Quasi-standard C*-algebras

Math. Proc. Camb. Phil. Soc. (1990), 107, 349 3 4 9Printed in Great Britain

Quasi-standard C*-algebras

BY R. J. ARCHBOLD

Department of Mathematical Sciences, The Edward Wright Building, Dunbar Street,Aberdeen AB9 2TY

AND D. W. B. SOMERSET

Queen's College, Oxford 0X1 4AW

(Received 5 April 1989)

Abstract

A necessary and sufficient condition is given for a separable C*-algebra to be*-isomorphic to a maximal full algebra of cross-sections over a base space such thatthe fibre algebras are primitive throughout a dense subset. The condition is that therelation of inseparability for pairs of points in the primitive ideal space should be anopen equivalence relation.

1. Introduction

From an algebraic point of view the theory of C*-algebras can be separated intothe study of primitive algebras (with particular attention to the simple case) and thestudy of how a general 6'*-algebra is related to its primitive quotients. The secondof these problems may be viewed as a search for non-commutative analogues of theGelfand-Naimark theorem for commutative C*-algebras, with an associatedrequirement for non-commutative Stone—Weierstrass results. This aspect of thetheory was studied by Kaplansky[17], and continued by Glimm[16], Fell [14],Tomiyama[22] and Lee [18], with the introduction of the notion of a maximal fullalgebra of cross-sections (or 'operator fields' in Fell's terminology) using varyingfibre algebras over a locally compact Hausdorff base space (see [14], pp. 235-242 forprecise details). The property of maximality may be viewed as a non-commutativegeneralization of the surjectivity of the Gelfand transform for a commutativeC*-algebra.

In attempting to represent a general C*-algebra in this way, whilst also involvingthe primitive quotients, Fell sought a base space such that the fibre algebras areprimitive throughout a dense subset. One such candidate (the Tg-closure of theprimitive ideal space Prim (A) in the notation of Section 2 below) turned out to bein some sense ' too large' in that the resulting full algebra of cross-sections is maximalif and only if Prim (̂ 4) is Hausdorff ([14], theorem 23), a severe restriction. On theother hand, Dauns and Hofmann[9] developed a theory which is valid for anarbitrary C*-algebra but at the expense of possibly losing continuity of the norm ofa section and also possibly losing control of the primitivity of the fibre algebras.

Fell's work indicated the existence of obstructions to the programme ofrepresenting an arbitrary C*-algebra as a maximal full algebra of cross-sections over

350 R. J. ABCHBOLD AND D. W. B. SOMERSET

a base space such that the fibre algebras are primitive throughout a dense subset, andso attention was turned to the other strand mentioned above, namely the study ofsimple C'*-algebras. This is a difficult area in which a complete classification is not yetin sight. A great deal of effort has been spent in the study of specific examplestogether with the development of general invariants such as K-groups and Ext-groups to distinguish isomorphism classes. The survey articles [7, 5] discuss theprogress that has been made towards a general theory.

In this paper we return to Fell's problem with the object of clarifying the natureand extent of the obstructions referred to above. In [3], a study was made of theminimal primal ideals of a (7*-algebra. These had been found to arise naturally in thetheory of limits of factorial states [4]. It was shown in theorem 5*3 of [3] that, undercertain natural hypotheses, a 6'*-algebra A is *-isomorphic to a maximal full algebraof cross-sections over the (topologized) space Min-Primal(^4) of all minimal primalideals. Moreover, if A is separable then Min-Primal(^4) contains a dense subset ofprimitive ideals.

We show here (Theorem 3-3) that the hypotheses of theorem 5-3 of [3] areequivalent to the condition that the relation of inseparability for pairs of points inthe primitive ideal space Prim (̂ 4) is an open equivalence relation. 6'*-algebrassatisfying this latter condition are said to be quasi-standard. The relevance ofconditions of this type was suggested by the work of Ching[6]. The class of quasi-standard O*-algebras may be viewed as a well-behaved enlargement of the sub-classconsisting of those C*-algebras with Hausdorff primitive ideal space.

