Bull. Kyushu Inst. Tech.
(M. & N. S.) No. 29, 1982, pp. 9-15
MULTIPLIER ALGEBRAS OF C*-CROSSED PRODUCTS AND CONDITIONAL EXPECTATIONS
By
Shigeru IToH
(Received Oct. 2, 1981)
1. Introduction
Itoh [8, 9] treated conditional expectations in (reduced) C"-crossed products and obtained some existence theorems for them in several interesting cases.
Choda and Takehana [4] gave a theorem which enables to extend conditional ex- pectations in C*-algebras to their multiplier algebras under some addjtional conditions.
In this paper we first touch upon briefly the proof of the result of Choda and Takehana [4]. Then, using this result and existence theorems in Itoh [8, 9], we study condjtional expectations in multiplier algebras of (reduced) C"-crossed products.
2. Preliminaries
Almost all notations are the same as in Itoh [8, 9]. Let A be a C'-algebra and (p, H) be the universal representation of A. The second dual of A is isomorphic to the enveloping von Neumann algebra A"==p(A)" of A, where p(A)" is the double commutant of p(A) in B(H) (the set of bounded linear operators on H) (cf. Dixmier [5, 12.1.3], Pedersen [11, 3.7.8], Takesaki [14, III.2.4]). If B is a C"-subalgebra of A, then, since BcA" and the strong closure ofB in A" is isomorphic to B", we may consider as B"cA" (cf. Pedersen [11, 3.7.9]). An element x of B(H) is called a left, respectively right, multiplier of A if
xAcA, respectively AxcA. We say that x is a multiplier ofA ifx is both a left and right multiplier ofA. It is seen that all multipliers are contained in A". Let M(A) be the set of multipliers of A. We call M(A) the multiplier algebra of A (Busby [3], cf. Akemann, Pedersen and Tomiyama [1], Pedersen [11, 3.12.1, 3.12.4], Takesaki [14, III.6.22]). It is easily checked that M(A) is a C"-subalgebra ofA". IfA is unital, then M(A) =A.
Let (A, G, ct) be a C"-dynamical system and C"(G, A, ct) (respectively C."(G, A, ct)) be the (respectively reduced) C"-crossed product of (A, G, ct) (Doplicher, Kastler and Robinson [6], Zeller-Meier [17], Takai [12], cf. Takesaki [13], Landstad [10], Pedersen [11], Itoh [8, 9]). If G is amenable, then C,*(G, A, ct)==C"(G, A, ct) (Zeller-Meier [17], Takai [12], cf. Pedersen [il, 7.7.7]).
Let K(G, A) be the set of all A-valued continuous functions on G with compact sup- ports. Then, with appropriate multiplication, involution and norm using the left Haar
10 Shigeru lToH
measure on G, K(G, A) is a normed "-algebra and its completion is L'(G, A) (the set of all (equivalence classes of) A-valued Bochner integrable functions on G with respect to the left Haar measure) (Doplicher, Kastler and Robinson [6, II], cf. Landstad [10], Pedersen [11, 7.6,1], Bourbaki [2, Definition 3,4. 2, Corollaire 4.10.1, Corollaire 5.5.1, Th6oreme 5.6.5], Hille and Phillips [7, Theorems 3.5.3 and 3.7.4], Itoh [8, 9]). It is seen that L'(G, A) is a Banach"-algebra with an approximate unit (Doplicher, Kastler and Robinson [6]). Let (ll, K) be the universal representation of Li(G, A) (cf. Dixmier [5, 2.7.6],
Pedersen [ll, 7.6.5]). Then ll is faithful (Doplicher, Kastler and Robinson [6, glV]) and C"(G, A, ct) is isomorphic to the uniform closure of ll(Li(G, A)) in B(K) (cÅí Dixmier [5, 2.7.6], Pedersen [11, 7.6.5], Itoh [9]).
Let (p, H) be the universal representation of A and (ll,, K,) be the representation of L`(G, A) induced by p, where K,=L2(G, H) (Doplicher, Kastler and Robinson [6], cf.
Pedersen [11, 7.6.4], Itoh [9]). Then (I7,, K,) is extended to a representation of C"(G, A, ct) (cf, Dixmier [5, 2.7.4]) and is faithfu1 on Li(G, A). In fact, for any xeL'(G, A),
Iin,(x)ll = ]Ixll, (the norm of x in C.*(G, A, ct)) (Taka' i [12, Proposition 2.2], cf. Appendix).
