ADVANCES IN MATHEMATICS 67, 1-140 (1988) Algebraic K-Theory of Stable C*-Algebras NIGEL HICSON Department qf Mafhemarics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 Let A be a unital C*-algebra and let 9 denote the Calkin algebra (the bounded operators on a separable Hilbert space, module the compact operators Jr). We prove the following conjecture of M. Karoubi: the algebraic and topological K-theory groups of the tensor product C*-algebra A 02 are equal. The algebra A 01 may be regarded as a “suspension” of the more elementary C*-algebra A @IX; thus Karoubi’s conjecture asserts, roughly speaking, that the algebraic and topological K-theories of stable C*-algebras agree. t 1988 Academic Press, Inc. Let A be a C*-algebra and suppose, for the sake of simplicity, that A is unital. The general linear group of A of dimension n, denoted GL,,A, is the group of invertible elements in the n x n matrix algebra M,(A) over A. This paper is about comparing invariants of GL,,A, considered as a topological space (with the norm topology, inherited from A), to invariants of GL,,A considered as a discrete group. On the topological side, we are going to study the topological K-theory of A, denoted K’.+(A), which is nothing more than the homotopy, n,(GL,A), of GL,,A. To be precise, K’,(A) is the homotopy of the CL, A, the “limit” as n -+ IX) of the GL,,A: it turns out to be a great conveneince to study this “stable” group, rather that the non- stable groups CL,,. This has been the object of quite intense scrutiny by operator algebraists in recent years. The result of this attention has been the development of powerful techniques to compute K:(A) for a great variety of C*-algebras A, and numerous applications, both to the theory of operator algebras, and perhaps more importantly and more significantly, to various other disciplines, notably differential topology. The algebraic invariant of GL,A is called the algebraic K-theory of A, denoted K,(A). It can be defined for any ring, and from our point of view, it is an analogue of the homotopy of GL,A in a purely algebraic context. (Again, to be precise, we consider GL, A rather than GL,A.) For instance, in the topological set- ting, ;rr,(GL, A) = GL 35A /GLf,= A, where CL”, A denotes the connected component of the identity; in the algebraic setting, the corresponding group is GL, A/PGL r A, where PGL T A denotes the maximal perfect sub- OOOl-8708/88 $7.50 CopyrIght c 1988 by Academic Press. Inc. All rigbfs of reproduction in any form reserved. 2 NIGEL HIGSON group, which plays the role of the connected component of the identity. Next is the fundamental group rt ,: in the topological case this is obtained from the universal covering group of CL”, A; we obtain the algebraic group from the analog of this-the universal central extension of PGL,,A. Stated informally, the main theorem of this paper is as follows. (The statement is imprecise due to the fact that we will consider not stable C*-algebras, but “suspensions” of stable C*-algebras. We do not need to go into the details of this immediately.) If A is a stable F-algebra then the algebraic K-theory of A is equal to the topological K-theory of A. This is, we think, an interesting result for the following reasons. First, the algebraic K-theory of a ring is in general quite inaccessible. For example, the algebraic K-theory of say the integers Z is not yet known (although the first several groups K,,(Z) are). Again, the groups K,(C) have only recently been determined, as a result of very deep computations. Yet for a stable C*-algebra, the algebraic K-theory is the same as the relatively accesible topological K-theory. The second, and we think more interesting point regards the quite distinct natures of topological and algebraic K-theory. Topological K-theory is constructed by considering GL,A solely as a topological space; on the other hand, algebraic K-theory is constructed from GL,,A, considering it solely as a discrete group, without regard to topology at all. Yet these two different approaches lead to exactly the same group, in the case of a stable C*-algebra. This last point brings us to an interesting parallel with the Brown Douglas Fillmore theory of extensions of C*-algebras. Since this is in many ways the foundation of our work, we want to spend a few lines now acquainting the reader with the broad outlines of it. An essentiall”v normal operator N on a Hilbert space is