Theorem 3-4 provides a useful characterization of quasi-standardness in terms ofprimal ideals and leads to our main result, Corollary 3-5, which gives a completesolution in the separable case to the maximality problem arising from Fell's work in[14], section II : a separable C*-algebra is quasi-standard if and only if it is*-isomorphic to a maximal full algebra of cross-sections over a base space such thatthe fibre algebras are primitive throughout a dense subset. We also use Theorem 3"4to show that some interesting and diverse group C*-algebras are quasi-standard.Finally, we show in Section 4 that the class of quasi-standard C*-algebras strictlycontains the class of pre-standard C*-algebras studied by Ching[6].

2. Preliminaries

Let A be a C*-algebra. By an ideal of A we shall always mean a closed two-sidedideal. Recall that an ideal / of A is primal if whenever n ̂ 2 and J1,J2,...,Jn are idealsof A with zero product then Jt^I for at least one value of i: see [4]. The set Id (̂ 4)of all ideals of A can be equipped with strong and weak topologies, TS and TW. Theprecise definitions and the origins of these topologies are described in [3], section 2.The main features are that a net (/J is rs-convergent to / in Id (A) if and only if| |a+/J | -»• | |a+/| | for all aeA, whilst a base for rw is given by the family of sets of the

f ° r m U(F) ={IeId{A):J£I for all JeF),

where F is a finite set (possibly empty) of ideals of A. When restricted to Prim (̂ 4),the primitive ideal space of A, TW coincides with the Jacobson topology. Therestrictions of rg and TW to the set of primal ideals are generally distinct (TS isHausdorff whilst TW may not be), but the restrictions to Min-Primal (̂ 4) coincide. Wedenote by r this common topology on Min-Primal (̂ 4).

Quasi-standard C*-algebras 351

We now recall some properties of the complete regularization of Prim (A) for aC*-algebra A (see [9] for further details). For P, Q e Prim (.4) let P as Q if and only iff(P) =f(Q) for all /eC6(Prim(^)). Then as is an equivalence relation on Prim (.4)and the equivalence classes are closed subsets of Prim (̂ 4). It follows that there is aone-to-one correspondence between Prim (A)/ ss and a set of ideals of A given by

(PePvim(A)),

where [P] denotes the equivalence class of P. The set of ideals obtained in this wayis denoted by Glimm (A) (see below) and we identify this set with Prim (A)/ as via theabove correspondence. The quotient map <j>A : Prim (̂ 4) -> Glimm (̂ 4) is known asthe complete regularization map. Note that if P, QePrim (A), GeGlimm (A) andP=>G = C\[Q] then, since [Q] is closed, Pe[Q] and so $A{P) = 4>A(Q) = G. It followsthat if PePrim(.4), GeGlimm (4) and P 2 G then <pA(P) = G.

There are two natural topologies on Glimm (̂ 4): the quotient topology rq, which isHausdorff, and the completely regular topology rcr which is the weakest topology forwhich the functions on Glimm (̂ 4) induced by C6(Prim (A)) are all continuous. For ageneral topological space X it can happen than Tcr differs from the finer topology rq

on Xj as (see [15], 3J-3) but it is not clear whether this can occur for X = Prim (A).On the other hand, there are various conditions which ensure that rq = rcr onGlimm (̂ 4). For example, if A is unital then Prim (̂ 4) is compact and hence Glimm(A) is vycompact. Since the weaker topology Tcr is Hausdorff, it must equal rq. Thetopologies rq and rcr also coincide if fiA is either Tcr-open or 7yopen (and so we mayspeak unambiguously of <j>A being open). This is immediate if <f>A is Tcr-open. On theother hand, suppose that <f>A is 7g-open. Then the local compactness of Prim (A)passes to the Hausdorff space (Glimm (̂ 4), TQ). Thus TQ is equal to the weak topologydenned on Glimm (.4) by C0(Glimm (A),Tq). It follows that Tcr~^Tq and hence

Suppose that A is unital with centre Z. It follows from the Dauns-Hofmanntheorem that for P,QePrim (A):

Hence there is a bijection ^Gl imm (^4)->PrimZ such that i/r(G) = G()Z(6'eGlimm(^)) and f~\J) = AJ (JePrim(Z)) (AJ is closed by the Cohenfactorization theorem). Thus our usage of the terminology Glimm (A) arises from[16], section 4. Since the mapping Pt->P 0 Z of Prim (̂ 4) onto Prim (Z) is continuousfor the Jacobson topologies, it follows that \{r is continuous for jq (= Tcr). SinceGlimm (̂ 4) is compact and PrimZ is Hausdorff, ^ is a homeomorphism. The aboveresults hold also in the more general case of a quasieentral (7*-algebra (see [10] forthe definition). Hence the fact that the canonical isomorphism of (G + Z)/G ontoZ/(Z n G) is isometric for GeGlimm (̂ 4) yields a short proof of a strengthened versionof the main result of [8]: if A is quasieentral and K is a compact subset of Glimm (-4)then there exist zsZ and A > 0 such that

A}:see [21].