Thus, C,ee(G, A, ct) is isomorphic to the uniform closure of fl,(Li(G, A)) in B(K,) which is nothing but ll,(C"(G, A, ct)) (cf. Landstad [10, p. 254], Pedersen [11, 1.5,7, 7.7.4], Dixmier [5, 2.7.4]).
Now we give a short proof of a result in Choda and Takehana [4].
PRoposmoN 2.1 (Choda and Takehana [4, Theorem 5]). Let A be a C*-algebra and B be a C"-subalgebra of A. Suppose there exist an approximate unit for A contained in B and a conditional expectation P ofA onto B. Then there exists a conditional expectation O.- of M(A) onto M(B).
PRooF. Let P*" be the double transpose of P. Since B"cA" and A (rexpectively B) is a-•weakly dense in A" (respectively B"), P** is a projection of norm one, hence a con- ditional expectation of A" onto B" by Tomiyama [15] (cf. Takesaki [l4, III.3.4]). We have P""(M(A)) =M(B). In fact, for each xEM(A) and beB,
P**(x)b =P**(x b) == P(x b) G B and
bP""(x) == P*"(bx) = P(bx) E B
because xb, bxEA. Thus P""(x)eM(B). By Akemann, Pedersen and Tomiyama [1]
(cf, Pedersen [11, 3.12,12]) M(B) is a C"-subalgebra of M(A). Therefore P""(M(A))==
M(B). If we denote by e the restriction of P"* to M(A), then 2 is a conditional expecta- -tion of M(A) onto M(B).
REMARK 2.2 (cf. Choda and Takehana [4, Theorem 5]). The above 9 is unique in the sense that if the restriction of 9 to A is equal to P, then 2 conincides with the re-
Multiplier Algebras of C"-Crossed Products and Conditional Expectations 11
striction of P"" to M(A).
PRooF. Let {uz} be an approximate unit for A in B. Then {uz} converges strongly to 1in B". For any xeM(A), utEB* (the dual of B), 9(x)EB" and {9(x)ua} converges strongly to 9(x) in B". Since {9(x)uz} is bounded and W is ff-weakly continuous on B", {Åq9(x)u2, WÅr} converges to Åq9(x), utÅr (cf. Pedersen [11, 3.6.4], Takesaki [14, II.2.6.]).
Let P" be the transpose of P. Then
Åqe(x)u,, utÅr =Åqe(xu,), wÅr = ÅqP(xuz), WÅr =Åqxuz, P"WÅr •
Since the bounded net {xuz} converges strongly to x in A", {Åqxuz, P"UÅr} converges to Åqx, P"WÅr =ÅqP""(x), utÅr. Therefore Åqe(x), WÅr = ÅqP*"(x), WÅr, hence 9(x) ==P*"(x).
3. Multiplier algebras ef C*-crossed products
We show the existence of conditional expectations in multiplier algebras of (reduced) C*-crossed products by using Proposition 2,1 and existence theoremg. for conditional expectations in (reduced) C*-crossed products obtained in Itoh [8, 9].
THEoREM 3.1. Let (A, G, ct) be a C*-dynamical system and B be a C"-subalgebra of A. Suppose that there exist an approximate unit for A contained in B and a conditional expectation P ofA onto B such that for any t e G, ct,P = Pct,. Then there exists a conditional expectation of M(C"(G, A, ct)) onto M(C"(G, B, ct)).
PRooF. By Itoh [8, Theorem 3.2], there exists a conditional expectation of C*(G, A, ct) onto its C"-subalgebra C"(G, B, ct). We show the existence of an approximate unit for C"(G, A, ct) in C*(G, B, ct). Let {uz} be an approximate unit for A in B. Choose an approximate unit {f,} of Li(G)=L'(G, C). Then {f,uz}(,,a) is an approximate unit for L'(G, A) contained in L'(G, B), where f.uA is defined by (f,ua) (t) =f,(t)ua (t e G) (Doplicher, Kastler and Robinson [6]). It is also an approximate unit for C*(G, A, ct) in C"(G, B, ct).
By Proposition 2.1 there exists a conditional expectation of M(C"(G,A,ct)) onto M(C*(G, B, ct)).
IfA has the unit 1, then we may identify the complex numbers C with Cl. We easily obtain the following corollary by Proposition 2.1 and Itoh [8, Corollary 3.3].
