PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 10, October 1998, Pages 2969-2978 S 0002-9939(98)04598-5

DUALITY FOR FULL CROSSED PRODUCTS OF C*-ALGEBRAS BY NON-AMENABLE GROUPS

MAY NILSEN

(Communicated by Palle E. T. Jorgensen)

ABSTRACT. Let (A, G, 6) be a cosystem and (A, G, a) be a dynamical system. We examine the extent to which induction and restriction of ideals commute, generalizing some of the results of Gootman and Lazar (1989) to full crossed products by non-amenable groups. We obtain short, new proofs of Katayama and Imai-Takai duality, the faithfulness of the induced regular representation for full coactions and actions by amenable groups. We also give a short proof that the space of dual-invariant ideals in the crossed product is homeomorphic to the space of invariant ideals in the algebra, and give conditions under which the restriction mapping is open.

INTRODUCTION

Let a locally compact group G coact, or act, on a C*-algebra A and denote the full crossed product by A x G. There are three well known maps between the ideal spaces of A and A x G. Every ideal I of A generates an ideal in A x G, denoted by Ext(I). Every representation of A x G restricts to a representation of A; this is well-defined on ideals, so for an ideal K in A x G we obtain the ideal Res(K) in A. A representation of A can be induced to a representation of A x G and it turns out that this gives a well-defined process on ideals, so for an ideal I in A, there is an ideal Ind(I) in A x G.

In ?2, we examine the extent to which the maps Res and Ind commute. Our results generalize some of the results of Gootman and Lazar [1] to full crossed products by coactions and actions of non-amenable groups. From these results we obtain the following corollaries. We give a short, self-contained proof of the faithfulness of the induced regular representation for full coactions [10, Corollary 2.7] (Corollary 2.4). It is suprising to find such a simple proof of Quigg's result, since the proofs in [10, Corollary 2.7] and [12, Theorem 4.1(2)] seem quite deep. We give a short proof of the analogous result for full crossed products by actions of amenable groups. Also we provide short, elegant proofs of both Katayama and Imai-Takai duality for full crossed products [5, Theorem 4], [3, Theorem 3.6], [11, Theorem 6] (Corollaries 2.6 and 2.12).

In ?3 we give key properties of the maps Res, Ext and Ind for both coactions and actions. In particular, we show that the dual-invariant ideals of A x G are exactly the ideals induced from A (Propositions 3.1(iv) and 3.3(iii)). In the coaction case,

Received by the editors May 15, 1996 and, in revised form, March 10, 1997. 1991 Mathematics Subject Classification. Primary 46L55. This research was supported by the Australian Research Council.

(@)1998 American Mathematical Society 2969

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this is a generalization (to non-amenable G) of a version of Mansfield's imprimitivity theorem for induction from the trivial subgroup [6, Theorem 28]. It follows easily from this that Res is a homeomorphism from the dual-invariant ideals of A x G onto the invariant ideals of A (Corollaries 3.2(i) and 3.4(i)). Although these results may be well known in certain cases, our emphasis here is to provide self-contained proofs where the key ideas work for both actions and coactions.

Then we examine the openness of Res and Res Ind. For coactions, Res is open in general but we need to assume G is amenable in order to prove that Res Ind is open; on the other hand, for actions the reverse is the case: Res Ind is open in general, and we assume amenability in order to prove that Res is open (Corollaries 3.2(ii),(iii) and 3.4(ii),(iii)).

Our main aim in this work has been to present coactions and actions in a unified framework, so as to make clear their similarities and differences, which can only be examined when G is allowed to be non-amenable. Gootman and Lazar do not investigate the issue of amenability, choosing rather to assume amenability for ideal theoretic results [1, ?3]. Furthermore, we have avoided the technicalities of the imprimitivity bimodule techniques employed in [4], [6]. Our overall approach has similarities to the work of Nakagami and Takesaki on duality of von Neumann algebras [7, Chapter 1, ?2].