Returning to the case of a general C*-algebra>4, let a e A. The argument in the firstparagraph of the proof of theorem 3-1 of [22] shows that the function GH> ||a + 6?|| onGlimm (̂ 4) is upper semi-continuous for both rq and Tcr (see also [8] and, in the unital

352 R. J. ARCHBOLD AND D. W. B. SOMERSET

case, [20, 16, 23]). However this function need not be continuous. The followingresult is essentially an amalgam of [22], theorem 3-1 and [18], theorem 4 (see also[8], corollary 1-9 and [23], lemma 4-1 for the equivalence of (i) and (ii)).

THEOREM 2 1 . Let A be a C*-algebra. The following conditions are equivalent:(i) <f>A is an open map;(ii) for each as A the function G>-> ||a + (?|| is Tcr-continuous on Glimm (A);(iii) for each as A the function G^- \\a + 0\\ is rQ-continuous on Glimm (̂ 4);(iv) the topology rq is given by the hull-kernel process on Glimm (̂ 4);(v) (Glimm (A),TQ) is locally compact and the mapping at^-d, where d(G) =a + G

(Ge Glimm (A)), is a *-isomorphism of A onto a maximal full algebra of cross-sectionsover (Glimm (A),Tq).

Furthermore, when these conditions hold the topologies rq and Tcr coincide onGlimm (A).

Proof, (i) => (iv). This follows from the second part of the proof of [18], theorem 4.(iv) => (v). This follows from [22], theorem 3-1.(v) => (iii). This is immediate.(iii) => (i). This follows from the first part of the proof of [18], theorem 4.

This shows that (i), (iii), (iv) and (v) are equivalent. Clearly (ii) => (iii). Finally, if (i)holds then rcr = rq and (iii) holds and so (ii) holds. I

We note in passing that it is possible to give a direct proof of the equivalence of(i) and (iii) in a more general setting: for any equivalence relation on Prim (̂ 4) forwhich the equivalence classes are closed, the quotient map is open if and only if thequotient topology is given by the hull-kernel process on the quotient space (regardedas a set of ideals). Some other necessary and sufficient conditions for the openness of<f>A in the unital case are given in [23], theorem 4-3. If A is a von Neumann algebrathen <f>A is open [16, 23] but there is an example of a quotient B of a type I vonNeumann algebra for which <j>B is not open [21]. Another weakly central C*-algebrafor which condition (iii) of Theorem 3'1 fails (contrary to the claim in [20]) is givenin [3], example 48.

In the next section we shall consider in detail the possible relations betweenGlimm (̂ 4) and Min-Primal (.4). The following result will be required.

LEMMA 2-2. Let J be a proper primal ideal in a C*-algebra A. Then there exists aunique Glimm ideal G such that G <= J.

Proof. Let PePr im (A/J). By [4], proposition 3-2, P « Q for each QePrim (A/J).Let G = <MP). Then G

= J.

Finally, if G1 e Glimm (A) and G1^ J then Gt<^P and so, as noted after the definitionof Glimm (A), G1 = <j>A(P). Hence Gx = 0. I

In certain situations Glimm (̂ 4) and Min-Primal (̂ 4) may coincide, as sets. If thisoccurs then necessarily Tcr ^ iq =% T (see Lemma 3-1) and in favourable circumstancesthe three topologies coincide (see Theorem 33). However, in [9], example 92,

although Glimm (̂ 4) = Min-Primal (.4) (as sets) the topology r is locally compactwhilst the topology rq ( = Tcr) is not. On the other hand, by generalizing theconstruction in [3], example 4-8 it is possible to construct a unital C*-algebra A (sothat rq coincides with rcr and makes Glimm (A) a compact Hausdorff space) such thatGlimm (̂ 4) = Min-Primal (̂ 4) (as sets) but T is not locally compact.