an operator for which the self-com- mutator [N, N*] is compact (as opposed to zero, in which case N would be normal). It is quite clear from the definition that every compact pertur- bation of a normal operator is essentially normal, and the question arises: does this exhaust the class? It is not hard to see that the answer is “no.” For example, the unilateral shift is essentially normal but not of this form. Two operators N,, N, are essentially unitarily equivalent if there exists a unitary U such that UN1 U* - N2 is compact. An obvious invariant of this equivalence relation is the essential spectrum a,(N), that is, the spectrum of the image of N in the quotient C*-algebra 93/X, and the natural question to ask is: what are the possibilities (up to essential unitary equivalence) for an essentialy normal operator with given essential spectrum X? It turns out that they are classified by elements of an abelian group denoted Extt’(C(X)). The construction of Ext -‘(C(X)) and the fact that it is a group is in itself remarkable, but it is the even more remarkable description of Ext ‘(C(X)) given by Brown, Douglas, and Fillmore that we are ~mm~rcK-THEORYOF C*-ALGEBR.~S 3 interested in. Recall for a moment the classification of normal operators up to unitary equivalence on a separable Hilbert space. A complete list of invariants is: the spectrum X of the operator, an equivalence class of measures on X, and a “multiplicity function” X+ { 1, 2,..., co}. (If the multiplicity function is constantly 1 then the operator is simply multi- plication by x on L’(X); in general it is a direct sum of pieces of this, as dictated by the multiplicity function.) In contrast, if N is essentially normal then the essential unitary equivalence class of N is the set of all essentially normal operators N’ with the same essential spectrum X as N, for which index(i - N) = index(i -N’) for every complex number i in the complement of X. Thus N is charac- terized by X and the “multiplicity function” 3.F + index(>” - N), defined on the complement of X. This is a result of the following beautiful fact: Ext ‘(C(X)) is equal to the (odd-dimensional) K-homology qf A’. It is not important for us to describe exactly what the K-homology of a space X is. The point we want to make is that the purely algebraically defined group Ext .‘( C(X)) turns out to be completely topological in character. This is a very intriguing and remarkable phenomenon, and the results of this paper are offered as another illustration of it. Besides the theorem mentioned, we present a number of other results, mostly on the same theme of comparing algebraic and topological K-theory, but occasionally as minor digressions from it. Most sections begin with a brief summary of their contents; however, let us give here an outline of the contents of the work as a whole. Section I Almost the whole of the paper relies in a very crucial way on the technical underpinings of extension theory, as developed in its general form by Kasparov. These are results on the structure of multiplier algebras and the outer multiplier algebras, or as we shall call them, “generalized Calkin algebras” J?‘(X@B)/X@ B. The two main results are a separation theorem of Kasparov, concerning orthogonal subalgebras of a Calkin algebra (Theorem 1.1.1 1) , and a type of stabilization theorem, which com- pares the multiplier algebras ..&‘(X@ B) and .X(XOJ), where J is an ideal of B (Theorem 1.3.14). Various other C*-algebra preliminaries are also given. Section II There are basically three topics covered. The first is the introduction of topological K-theory, about which we need say nothing here. The second is 4 NIGEL HIGSON the introduction of algebraic K-theory. The definition of the higher algebraic K-theory groups is due to Quillen; it is a beautiful illustration of the interplay possible between algebra and algebraic topology. A certain amount of familiarity with topology is necessary to work with it, and in an attempt to make the paper accessible to non-topologists we have included most of the background needed. The third topic of the section is the exten- sion theory of C*-algebras. We have included it partly because of the close parallels between our results, as we have described; partly because exten- sion theory provides a good illustration of some of the techniques we develop; and partly because, by means of these techniques, we are able to contribute a little to