an ct-invariant state, then there exists a conditional expectation of C."(G, A, ct) onto C,"(G).

REMARK4.3 (Example). Let G be a discrete non-amenable group (e.g. the free group on two generators (cf. Greenleaf [7, Example 1.2.3])). Let U: G.B(H) be a unitary representatjon of G on a Hilbert space H, where B(H) is the set of bounded linear operators on H. Denote K==HeH (the direct sum Hilbert space of two copies of H). Then B(K) is the set of 2Å~2 matrices with entries in B(H). Define V: G-ÅrB(K) by

v(t)- (U,(`) O,l (teG),

where I is the identity operator on H. Then Vis also a unitary representation of G on K.

Now, let A=B(K) and define ct:G-•Aut(A) by ct,(a)=V(t)aV(t")(teG,aeA), Then (A, G, ct) is a C"-dynamical system. Choose any xEH with llxll=:1 and define ip: A-C by ip(a)==(a(z), z)(aGA), where z=(O, x)eK. Then di is an ct-invariant state. Thus, by Itoh [9, Proposition 3.1] and Proposition 3.2, C"(G) is a C"-subalgebra of C"(G, A, ct) and C,"(G) is a C*' -subalgebra of C,"(G, A, ct) respectively. Since G is not amenable, C,*(G)

:C"(G) (cf. Dixmier [4, g18.3], Greenleaf [7, g3.5], Pedersen [11, g7.3]). Hence C#(G, A, ct)7EC*(G, A, ct). By Coroliary 4,2, there exists a condjtional expectation of C.*(6,A,ct) onto C,"(G). Alfo, by Itoh [9, Corollary3.3•], there existg. a conditional expectation of C"(6, A, ct) onto C"(G).

References

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Pures AppL 47 (1968). 101-239.

Department of Mathematics,

Kyushu Institute of Technology