I would like to thank my advisor Iain Raeburn, and his colleague, John Quigg, for their helpful discussions.

1. DEFINITIONS

Throughout A is a C*-algebra and G is a locally compact group. We denote the full group C*-algebra by C*(G) with canonical embedding iG, and Co(G) is the continuous functions on G disappearing at infinity. All representations a are nondegenerate, tensor products 0 are minimal, and identity maps are denoted by id. Given a nondegenerate homomorphism q into the multiplier algebra M(C), there is nondegenerate homomorphism q$ 0 id: A 0 B -+ M(C 0 B). Let I(A) denote the space of closed ideals of A, with the topology which has subbasic open sets of the form OJ = II: I 2 J}, where J is a closed ideal in A. The relative topology on the space of primitive ideals PrimA in I(A) is the usual hull-kernel topology.

We now recall some definitions from [9, Lemma 1.1]. Let B be a C*-algebra and t: A -+ M(B) be a homomorphism. Define

Res,: I(B) I(A) by Res,(kera) = ker(u- o t), and

Ext,: I(A) I(B) by Ext,(I) = Bt(I)B.

We showed there that Rest is continuous and containment preserving, and that Rest is open onto its range if and only if Ext IrIm Res is continuous. Also I C Res Ext(I) since I C Res(K) * Ext(K) C I. These are key elements in this work.

The map in Cb(G, M(C*(G))) defined by s * iG(S) is unitary. When this element is considered to be in M(Co(G) 0 C*(G)), we denote it by WG; as an element of M(C*(G) 0 Co(G)), it will be denoted by VG.

The map SG: C*(G) - M(C*(G) 0 C*(G)) is the unique nondegenerate homo- morphism such that 6G(iG(S)) = iG(S) 0 iG(S). It satisfies the relation (bG 0 id) o 6G = (id ?SG) ? 6G, as maps of C*(G) into M(C*(G) 0 C*(G) 0 C*(G)).

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Define )G: Co (G) -> M(Co (G) 0 Co (G)) by aG (f) (S, t) = f (st) . It satisfies the relation (aG 0 id) o aG = (id a0lG) o aG*

We use the same definitions for full crossed products by coaction as in [8] (min- imal tensor products and full group C*-algebras). A covariant homomorphism of a cosystem (A, GI ) into M(B) is a pair (q, 4) of nondegenerate homomorphisms q: A -+ M(B) and 4: Co(G) - M(B), such that

q 0 id(6(a)) = 4 0 id(wG)(q(a) 0 1)4' 0 id(w*),

as elements of M(B 0 C*(G)). A covariant representation is a pair (7r, ,a) of rep- resentations into B(H) = M(K(H)), so that the covariance condition holds in M(K(H) (0 C*(G)).

An action a of a locally compact group G on a C*-algebra A can be considered to be a nondegenerate injection a: A -> M(A 0 Co(G)) which satisfies:

a (A)(1 0 Co(G)) = A 09 Co(G), and (a 0 id) o a = (id O0G) o a.

Then a covariant homomorphism of a dynamical system (A, G, a) into M(B) is a pair (q, 4) of nondegenerate homomorphisms q: A -> M(B) and P: C*(G) M(B), such that

q 0 id(a(a)) = 4' 0 id(vG)(q(a) 0 1)4' 0 id(v*),

as elements of M(B 0 Co(G)). A covariant representation is a covariant homomor- phism (7r, U) into B(H) = M(K(H)), so that the covariance condition holds in M(K(H) 0 Co (G)).

Let A and p be the left and right regular representations of G on L2(G), M be the representation of Co(G) on L2(G) as multiplication operators, and define a representation M-1 by M-1(f)(()(s) = f(s- 1)(s). There are many covariance identities which are readily verified: for example,

(1) A 0 id(6G(s)) = [M 0 id(WG)] [A(s) 0 1] [M 0 id(w*)],

(2) id 0A (6G(S)) = [id OM1(V)] [1 A(s)] [id OM-1(VG)], and

(3) id 0M(aG(f)) = [id 0A(w*)] [1 0 M(f )] [id oA(WG)].