the simplification of the subject. Section III The main topic of the section is a homotopy invariance theorem, proved in a general context. The techniques in the proof are for the most part borrowed from Kasparov’s treatment of the homotopy invariance of the extension groups. However, we use ideas due to Cuntz to put Kasparov’s work in an abstract setting, and the result is quite surprising: any functor from C*-algebras to abelian groups which is “matrix stable” and which preserves split exact sequences is homotopy invariant. Section IV We prove that if A is a stable C*-algebra then the following three objects are equal: (i) The universal connected covering group of GL:A. (ii) The Steinberg extension of the group E, A of elementary matrices in GL,,A. (iii) The universal central extension of the maximal perfect subgroup of GL, A. As a result, the algebraic K,-group of A is equal to topological K2. Section V This is the main section in the paper. We prove the theorem already stated that the topological and algebraic K-theory groups for Calkin algebras are equal. Section VI There are two main topics. The first is what might be called non-stable K-theory-the study of the group GL, A instead of the stable version GL, A. As we mentioned earlier, it is a considerable simplification to work with GL, rather than GL, for some fixed n: this section should illustrate ALGEBRAIC K-THEORY OF C*-ALGEBRAS 5 the point. However, we are able to show that if B is unital then the non- stable algebraic K-theory of &(X @ B)/X@ B is equal to its non-stable topological K-theory. By results already known in topological K-theory, this implies that the non,stable and stable algebraic K-theories agree for these algebras. The other topic is the Karoubi-Villamayor algebraic K-theory, which is another possible definition for the homotopy of the dis- crete group GLA. We prove that for a stable C*-algebra, this too is equal to the topological K-theory. This paper is a modification of the author’s Ph.D. thesis (Dalhousie University, 1985). He would like to take this opportunity to thank his supervisor, Peter Fillmore, for his patience and support, as well as Dick Kadison for his encouragement and his interest in this work. I. MULTIPLIER ALGEBRAS AND TENSOR PRODUCTS 1.1. Multiplier Algebras Let A be a C*-algebra. There are a number of definitions of the mul- tiplier algebra, d(A), of A, of which the following (the original one, due to Johnson [28]) is perhaps the most concrete. DEFINITION 1.1.1. A (double) centralizer of A is a pair (L, R) of linear maps from A to itself, such that: L is a right A-module homomorphism (i.e., L(xy) = L(x)y); R is a left A-module homomorphism; and R(x) y = XL(JJ) for all x and y in A. The composition of two centralizers (L,, R,) and (L2, R,) is given by (L,, R,NL,, &I= (L,L,, &RI) (1.1.1) (which is easily seen to be a centralizer itself), and the resulting algebra is the multiplier algebra of A, denoted ,&‘(A). This rather odd definition is made with a simple idea in mind: if A is an ideal in an algebra B, and if b E B, then b defines a multiplier (L, R), by L(x) = bx, R(x) = xb. (1.1.2) We note that the composition law (1.1.1) corresponds to multiplication of elements in B. Of course, A is contained as a trivial ideal in itself, and since for any x E A there is some y E A (namely y = x*, for example) such that xy # 0, the element of &‘(A) corresponding to x is non-zero. Thus A is embedded as a subalgebra of &‘(A), and it is easily verified that A is in fact an ideal in d(A). (From this we see that every element (L, R) of ,+&‘(A) is 6 NIGEL HIGSON obtained in the manner of (1.1.2), namely take b = (15, R) and B=&(A).) Continuing along these lines, we arrive at the following useful (and, of course, well known) characterization of .&(A). THEOREM 1.1.2. If B is any algebra containing A as an ideal then there is a unique homomorphism from B to A(A) which extends the inclusion of A into &I( A). The algebra &2’(A) is in fact a C*-algebra, for it turns out that L and R are both bounded linear maps, of equal norm (see [ 141) and we may set ll(L R)Il = IILII (= IIRIIL (1.1.3) which makes .&‘(A) into a Banach algebra. We define an involution on .#(A) by (L, R)* = (L*, R*), where L*(x)= (R(x*))* and R*(x)= (L(x*))*. With respect to this and the norm (1.1.3), ,&(A) is a C*-algebra. In Theorem 1.1.2, if A is a closed ideal in a C*-algebra B then the canonical homomorphism from B to ,&(A) is a *-homomorphism. Apart from obtaining elements of &‘(A) via algebras containing A as an ideal, the main source of supply is from certain limits. For this it is useful to introduce the strict topology on .