3. The main results

In this section we introduce a second relation ~ on Prim (.4) and study itsconnection with primal ideals and also with the equivalence relation as of Section 2.

For P, QePrim (A) we say that P ~ Q if P and Q cannot be separated by disjointopen subsets of Prim (̂ 4). It is immediate that if P ~ Q then P a; Q but it is possiblefor the converse to fail. For example, in the C*-algebra of sequences x = (xn)n?.1 of2 x 2 complex matrices such that, as n^-co,

Mx) ° 1 and x JMX) C0 Mx)\ 2 n + 1 0 a(.

we have ker A ~ ker/t and ker/i ~ kercr but ker A <+- ker a.Let A be a C*-algebra for which ~ is an equivalence relation and let qA: Prim (̂ 4) -»

Prim (A)/~ be the quotient map. Since each equivalence class is a closed subset ofPrim (A), Prim (A)/~ is a Trspace. It is not clear whether Prim (A)/ ~ need beHausdorff, but this is certainly the case if qA is open (see the proof of [6], lemma 7).If A has a unit then Prim (A)/ ~ is compact since qA is continuous. If, in addition,Prim (^4)/~ is Hausdorff then the functions/oqA (/eO(Prim (A)/~)) separate theequivalence classes in Prim (̂ 4), from which it follows that ~ is equal to as. Weremark that we do not know of an example in which ~ is an equivalence relation butnot equal to x.

LEMMA 3-l. Let A be a C*-algebra such that every Glimm ideal is primal. Then(i) the relation ~ on Prim (.4) is equal to as (and hence ~ is an equivalence relation);(ii) Glimm (A) = Min-Primal (A), as sets;(iii) the topology T on Min-Primal (̂ 4) is finer than the topology rq on Glimm (̂ 4).

Proof, (i) Suppose that P a; Q in Prim (A). Then <pA{P) = <j>A{Q) in Glimm (A). Sincethe primal ideal <j)A{P) is contained in both P and Q, it follows from [4], proposition32 that there is a net (Pa) in Prim (̂ 4) which is convergent to both P and Q. HenceP ~ Q as required.

(ii) Let J be a minimal primal ideal of A. Since J is primal there exists GeGlimm (̂ 4) such that G c J (Lemma 22). By hypothesis, G is primal and so G = Jby minimality. Hence JeGlimm (̂ 4).

Conversely, suppose that Ge Glimm (̂ 4) and suppose that J is a primal ideal suchthat J £ G. By Lemma 2-2 there is a Glimm ideal G' such that G' £ J £ G. Since eachprimitive ideal contains a unique Glimm ideal (see Section 2), G' = G and so G = J.

(iii) Let U be a non-empty 7yopen subset of Glimm (̂ 4). Then

0-i([/) = {PePrim(,4):P2Gforsome<?eC/}

is open in Prim (.4) and so there is an ideal J of A such that

0-i(£7) = {PePrim (A) :P $ J}.

354 R. J. ARCHBOLD AND D. W. B. SOMERSET

Suppose that/eMin-Primal (̂ 4) and/ $ J. Then there exists PePrim (A/I) such thatP $ J . Hence / = <f>A(P) e U. Conversely, it is easy to check that if Ie U then / $ J.Thus

U = {I e Min-Primal (A): 1 $ «/}which is r-open. I

I t is not necessary that every Glimm ideal should be primal in order to have ~equal to « . In [3], example 412, ~ is equal to « , but the Glimm ideal QlnQ2[\ Q3

is not primal. However, the next result gives a useful partial converse to Lemma 3-1.We are grateful to Charles Batty for supplying the inductive argument in the proofbelow, thereby enabling us to remove a separability condition that was present in anearlier version.

PROPOSITION 3-2. Let A be a C*-algebra such that ~ is an open equivalence relation.Then ~ is equal to « and every Glimm ideal of A is primal (and hence minimal primal).

Proof. Since P r i m ^ ) is locally compact and qA :Prim (^4)-s-Prim ( 4 ) / ~ iscontinuous and open, Prim (A)/ ~ is locally compact. Furthermore, as noted above,the fact that qA is open also implies that Prim (A)/ ~ is Hausdorff. Thus if P, QePrim (.4) andP + Q then there exists/eC*(Prim (A)/~) such that/og,,(P) =¥foqA(Q)and hence P $ Q. This shows that ~ is equal to ss. It follows that <f>A: Prim (A) -*•Glimm (̂ 4) is r^-open and hence Tg and rcr are equal on Glimm (̂ 4) (see Section 2).