Let (A, G, 6) be a cosystem and ir: A -> B(H) be a representation. Then the induced regular representation Ind6 7r: A x6 G -> B(H 0 L2(G)) is the unique nondegenerate homomorphism arising from the covariant pair (7r 0 A o 6, 1 0 M).

An ideal I = ker ir is 6-invariant if

ker(ir 0 A o 6) = ker 7r.

It will follow from Proposition 2.1 that invariance is independent of the choice of representation. Denote by L, (A), the subspace of 6-invariant ideals of A with the relative topology. The dual action 6: A x6 G -> M((A x6 G) 0 Co(G)) satisfies

6(JA(a) Jco(G)(f)) = [jA(a) 09 1] [Jco(G) 09 id(aG(f))].

Let (A, G, a) be a dynamical system and let 7r: A -> B(H) be a representation. Then the induced regular representation

Indlr: Ax G -B(H 0L2(G))

is the unique nondegenerate homomorphism arising from the covariant pair (7r 09 M-1 o a, 1 09 A). The ideal I in A is ca-invariant if ker(ir 0 M-1 o a) = ker7r.

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Denote the space of a-invariant ideals of A, with the relative topology, by I, (A). The dual coaction a: A xc, G -> M((A xc, G) 0 C*(G)) satisfies

a'(kA(a) kG(S)) = [kA(a) 0 1] [kG 0 id(6G(s))]I

Remark 1. If 7r is a faithful representation of A, then 7r0M-1oa is also faithful, and this implies that the canonical embedding kA of A into the crossed product A x, G is faithful. For coactions, however, the situation is different because faithfulness of ir is not enough to ensure the faithfulness of 7r 0 A o 6. Thus, in general, we cannot assume that the embedding jA of A into A x6 G is faithful. A cosystem where IA is faithful is called normal [10, Definition 2.1]. Notice, if G is amenable then A x6 G is normal.

We note that the ideal kerjA is 6-invariant. To see this, take a faithful represen- tation q of A x6 G; then (7r, ,u) = (q o jA, q$ 0jCo(G)) is a covariant representation of (A, G, 6). The covariance condition says

ir 0 id(6(a)) = Ht (0 id(wG)7r(a) 09 ltu 0 id(wG),

as elements of M(K(H) 0 C*(G)). Since ,u 0 A(WG) is unitary, by applying id 0A to both sides we have that ker(ir 0 A o 6) = ker ir. Thus ker 7r is invariant, and since

= 0 ?jA and 0 is faithful, ker jA is invariant.

2. RESTRICTION AND INDUCTION OF IDEALS

Let (A, G, 6) be a cosystem. Given the canonical embedding jA of A into A x6 G, we have ResjA: I(A x6 G) -> f;(A) (defined in ?1). But we choose to denote this map by Res6. Also we have Ext6: 1(A) -- I(A x6 G). Similarly for a dynamical system (A, G, a) we have Res,: I(A x, G) -+ 1(A) and Ext,: 1(A) -+ I(A x, G).

Define Ind6: 1(A) -> I(A x6 G) by Ind6 (ker ir) = ker(Ind6 ir), where Ind6 ir is the induced regular representation. It will follow from our first proposition that Indb is well-defined on ideals. Define Ind,: 1(A) -> I(A x, G) similarly. Now define

kA: A - M(A 0 K(L2(G))) by kA(a) = id 0Q(6(a)),

kco(G): Co(G) - M(A ? K(L2(G))) by kco (G)(f) = 1 0 M(f)) and

kG: G - M(A 0 K(L2(G))) by kGG(S) = 1 0 p(s).