&‘(A), which is characterized by the following: a net i-u,} in .&(A) converges in the strict topology to ,YE ,X(A) if and only if for every a E A the nets { x,a} and {ax,} converge in the norm topology to xa and a.x, respectively. For details, see [14]. The following simple fact is very useful: the strict topology on ,&(A) is com- plete, in the sense that if for every aE A, the sequences (x%a} and {ax,} are Cauchy (in the norm topology), then {?c,} converges in the strict topology. EXAMPLE 1.1.3. Denote by x the C*-algebra of compact operators on a separable Hilbert space. Then the C*-algebra B of all bounded operators on the Hilbert space contains Z” as an ideal, and so by Theorem 1.1.2 there is a canonical *-homomorphism ~8 -+ &(xX). This map is in fact a *-isomorphism since it follows from elementary representation theory that &I enjoys the universal property described in Theorem 1.1.2. The notations 58’ and % for bounded and compact operators on a separable Hilbert space will be used throughout the rest of the paper without further explanation. There is an interesting genralization of this, which is useful to bear in mind (for counterexamples, and so on). If C(X, X) denotes the C*-algebra of norm continuous functions from a compact space X to x then Jz’(C(X, ,X)) is equal to C,,,(X, g), the C*-algebra of bounded functions A(x) from X to g which are continuous in the *-strong topology (i.e., both AuxBR~rc K-THEORYOF P-ALGEBRAS 7 A(x) and A*(x) are strongly continous; the *-strong topology on 9J is equal to the strict topology on bounded subsets); see [l]. Note that C(X, X) and C,,,(X, 9) are, respectively, the (pointwise) compact and the bounded endomorphisms of the trivial field of Hilbert spaces over the space X. In general, it is very convenient to regard elements of A as “compact operators” and elements of eg (A) as “bounded operators.” We consider now the functorial properties of the multiplier algebra, beginning with what we shall call “restriction.” Suppose that A’ is an ideal in A. Then from the fact that A’. A’ = A’, it follows that A’ is also an ideal in J!(A). DEFINITION 1.1.4. The restriction homomorphism r: ..&‘(A ) + =M( A’) is the unique *-homomorphism that extends the inclusion of A’ into .&(A’). Notice that if A’ is an essential ideal of A. in other words, if the annihilator ideal Ann(A’)= (aEA / aA’=A’a=O). (1.1.4) is zero, then the restriction homomorphism is injective. Because of this, whenever Ann(A’) = 0 we will regard &‘(A) as a C*-subalgebra of .&(A’). It is useful to have the following characterization of this subalgebra: if x E mK ( A’) then x E ,.N( A ) if and only if x.AcA and A..rc A. (1.1.5) Indeed, if x satisfies ( 1.1.5) then .Y defines a double centralizer of A, as in Definition 1.1.1. and the image of this centralizer in .&‘(A’) under restric- tion returns .Y. Let us turn from the restriction homomorphism to a discussion of the covariant functoriality of .&‘(A). Unfortunately it is not true that every *-homomorphism f: A, -+ A, extends to a *-homomorphism from .&‘(A,) to J?‘(A?). However, by means of the next three results we are able to get by. LEMMA 1.1.5. (See [43, Proposition 3.12.121.) Q” f[A,] contains an approximate unit for A, therz the map j A, + A, extends uniquel~~ to a *-homomorphism f: -U( A, ) -+ -.k!( AZ). All the approximate units that we deal with in this paper are assumed to be positive and increasing. The condition that f[A,] contain an approximate unit for A, is equivalent tof[A,] A, being dense in A,, or in the other words, it is equivalent to the hereditary subalgebra generated by ./-[A,] in A, being equal to AI. 8 NIGEL HIGSON Proof. First, if an extension f: &!(A,) -+ &(Az) exists, then it is cer- tainly unique because, for x E .&‘(A,) and a E A, we have f(x) a =f(x) j,l&fM a (1.1.6) = lim f(xuJ a, j. + cc and similarly, of(x) = lim (uf(-x))f(ud i. - rl (1.1.7) = lim uf(x~,), i. -+ r where {U;. jj.E n is an approximate unit for A, (and so {f(uj,)} is an approximate unit for A,). In other words, f(x) is the limit in the strict topology of the net {~(xu;.)