CoRoLLARy 3.2. Let (A, G, ct) be a C*-dynamical system. IfA is unital and has an ct-invariant state,; then there exists a conditional expectation of M(C"(G, A, ct)) onto M(C*(G)).
12 Shigeru lToH
REMARK 3.3. If (A, G, ct) is a C*-dynamical system with A unital and G amenable, then there always exists a conditional expectation of M(C"(G, A, ct)) onto M(C"(G)) (cf.
Itoh [8, Corollary 3.4]).
The following two results for reduced C"-crossed products are similarly proved as above by using Proposition 2.1 and Itoh [9, Theorem 4.1, Corollary 4.2].
THEoREM 3.4. Let (A, G, ct), B and P be as in Theorem 3.1. Then there exists a conditional expectation of M(C,"(G, A, ct)) onto M(C,"(G, B, ct)).
CoRoLLARy 3.5. Let (A, G, ct) be as in Corollary 3.2. Then there exists a conditional expectation of M(C,"(G, A, ct)) onto M(C,*(G)).
If G is abelian, then we have the following theorem. We note that G is the dual group of G and (C"(G, A, ct), G, a) is 'the dual C"-dynamical system of (A, G, ct) (Takai [12], cf. Landstad [10], Pedersen [11, 7.8.3], Itoh [8]).
THEoREM 3.6. Let (A, G, ct) be a C"-dynamical system with A unital and G abelian.
Then there exists a conditional expectation of M(C"(G, C*(G, A, ct), a)) onto M(C"(G, C*(G), a)).
PRooF. Let 1 be the unit ofA and identify C with Cl. As in the proof of Theorem 3.1, there exists an approximate unit for C"(G, A, ct) contained in C"(G). Then there also exists an approximate unit for C"(G, C"(G, A, ct), a) in C'(G, C"(G), a). By Itoh [8, Corollary 3.5] and Proposition 2.1, there exists a conditional expectation of M(C"(G, c*(G, A, ct), a)) onto M(c*(G, c*(G), a)).
If G is discrete, then the following holds.
THBoREM 3.7. Let (A, G, ct) be a C*-dynamical system with G discrete. Then there exists a conditional expectation of M(C"(G, A, ct) onto M(A).
PRooF. By the correspondence a.6.a(a G A), A may be considered as a C*-subalgebra of C*(G, A, ct), where 6. is a function on G defined by 5.(t)=1 if t :e (the identity of G), or
O if t#e (Zeller-Meier [17, p. 146]). Thus, an approximate unit ofA is also an approximate unit of C"(G, A, ct). Then the conclusion follows from Proposition 2.1 and Itoh [8, Theorem 4,1, (ii)].
Similarly we obtain the following result by making use of Proposition 2,1 and Itoh [8, Theorem 4.1. (i)].
Multiplier Algebras of C"-Crossed Products and Conditional Expectations 13
THEoREM 3.8. Let (A, G, ct) be as in Theorem 3.7. Then there exists a conditional expectation of M(C,*(G, A, ct)) onto M(A).
REMARK3.9. Let (A,G,ct) be a C"-dynamical system. Then M(A) is a CX:-
subalgebra of both M(C"(G, A, ct)) and M(C."(G, A, ct)) (cf. Pedersen [11, 7.6.2•], Ap- pendix). However, it is not known whether there exists a conditional expectation of M(C"(G, A, ct)) (or M(C.*(G, A, ct)) onto M(A) for non-discrete G.
4. Appendix
For notations used in this section, we refer to Itoh [9].
PRoposiTioN 4,1. Let (A, G, ct) be a C"-dynamical system. Then ll • ll, is a C"-norm on L'(G, A).
PRooF. For any state ip ofA, let (pdi, Hip, 4di) be the GNS representation of A induced by ip and (lldi, Kip) be the representation of Li(G, A) induced by pto, where Kdi=L2(G, HÅë).
Take an arbitrary xELi(G, A). It is sufficient to show that x=O in Li(G, A) whenever llip(x)=O for all state di of A. For every f, gEK(G)= K(G, C), we have
o == (lldi (x) (f4ip), g4di)
=jSf(t-'s)gM/s)Åqct,-i(x(t)), ipÅrdsdt
= S (Sf(t-is)Åqct,-i(x(t)), ipÅrdt) g(s)ds
(Takai [12], cf. Itoh [9, Lemma 3.1]). Since K(G) is dense in L2(G)=L2(G, C) and the function in s
Sf(t'is)Åqct,..,(x(t)), ipÅrdt
is continuous and belongs to L2(G), it follows that for any se G, jf(t-is)Åqct,-,(x(t)), ipÅrdt ,=O.