One can check using standard covariance relations that (kA, kc0 (G)) is covariant for 6, (kA X kco(G) I kG) is covariant for 6, and that

0:= (kA X kco(G)) x kG: (A x6 G) xS G -4 A 0 K(L2(G)),

is a well-defined surjection. Let h be the homeomorphism h: 1(A) -

I(A 0 K(L2(G))) satisfying h(kerir) = ker(ir 0 id). The following diagram is a useful summary of the maps involved.

(4) I(A x,G

IndS Ind; || ResR

-4 , esS

1(A) -e I (A x6 G xS G) Res~p oh

In this section we examine the extent to which this diagram commutes. Our re- sults are analogous to Gootman and Lazar, which are for spatial crossed products

DUALITY FOR CROSSED PRODUCTS OF C*-ALGEBRAS 2973

by coaction (compare [1, Corollary 2.12, Theorem 2.9, Corollary 2.10]). Our tech- niques lead to new self-contained proofs of the faithfulness of the induced regular representation for full coactions, the faithfulness of the induced regular representa- tion for actions by amenable groups, and both Katayama and Imai-Takai duality.

Proposition 2.1. Let (A, G, 6) be a cosystem and I be an ideal in A. Then

Ind6 (I) = ResS Resk, oh(I).

Proof. Using the definitions:

ResS Resk oh(kerir) = Rest Resk(ker(r 03 id)) = ResS(ker(ir 0 id) o )

= ker((ir 0 id) o X ol AxG) = ker(r, 0 id(kA x kco(G)))

= ker(r, 0 id((id OA o 6) x 1 M))

= ker(ir0Ao6x10M). ED

Remark 2. It follows from this proposition that Ind6 is well-defined. It is continu- ous, and containment preserving, since all Res maps have these properties (?1).

Proposition 2.2. Let (A, G, 6) be a cosystem and K be an ideal in A x6 G. Then

IndS (K) = Res?, oh o Res6 (K).

Proof. Let a: A x6 G -> B(H) be a representation, ir = 0 jA and ,u = 0JCo (G) so that a = ir x ,t. Then Indo a(lAx6G(jA(a))) = ir(a) 01,

IndS U(lAx6G(jco(G)(f ))) = ,u ( M-1(axG(f)), and IndS o(lG(s)) = 1 0 A(s). Also Res? oh o Res6(ker a) = Res?, oh(kerir) = Res?,(ker(r, 0 id)) = ker((r, 0 id) o 0). On the generators, (ir 0 id) o q does the following:

(ir 0 id) a q$(lAx6G(jA(a))) = ir 0 A(6(a)),

(ir 0 id) o t (lAx6G( kco(G)(f))) = 1 0 M(f), and (ir 0 id) o 4)(lG(S)) = 1 0 p(S). We want to know that IndS a has the same kernel as (ir 0 id) o q. We claim that

Ad([t0A(wG)1 0S)oIndSa = (rx0id) oaq,

where S is the operator in B(L2(G)) such that S(Q)(t) = ((t-1). Using the covari- ance of (ir, ,u) we obtain Ad(,u 0 A(WG) 1 0 S) IndS o(lAx6G(jA(a))) = ir 0 A(6(a)). Using equation (3) one can verify that

Ad(,t 0 A(WG) 1 0 S) IndS o(lAx6G(jco(G) (f))) = 1 0 M(f)

Since the image of A commutes with the image of p we have:

Ad(,t 0 A(WG) 1 0S) IndS U(lG(S)) = 1 0 p(S). D

Remark 3. These calculations are essentially those referred to in [5, Theorem 4] (cf. working in [1, Proposition 2.11]).

Corollary 2.3. Let (A, G, 6) be a cosystem and K be an ideal in A x6 G. Then

ResS IndS (K) = Ind6 Res6 (K).

Proof. ResS IndS (K) = ResS Res?, oh o Res6 (K) = Ind6 Res6 (K). 1

Corollary 2.4. Let (A, G, 6) be a cosystem and ir: A -> B(H) be a representation such that ker ir C ker jA. Then the induced regular representation Indb r: A x,6 G -

B(H 0 L2 (G)) is faithful.