Let Ge Glimm (̂ 4). Let% ^ 2 and suppose that JltJ2, ...,Jn are ideals of A such thatJx J2... Jn = {0}. In order to show that G is primal we require that Jt^ G for at leastone value of i. For each i let Vt = ^ ( ^ ( P r i m ^ ) ) , an open subset of Prim (A) since<j)A is open. We prove by induction on r that

(*) Vx 0 .. . fl Vr n Prim Jr+l n ... n Prim Jn = 0.

For r = 0 this follows immediately from the fact that Jt J2... Jn = {0}. Let

Uo = Prim J2 f| ... D Prim Jn,

Uk= F ,n . . . n Ffcn Prim Jfc+2(1...n Prim Jn (1 ^ k^n-2),

un_1 = v1n...nvn_l.Suppose that (*) is true when r = k for some k with 0 ^ k < n. Then Uk and Prim Jk+l

are disjoint open subsets of Prim (̂ 4). Thus if PeC/fc and QePrim Jk+1 thenP 4- Q andso <pA(P) 4= <f>A(Q)- Hence P ^ Vk+1 and so Uk 0 Vk+1 = 0 i.e. (*) holds for r = k+1. Byinduction, (*) holds for r = n and so HjLi Vi = 0.

Now suppose that Jt%G for each i = 1,2, ...,n. Fix PePrim^4/(r). For each ithere exists P(E Prim (A/G) such that Pt $ J4. Then PfGPrim J4 and <f>A(Pt) = ^X(P) soPeVt for each i, which contradicts the fact that DjLj F( = 0 . Thus Jt<=,G for at leastone value of i and so C? is primal. I

We note that for the conclusions of Proposition 3-2 to hold it is not sufficient toassume that « , rather than ~ , is open. For example, in the algebra of [2], p. 676 theequivalence relation « is open but ~ is not transitive and the Glimm ideal {0} is notprimal.

We now define a C*-algebra A to be quasi-standard if ~ is an open equivalence

relation on Prim (̂ 4). The reason for this choice of terminology is explained in Section4. Note that, as shown in the proof of Proposition 32, if̂ 4 is quasi-standard then <j>A

is open and so Theorem 2-1 applies. We shall use this in the next theorem, which givessome characterizations of quasi-standard C*-algebras. Condition (v) is the hypothesisused in [3], lemma 5-1.

THEOREM 3-3. Let A be a C*-algebra. The following conditions are equivalent:(i) A is quasi-standard;(ii) the complete regularization map <fiA is open and every Glimm ideal of A is primal;(iii) every Glimm ideal of A is primal and the topologies iq and rcr on Glimm (̂ 4)

coincide with the topology T on Min-Primal (A);(iv) every Glimm ideal of A is primal and the topology rg on Glimm (A) coincides with

the topology T on Min-Primal (̂ 4);(v) Min-Primal (A) is rs-closed in Primal' (A) (the set of proper primal ideals of A)

and each primitive ideal of A contains a unique minimal primal ideal.

Proof, (i) => (ii). This follows from Proposition 3-2 and its proof.(ii) => (iii).Assuming(ii),itfollowsfromLemma3-lthatGlimm (̂ 4) = Min-Primal (A)

and T ^ TQ = rcr. Since <j>A is open, it follows from Theorem 2-1 ((i) => (iii)) that TQ

is finer than T, and so T = rg = TCT.(iii) => (iv). This is immediate.(iv) => (v). Let (Ja) be a net of minimal primal ideals which is rs-convergent to a

proper primal ideal / . Let aeA\I. By [13], 3-3-6, the set

X = {GeGlimm (A):\\a + G\\ > | | |o+/ | |} ,

is the image under <j>A of the compact subset

of Prim (A). Thus X is rg-compact and, assuming (iv), contains Ja eventually. Hencethere is a subnet of (JJ which is 7Q-convergent to some GeX. By (iv), this subnetis rs-convergent to G and so G = I by uniqueness of rs-limits in Primal (̂ 4). ThusIeMin-Primal (̂ 4) by (iv).