This in turn implies that Åqx(•), ipÅr=O in L'(G) for every di eA" (cf. Dixmier [5, 13.2.5]).
Since x is Bochner integrable, x is almost separably-valued (cf. Hille and Phillips [7, Definition 3.7.3]). Therefore,x==O in L'(G, A).
PRoposmoN 4.2 (cf. Pedersen [11, 3.12.3, 7.6.2, 7.6.3]). Let (A, G, ct) be a C"-
dynamical system, Then M(A) is a C"-subalgebra of both M(C*(G,A,ct)) and
M(Cr"(G, A, ct))•
PRooF. We first show that M(A) is a C"-subalgebra of M(C"(G, A, ct)). For each aGM(A),' define bounded linear operators L(a) and R(a) on L'(G, A) by
(L(a)x) (t) = ax(t) , (R(a)x) (t) = ct ,( ct, - i(x(t))a)
for xeL'(G, A), tEG. Then L(a) (respectively R(a)) is a left (respectively right) cen- tralizer and (R(a), L(a)) is a double centralizer of L'(G, A), that is,
L(a) (xy) = (L(a)x)y,
R(a) (xy)=x(R(a)y) •
and
(R(a)x)y =- x(L(a)y)
for all x,yeL'(G, A) (cf. Pedersen [11, 7.6.2, 7.6.3]). Let (n, K) be the universal re- presentation ofLi(G, A). Then C"(G, A, ct) is equal to the uniform closure of il(Li(G, A)) in B(K). Let {ua} be an approximate unit of Li(G, A). Since {I7(L(a)uz)} is bounded, it has a weak limit point 7i(a) in C"(G, A, ct)". For any xELi(G, A), {L(a)(uAx)} con- verges to L(a)x in L'(G, A). On the other hand,
ill(L(a) (u,x)) = : I7(L(a)uA)III(x) .
Taking the limit, we have
7i(a)I7(x) = R(L(a)x) ,
hence 7i(a) is a left multiplier of C"(G, A, ct). If 71(a) is another element in C"(G, A, ct)"
obtained as above, then
71(a)LI(x) = I7(L(a)x)
for every xGL'(G, A). It follows that 7i(a)z=:71(a)z for all zEC*(G, A, ct), and 7i(a)=
71(a). Thus 7i(a) is uniquely determined by a. Similarly, for any aEM(A), there exists a unique element 72(a) in C"(G, A, ct)" such that
n(x)72(a) = 17(R(a)x)
for all xeLi(G, A). It is evident that 72(a) is a right multiplier of C"(G, A, ct). We show that 7i(a)= 72(a). In fact, since (R(a), L(a)) is a double centralizer of Li(G, A), for any x, yeLi(G, A), we have
' fl(x)yi(a)ll(y) == ll(x)U(L(a)y) == fl(x(L(a)y))
Multiplier Algebras of C"-Crossed Products and (i.londitional Expectations IJcr
== I7((R(a)x)y) : ll(R(a)x)II(y) == "(x)"r'2(a)n(y) ,
whence 7i(a)= 72(a). Denote by År'(a) this common element in C\'(G, A, ct)". Then 7(a) is a multiplier of C*(G, A, ct) and fi1 i's a "-homomorphism of M(A) into M(C"(G, A, ct)).
Since ll ig. faithfu1 on L`(G, A), 7 is an into *-isomorphism (cf. Dixmier [5, l..8.3], Pedersen [11, 1.5.7], Takesaki [14, I.5.4]). Identifying M(A) with 7'(M(A)), M(A) is a C"-subalgebra of M(C*(G, A, ct)).
Now we verify that M(A) is also a C*-subalgebra of M(C,"(G, A, ct)). Let (p, H) be the universal representation of A and (II,, K,) be the represeniation of L'(G, A) induced by p, where K,==L2(G, H). For this II,, we can proceed as above. Consequently we have a "-isomorphism of M(A) into M(C,*(G, A, ct)) because ll, is faithful on Li(G, A).
Thus M(A) may be considered as a C*-subalgebra of M(CY(G, A, ct)).
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Department of Mathematics, Kyushu Institute qf Technology