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Proof. Since 6 is an action, 0 is a 8-invariant ideal in A x6 G. Thus by Corollary 2.3, 0 = ResS IndS (0) = Ind6 Res6 (0) = Ind6(ker jA). Since Indb is containment preserving and ker7r C kerjA, Ind6(ker7r) C Ind6(kerjA) = 0. L1

Remark 4. Corollary 2.4 was first proved by Quigg [10, Corollary 2.7], an extension of a result of Raeburn [12, Theorem 4.1(2)]; our method is competely new and significantly shorter.

It follows from Corollary 2.4 that ker jA = ker(id OA o 8). To see this, let 7r be a faithful representation of A, so that ker(id 0Ao8) = ker(7r0Ao8). Corollary 2.4 says that Ind6 (7r) is faithful, so ker jA = Res6 (0) = Res6 (Ind6 (ker 7r)) = ker(id OA o 8).

Corollary 2.5. Let (A, G, 6) be a cosystem and I be a 6-invariant ideal in A. Then

IndS Ind6(I) = Rest oh(I).

Proof. Since I is 6-invariant, Res6 Ind6(I) = I by definition. By Proposition 2.2,

IndS Inds(I) = Resk o h(Ress Ind6(I)) = Resk oh(I). L1

Corollary 2.6 (Katayama Duality). Let (A, G, 6) be a normal cosystem. Define

kA: A M(A 0 K(L2(G))) by kA(a) = id 9A(6(a)),

kc0(G): Co(G) - M(A X K(L2(G))) by kc0o(G)(f) = 1 ?9 M(f), and

kG: G >M(A0K(L2(G))) by kG(s)=1 = p(s).

Then q := (kA x kc0(G)) x kG: (A x5 G) xS G -> A 0 K(L2(G)) is a subjection, which defines an injection on the reduced crossed product (A x6 G) XS;r G.

Proof. Since (A, G, 6) is normal, kerjA = 0 is invariant and Corollary 2.5 implies

IndS Ind6(0) = ker q. By Corollary 2.4, Ind6(0) = 0 and so ker q is equal to the kernel of the regular representation. El

Remark 5. One can verify that q intertwines the double dual coaction 6 of G on (A xE G) XESr G with a coaction S on A 0 K(L2(G)) defined by

S = Ad(1 0 M o id(w*))[id ?E(6 0 id)],

where E is the flip isomorphism of T: C*(G) 0 K -> K 0 C*(G) (use equation 1). It follows from Corollary 2.6 and A x6 G - (A/ ker(jA)) X,5ker(jA) G [8, remark

after Theorem 2.3] that for any cosystem (Ax6G) XESrG - (A/ker jA)0K(L2(G)).

We now state the analogous results for the case of crossed products by actions. There is no need to repeat the arguments, since they are very similar (that's the whole point of our approach). The comparable results of Gootman and Lazar for reduced crossed products by actions are [1, Corollary 2.7, Theorem 2.4, Corollary 2.5]. Let (A, G, a) be a dynamical system, and define

jA: A - M(A?3K(L2(G))) by jA(a) =idOM-1(a(a)), G: Co(G) --M(AOK(L2(G))) by jG(f) = 1)A(s)) and

3co(G): G -+ M(A 0 K(L2(G))) by iCo(G)(f) = 1 0 M(f)-

One can show that (iA, iG) is covariant for a, (iA X iG, iCo(G)) is covariant for a', and that 0: (jA X iG) X iCo (G): (A x C G) x & G -> A 0 K(L2(G)) is a well-defined surjection.

DUALITY FOR CROSSED PRODUCTS OF C*-ALGEBRAS 2975

Proposition 2.7. Let (A, G, a) be a dynamical system and I be an ideal in A. Then Ind, (I) = Res& Restp oh(I).

Proposition 2.8. Let (A, G, a) be a dynamical system and K be an ideal in Ax, G. Then Ind& (K) = Restp oh o Res, (K).