Suppose that P is a primitive ideal containing minimal primal ideals Jt and J2.Then (iv) yields that Jx = <j>A(P) = J2.

(v) => (i). Assuming (v), it is shown in [3], lemma 5-1 that the map O, which takeseach primitive ideal to the unique minimal primal ideal that it contains, iscontinuous and open. Let P,QePrim (A). We shall show that P ~ Q if and only if<D(P) = <$>(Q), from which it will follow that ~ is an open equivalence relation.

Suppose that P ~ Q. Then P [) Q is primal by [4], p. 60. Let J b e a minimal primalideal contained in P n Q. By uniqueness, <&(P) = J = Q>(Q). Conversely, suppose that<&(P) = Q>(Q). By [4], proposition 3-2 there is a net in Prim (̂ 4) which is convergent toboth P and Q. Hence P ~ Q. I

Examples of quasi-standard 6'*-algebras are discussed in [3], section 5, usingcondition (v) of Theorem 33. There are some interesting examples of quasi-standardgroup C'*-algebras. For example, the C*-algebra of the discrete Heisenberg group isquasi-standard ([3], example 5-4), as is the C*-algebra of the continuous Heisenberggroup (described in [19], section III). Another interesting example of a quasi-

standard C*-algebra is given in [19], section IV. The easiest way to see that thesealgebras are quasi-standard is to use the characterization given below in Theorem34.

If / is a non-zero ideal of a C*-algebra A then Prim (/) is (homeomorphic to) anopen subset of Prim (A) and so it follows directly from the definition that if A is quasi-standard then so is the G'*-algebra / . On the other hand, although every vonNeumann algebra is quasi-standard (see [3], section 5) there is an example of aquotient of a type I von Neumann algebra which is not quasi-standard [21]. It alsofollows from Proposition 3-6 below that a quotient of a quasi-standard C*-algebraneed not be quasi-standard.

Recall that if a C*-algebra A is a maximal full algebra of cross-sections over a basespace X with fibre Ax at xeX then there is a one-to-one correspondence between Xand a set of ideals of A given by

#<->/x = {xeA:ax = 0} (xsX).

For each xeX there is an (isometric) *-isomorphism of A/Ix onto Ax given by

THEOREM 3-4. Let A be a C*-algebra. Then the following conditions are equivalent:(i) A is quasi-standard;(ii) Min-Primal (̂ 4) is locally compact and A is *-isomorphic to a maximal full algebra

of cross-sections over Min-Primal (^4);(iii) A is *-isomorphic to a maximal full algebra of cross-sections in which the base

space contains a dense set of proper primal ideals.

Proof. (i)=>(ii). This follows from Theorem 3-3 and either Theorem 2-1 or [3],theorem 53 .

(ii) => (iii). This is immediate.(iii) => (i). We may suppose that A is a maximal full algebra of cross-sections over

a locally compact Hausdorff space X and that a dense subset of X consists of properprimal ideals. The topology on X is stronger than TS and hence stronger than TW. SincePrimal (̂ 4) is rw-closed in Id (A) (by [3], proposition 31), each ideal in X is primal. By[18], theorem 4 there is a continuous open map <f> of Prim (̂ 4) onto X such that<fi(P) e P (PePrim (.4)). By the maximality of the full algebra, for each PePrim (̂ 4)the ideal (f>(P) is the unique ideal of X contained in P (see, for example, the proof of[3], theorem 5-3). Let P,QePrim(A). We shall show that

Suppose that <j)(P) = <j>(Q). Since <f>{P) is primal and P,Q^ (P) it follows from[4], proposition 3-2 that P ~ Q and hence P K Q. Conversely, suppose that<j){P) #= ((>{Q). Without loss of generality we may assume that there existsae<f>(P)\0(Q). For RePrim (̂ 4) let f(R) = \\a + <fi(R)\\. Since <j> is continuous we have/eCb(Prim (A)) and hence P 4 Q.

It now follows that for P e Prim (A)

Thus X = Glimm (^4), as sets, and <f> = <f>A. Since <j> is continuous and open, the giventopology on X equals the quotient topology Tq on Glimm (A). Furthermore, 0 4 is openand so A is quasi-standard by Theorem 3-3 ((ii) => (i)). I

COROLLARY 3-5. Let A be a separable C*-algebra. The following conditions areequivalent:

(i) A is quasi-standard;(ii) A is *-isomorphic to a maximal full algebra of cross-sections in which the base

space contains a dense set of primitive ideals.