Proof. The key equation to check is Ad(U 0 M1(VG)) o Ind& a = (ir 0 id) o o (use equation 2). El

Corollary 2.9. Let (A, G, a) be a dynamical system and K an ideal in A x, G. Then Res& Ind& (K) = Ind, Res, (K).

Corollary 2.10. Let (A, G, a) be a dynamical system and I be an a!-invariant ideal in A. Then Ind& Ind,(I) = Restp oh(I).

Corollary 2.11. Let (A, G, a) be a dynamical system, G be an amenable group and 7r: A -? B(H) be a faithful representation of A. Then the induced representation Indc r: A x, G - B(H X L2(G)) is faithful.

Proof. Let lAxG denote the embedding of A x, G into the double crossed product A xG x&G, so ker(lAxG) is an a&-invariant ideal. Since G is amenable, the cosystem (A xc, G, G, a') is normal, which means ker(lAxG) = 0 and 0 is an a'-invariant ideal of A x, G. Thus by Corollary 2.9,

0 = Res& Ind& (0) = Ind, Res, (0) = Ind, (ker kA) = Ind, (0).

So if ir is a faithful representation of A, then IndQ ir is a faithful representation. E[

Corollary 2.12 (Imai-Takai Duality). Let (A, G, a) be a dynamical system. De- fine jA: A-M(A 3K(L2(G))) by jA(a) =idOM (a (a)),

jG: G - M(A 0 K(L2(G))) by jG(S) = 1 0 A(s), and

iCo(G): Co(G) - M(A 0 K(L2(G))) by 3co(G)(f) = 1 0 M(f).

Then ):= (jA XjG) Xico(G): (AxaG) x&G - AOK(L2 (G)) is an isomorphism.

Proof. Since 0 is an invariant ideal of A, Ind& Ind, (0) = kerL (Corollary 2.10), and Ind&(K) = 0 if K C ker(jAXCIG) (Corollary 2.4), it is enough to show that Ind,(O) C ker(jAxaG). Because Ind, preserves containment, Indc(0) is contained in Ind, Resc, (ker(jAxaG)), and using the fact that ker(jAxaG) is &-invariant, from Corollary 2.9 we deduce

Ind, (0) C Ind, Res, (ker(jAx yG)) = Res& Ind& (ker jAx xG) = ker jA xG. E

3. INVARIANT IDEALS

The definition of 6-invariance used in this paper is different than that used in [8]. In that paper we needed to restrict a coaction to give a coaction on an ideal. It turns out that the condition required for this is different from that required to ensure there is a coaction on the quotient. However, there is a coaction on the quotient by a 6-invariant ideal by the following argument. If I E 2% (A), that means I = Res Ind6 (I), which implies I is in the image of Res6. Thus there exists a covariant pair (ir, ,) such that ker ir = I. The covariance relation says ir 0 id(6(a)) = a 0 id(wG) ir(a) 09 1 Ia 0 id(w*), where 7r 0 id means the map into M(K(H)0C*(G)). Since we are using minimal tensor products, M(7r(A)0C*(G)) is mapped faithfully into M(K(H)0C*(G)), so the map ir Oid into M(K(H)0C*(G))

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has the same kernel as the map into M(7r(A) 0 C*(G)). Since 7r(A) ,v A/I, we have ker(q 0 id o6) = ker q, where q is the quotient map. The coaction 6' on the quotient A/I is then defined by 6'(q(a)) = q 0 id(6(a)) [8, Proposition 2.2].

Proposition 3.1. Let (A, G, 6) be a cosystem and I an ideal of A. Then

(i) the following are equivalent: (a) Res5(Ind65(I)) = I, (b) I = Res6 (Ext5(I)), and (c) I EJJ% (A);

(ii) if G is amenable, then Res65(Ind65(I)) C I C Res6 (Ext65(I)); (iii) Ext6 (I) = Ind6 (I) for I E 7:5 (A); (iv) (a) 17% (A) = Im Res6 and

(b) IS (A x6 G) = ImInd6.