Proof, (i) => (ii). This follows from Theorem 3-4 ((i) => (ii)) and the fact that if A isseparable then Min-Primal (.4) contains a dense subset of primitive ideals (see [3],corollary 4-6).

(ii) => (i). This follows from Theorem 3-4 ((iii) => (i)) since primitive ideals areprimal. I

There is a construction due to Dixmier[12], proposition 4 of an inseparable non-unital C*-algebra such that no minimal primal ideal is primitive [3], pp. 533-4. It canbe checked directly that this algebra is quasi-standard. Thus the implication(i) => (ii) in Corollary 3-5 need not hold without the assumption of separability.

We now illustrate the application of Theorem 3-4 to three of the examplesmentioned after Theorem 3-3. Let A be the C*-algebra of the discrete Heisenberggroup [1]. Then A is *-isomorphic to a maximal full algebra of cross-sections over theunit circle T. If A e T is not a root of unity then the fibre at A is the correspondingirrational rotation algebra and so the ideal of A corresponding to A is primitive andhence primal. By Theorem 3-4 ((iii) => (i)), A is quasi-standard (see [3], example 5'4for another proof).

Next, let A be the C*-algebra of the continuous Heisenberg group. Then A is*-isomorphic to a maximal full algebra of cross-sections over IR (see [19], section III).For A 6 IR\{0}, the fibre at A is *-isomorphic to the algebra of compact linear operatorson a separable infinite-dimensional Hilbert space. Thus the ideal of A correspondingto A is primitive and so A is quasi-standard by Theorem 3-4.

Finally, let A be the algebra of [19], section IV, which is *-isomorphic to a maximalfull algebra of cross-sections over the one-point compactification of the positiveintegers. The fibre at the integer n is *-isomorphic to the algebra of nxn complexmatrices and so A is quasi-standard by Theorem 3-4.

The next result is a development of [4], p. 59 and [3], proposition 4-7 which willbe useful in Section 4.

PROPOSITION 3-6. Let B be an arbitrary C*-algebra. Then there is a quasi-standardC*'-algebra A with a minimal primal ideal J such that B = A/J. If B is separable then Acan be constructed so as to be separable.

Proof. We may assume that B is a C*-algebra of operators on a Hilbert space H.Let A be the C*-algebra of all sequences x = (xn)n>1 of elements in L(H) which arenorm-convergent to an element a(x) in A. Then A is a full algebra of cross-sectionsover the one-point compactification of the positive integers and A is maximal by[14], corollary on p. 242. The ideal corresponding to a positive integer is primitiveand so the proof of (iii) => (i) in Theorem 34 shows that A is quasi-standard and theideal J corresponding to the point at infinity (the ideal of null sequences) is a primalGlimm ideal and hence minimal primal. Clearly B S A/J.

358 R. J . ARCHBOLD AND D. W. B. SOMERSET

Suppose that B is separable. Then we can construct A to be separable as in [3],proposition 4-7 and we can apply Theorem 34 as in the previous paragraph. I

4. Comparison with Ching's results

In this section we compare the quasi-standard C'*-algebras of Section 3 with thepre-standard C*-algebras introduced by Ching[6]. We show that the class of pre-standard algebras is strictly included in the quasi-standard class. Thus the theory ofquasi-standard O*-algebras, which requires no assumptions concerning the existenceof minimal primitive ideals, may be regarded as an effective generalization of the pre-standard theory. We begin with some terminology from [6].

A C'*-algebra A is said to be bounded if every primitive ideal contains a minimalprimitive ideal. If A is separable then every prime ideal is primitive [11] and so itfollows from Zorn's lemma that A is bounded. The basic structure space Ab of abounded C*-algebra^4 is defined to be the set of all minimal primitive ideals with therestricted Jacobson topology. It is easy to check that if A is bounded and P,QeA"then P +• Q if and only if P and Q can be separated by disjoint open sets in Ab. Therestriction of ~ to Ab is not necessarily an equivalence relation, as is shown by theexample for ~ itself (see Section 3) in which Ab coincides with Prim (̂ 4).