Proof. (i) By definition, I E 7:%(A) if and only if Res6 Ind6(I) = I, so (a) is equiv- alent to (c). Before we verify that (b) is equivalent to (c), we will show that li%(A) = ImRes6.

If I E 7E (A), then I is in the image of Res6. Conversely, suppose I E ImRes6. That means there is a covariant pair (ir, A) with ker ir = I, and the covariance relation says ir 0 id(6(a)) = [j 0 id(wG)][7r(a) 0 1][ji 0 id(w*)]. Apply id0A to both sides to see that ker((ir 0 A) o 6) = kerir. Since

Indair= ((ir0A)o6) x (1OM),

Res Ind6 (ker ir) = ker((ir 0 A) o6), and so Res Ind6 (I) = I. Thus 2E (A) = Im Res6. Now suppose Res6 (Ext6 (I)) = I. Then I E Im Res6 = 165(A). Conversely,

suppose I E 7% (A) so that I E Im Res6. Then I = Res6 (K) for some ideal K of A x6 G, so Res6(Ext6(I)) = Res6 Ext6 Res6(K) = Res6(K) = I, which shows that (c) is equivalent to (b).

(ii) We have I C Res6 Ext5(I) from [9, Lemma 1.1](ii). Now suppose G is

amenable. Then A is faithful, so that id OA: A 0 C*(G) -> M (A 0 K(L2(G))) is

faithful. We want to show that Res Ind6 (I) C I, so let ir be a representation of A such that kerir = I. Suppose a E Res6 Ind6(kerir), that is, ir 0 A(6(a)) = 0. Then

(id OA) o (Ir 0 id)(6(a)) = 0.

Since id OA is faithful, ir 0 id(6(a)) = 0, and since id 01(6(a)) = a, it follows that ir(a) = 0.

(iii) Let I E 1,5(A) and ir: A -> B(H) be a representation of A with ker ir = I. Let r: A/I -> B(H) be the associated faithful representation. Since I is 6-invariant, there is a coaction 6' on the quotient A/I. By Corollary 2.4, the induced repre- sentation Ind6 i: (A/I) x Ei G -> B (H 0 L2 (G)) is faithful. By [8, Theorem 2.3], there is an isomorphism +: (A x6 G)/ Ext6 (I) -> (A/I) X6i G, so there is a faithful representation of (A x6 G)/ Ext6 (I) given by 77 = Ind6 r o b. The representation

ye': Ax6G - B(H oL2(G)) defined by 7y'(a) = 77(a+Ext6(I)), has kernel Ext6(I). To show that Ind6(I) -. Ext6(I), one can show that Ind6 ir = ye', by checking they agree on the generators.

(iv) We showed (a) in part (i), so it remains to show (b). Now suppose K E

IS (A x E G). By Corollary 2.3, K = Ress Indi (K) = Ind6 Res6 (K), so IS (A x6E G) C

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ImInd6. For the reverse containment, we need to know that Md6(kerir) is a 6- invariant ideal of A x6G. Routine calculations show that (Ind6 r, 1 0 p) is covariant for 6; thus Ind6(kerir) E ImResS = IS (A x6 G), by Proposition 3.3(iv) (a). D

Corollary 3.2. Let (A, G, 6) be a cosystem. Then (i) Res6 : ZS(A x6 G) t-> 2%(A) is a homeomorphism,

(ii) Res6: I(A x6 G) ---> 2(A) is a continuous, open subjection, and, (iii) if G is amenable, then Res mInd: 1(A) -> 1 (A) is continuous, open and

surjective.

Proof. (i) Let I E 176(A). To show that Res IXS(AxG) is onto, we must find an invariant ideal of A x6 G which maps to I. We claim Ind6(I) will do. Firstly it is 6-invariant by Proposition 3.1 (iii), and since I is invariant, Res6 Ind6 (I) = I by definition.