A C*-algebra A is said to be *-bounded if it is bounded and whenever S is a non-empty subset of Ab such that kerS^P for some PePrim (̂ 4) then there existsPo eA" such that ker S cr Po c p . A C*-algebra A is said to be normal if it is *-boundedand each primitive ideal of A contains only one minimal primitive ideal of A. Toillustrate the differences between the above notions, we note that in [3], example 4-8A is bounded but not *-bounded (consider the inclusion / 2 nB ; s lP2 n) , whilst theexample in [2], p. 676 is *-bounded but not normal (see [3], p. 537). This shows thata postliminal C*-algebra need not be normal, thus answering a question raised in [6],p. 141.

We can now give the key definitions from [6]. A C*-algebra A is standard if it isnormal and Ab is a Hausdorff space. A is pre-standard if it is normal and therestriction of ~ to Ab is an open equivalence relation. It is shown in [6], theorem 1that if A is standard then A is *-isomorphic to a maximal full algebra of cross-sectionswith each fibre a primitive C*-algebra. It is not difficult to show that, conversely,each such maximal full algebra of cross-sections is standard. These facts should becompared with Theorem 3-4 and Corollary 35.

Using Proposition 3-6 we are able to construct examples of quasi-standard C*-algebras which are not pre-standard. Let B be any C*-algebra with a primitive idealcontaining two distinct minimal primitive ideals (a separable example is given in [2],p. 676). Then the C*-algebra A, as constructed in the proof of Proposition 3-6, isquasi-standard but not normal. As indicated in Proposition 3-6, if B is separable theconstruction can be modified so as to ensure that A is separable.

We now show that every pre-standard C*-algebra is quasi-standard.

THEOREM 4-1. Let A be a pre-standard C*-algebra. Then A is quasi-standard.

Proof. Let X = Ab/~ and let q:Ab-+Xbe the quotient map. By [6], lemma 7, X isHausdorff. For each PePr im (̂ 4) let Po denote the unique minimal primitive ideal ofA contained in P.

We define a map <D:Prim (A)-^X by <J>(P) = q(P0) (PePrim (A)). If U is an opensubset of Prim (̂ 4) then, since A is bounded,

®(U) = ®(U fi A") = q(U fl 4 6 )

which is an open set of X. Thus iscontinuous.

Let Y be a closed subset of X. As in [6] we may identify X with a set of ideals inA via the one-to-one correspondence:

x++r\{PeA":q(P) = x] (xeX).

Then ker Y = kerO"1^) and so for the continuity of <f> it suffices to show that

O-^y) = hull ker Y,

where the hull is taken in Prim (A). So suppose that P e Prim (̂ 4) and P 2 ker Y. Sinceker Y = kerq~1(Y) and A is *-bounded, there exists P0EAb such that

Thus Po lies in the closure of q^iY) in Ab and so q(P0) e Y — Y by the continuity of q.Conversely, if P e Prim (A) and O(P) e Y then ker Y^<S>(P)^P and so P e hull ker Y.Thus (P) #= <D(Q) then P ^ Q since X is Hausdorff and <1> is continuous. Conversely, ifO(P) = (<2) then <7(P0) = g(Q0) and so Po ~ Qo. Since every neighbourhood of P(respectively Q) in Prim (̂ 4) contains Po (respectively Qo), it follows that P ~ Q.

Since P ~ Q if and only if (P) = O(Q), it follows from the fact that is acontinuous open mapping that ~ is an open equivalence relation as required. I

The results of this section show that, as contended in [3], p. 541, the notion ofnormality used in the definition of a pre-standard algebra is stronger than isnecessary for a maximal representation by cross-sections. One may weaken thenotion of normality to: every primitive ideal contains a unique ideal in the quotientspace X. It can then be proved that a bounded C*-algebra is quasi-standard if andonly if it is 'pre-standard' (using the amended version of normality).

We are grateful to Charles Batty for helpful discussions and particularly forenabling us to remove a separability condition from an earlier version of Proposition3-2. The second-named author gratefully acknowledges the support of the S.E.R.C.

Note added in proof. Concerning the remarks preceding Lemma 31 , the second-named author has recently shown in [21], proposition 3-2, that if A is a quasi-centralC*-algebra then ~ is an equivalence relation on Prim (A) if and only if ~ is equalto x on Prim (̂ 4).

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