Next we show Res IX:&(AxG) is infective. Suppose K and L are 6-invariant ideals of A x6G, and Res6 (K) = Res6 (L). Since they are invariant, there exist ideals I and J of A such that Ind6(I) = K and Ind6(J) = L. Thus Res6 Ind6(I) = Res6 Ind6(J), and applying Ind6 to both sides gives Ind6 Res6 Ind6(I) = Ind6 Res6 Ind6(J). FRom Proposition 3.1(iii) one can deduce Ind Res6 Ind = Ind6, so Ind6(I) = Ind6(J), which means K = L.

We know Res6 is continuous, so we show Res IlXS(AxG) is a homeomorphism by verifying that Ind6 is a continuous inverse. Well, Ind6 is continuous by Proposition 2.1, and Res6 Ind6(I) = I for invariant ideals by definition. Let K be a 6-invariant ideal of A x6 G, so it is of the form Ind6(I) = K for a 6-invariant ideal of A. Thus Ind6 Res6(K) = Ind6 Res6(Ind6(I)) = Ind6(I) = K.

(ii) We need to show Res6 is open. Proposition 3.1 (ii) says that Ext (I) =

Ind6(I) for I E 165(A); thus Ext6 I-:Z(A) is continuous. Now Res is open onto its range if and only if Ext rIm Res (see ?1) and since 165(A) = Im Res6 (Proposition 3.1(iii)), this implies that Res6 is open onto its range.

(iii) Since Res mInd is continuous, it remains to show Res mInd is open. In Corollary 3.4(ii) we show that Rests: 1((A x6 G) xS G) t-> IS(A x6 G) is open onto its range when G is amenable. That corollary also says that Rests restricted to 12- ((A x6 G) x S G) is a homeomorphism with inverse Indi, so Indi ITS (AX6 G) is open onto its range. Thus Ind IrIm ResS is open onto its range because I74 (A x6 G) =

Im Ress. We show that Res6 Ind6 is open by showing that

Res6 Ind = (Reso oh)-1 Indi ResS(Reso oh).

This makes sense because G is amenable, and Corollary 2.6 (Katayama duality), says that q is an isomorphism, so that Resd is a homeomorphism. From Corollaries 2.5 and 2.1 we conclude

(Reso oh) Res6 Ind6 = Indi Ind6 Res6 Ind6 = Indi Ind6 = Indi ResS(Reso oh). Cl

The first three parts of the next proposition are in [9, Lemma 5.2(ii), (iii)], and the proof of the last part is the same as Proposition 3.1 (iv). The proof of the corollary following mirrors Corollary 3.2.

Proposition 3.3. Let (A, G, ar) be a dynamical system and I an ideal of A. Then

(i) the following are equivalent: (a) Res,(Ind, (I)) = I, (b) I = Res,,(Ext,,(I)), and

2978 MAY NILSEN

(c) I E -Ea(A); (ii) Res, (Ind. (I)) C I C Res, (Ext, (I));

(iii) Ext, (I) C Ind. (I) for I E a, (A), with equality if G is amenable; (iv) (a) Z, (A) Im Resa and

(b) I&(A xc, G) = ImIndO.

Corollary 3.4. Let (A, G, a) be a dynamical system. Then

(i) Res,,: I& (A xa G) t-> la (A) is a homeomorphism, (ii) if G is amenable, then Res,: I(A x, G) t-> la (A) is a continuous, open

surjection, and, (iii) Resc, Ind,,: 1(A) t-> Ta (A) is a continuous, open surjection.

Remark 6. Gootman and Lazar proved that Res6 Ind6 is open for spatial crossed products and G amenable [1, Proposition 4.6], and Green proved that Res, Ind, is open [2, pp. 221-222]; our method of proof is completely new.

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DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF NEBRASKA-LINCOLN, LIN-

COLN, NEBRASKA 68588-0323 E-mail address: mnilsen~math.unl .edu