}. Since xu,~A,, the bottom lines of (1.1.6) and ( 1.1.7) do not depend on the extension off: On the other hand, it is readily verified that the limits (1.1.6) and (1.1.7) always exist: if a E~[A, ] A 2 then this is clear, while the case of a general a E A z is dealt with by approximating with elements off[A ,] AZ. It is easily seen that the limits define an extension off from J$‘(A ,) into &!(A,). 1 The following definition gives a class of *-homomorphisms which is large enough for our purposes, and all of whose elements are extendible. DEFINITION 1.1.6. A quasi-unitul *-homomorphism f from A i to A, is a *-homomorphism with the property that the hereditary subalgebra of A, generated by fC.4 i] is of the form pA,p, where p is a projection in .,4!(A2). Notice that the projection p in this definition is unique, if it exists at all, since, for example, 1 -p may be recovered from f as the unit of the C*-algebra of x E &+‘(A2) such that xf[A i] = 0 =f[A i] x. PROPOSITION 1.1.7. A quasi-unitul map f: A, + A, extends to a *-homomorphism from &?(A,) to &(A,). Proof. Let {uj,> be an approximate unit for A,. Define J ,&‘(A,) + &2’(A2) by the formulas f(x)u=/~~f(x%)u, (1.1.8) Uf(X) = lim Uf(XUj,), i. + cc ALGEBRAIC K-THEORY OF C*-ALGEBRAS 9 where a E A, and x E &‘(A ,). By writing a as pa + (1 -p) a, with p the pro- jection of Definition 1.1.6, since ~(xu~)( 1 -p) = 0, we see from Lemma 1.1.5 that the limits exist and define a *-homomorphism from .&Z(A,) to &(A,) as required. 1 Of course, the extension off! A, -+ A, is not necessarily unique, for we can add to the map defined by (1.1.8) any *-homomorphism from the quotient ,&‘(A,)/A, to (1 -p) &‘(A?)(1 -p). However, when we speak of the extension of a quasi-unital map we will always mean the one given by ( 1.1.8); we will denote it simply by,f: PROPOSITION 1.1.8. (i) The composition of t~‘o quasi-unital maps is quasi-unital. (ii) Extension of quasi-unital maps to multiplier algebras is functorial. Proof (i) Suppose that f, : A, + A, and fi: A, + A, are quasi-unital. Let pAzp be the hereditary subalgebra of A, generated by f,[A,], and let qA,q be the hereditary subalgebra of A, generated by f2(A2). If {uj,JiE,, is an approximate unit for A, then { fi(ui))- j.En is an approximate unit for pA2 p. Therefore, if {u;., >,.,E, 1, is an approximate unit for (1 -p) A,( 1 -p) then Cfi(uj.) + Di,l (>,.,.‘)E.j% .I’ is an approximate unit for A,, and so {f2fi(u1)+fZ(U1,)}(~,i,,E.,rA’ is an approximate unit for qA,q. Finally, from the fact that f2(p)f2(t);.,) =O, it follows that (f2 f,(u,)),,, is an approximate unit for f?(p) A,f,(p). Hence ,f2 f, [A, ] generates f2(p) A, f,(p) as a hereditary subalgebra. (ii) We have to show that the extension off2 f, is equal to the exten- sion off, composed with the extension of fi. Thus if x E .&‘(A,) we must show that f2 fi(x) andf,(f,(.u)) are the same element of dl(f,(p) A, f,(p)). Since f2 f,[A,] generates f2(p) A3,f2(p) as a hereditary subalgebra, it suf- fices to show that f2,fi(.u) h=,fi(f,(x)) b for any bEfif,[A,]. But if b=f2 f,(a) then f;fi(,x) b=f,f,(xa) (since fi f, is a homomorphism) =.fdf,(-xa) (since xa E A 1) =f2(fi(*x)fi(a)) (since f; is a homomorphism) =f?(fi(-x))fAf1(a)) (since fi is a homomorphism). 1 We close our discussion of functioriality properties by making note of a theorem of Akemann, Pedersen, and Tomiyama [l]. The following terminology is due to Pedersen [44]. DEFINITION 1.1.9. A C*-algebra is said to be o-unital if it possesses a countable approximate identity. 10 NIGEL HIGSON THEOREM 1.1.10. If A is a a-unital C*-algebra then any surjective *-homomorphism A -+ A/J extends to a surjective *-homomorphism ,4’(A) + ,X( A/J). Actually, the original version of this is for separable C*-algebras A only; the extension to o-unital algebras is carried out by Pedersen in [44]. Next, we state a separation theorem for subalgebras of J%‘(D) which will be used a great deal in the sequel. THEOREM 1.1.11. Let D be a C*-algebra; let E, and E, be C*-subalgebras of .4?(D), E, o-unital and E, separable; let E be a (closed, two-sided) ideal in E, ; and let 9 be a separable, linear subspace of A(D). Zf E,.EZcE, [F,E,]cE, and DcE,+E 2, then there exists an element NE,&(D) such that l>,N>O, (1-N).E,cE, N.E,cE, and [N, S] c E. (The symbol [S, E,], for example, denotes the set of all commutators [IF, e] =fe - ef, wherefE 9 and e E E, .) This extremely useful result is due to Kasparov [35] (for a shorter proof, see [26]). It is a basic tool in Kasparov’s bivariant K-theory, where it is the principal technical component in the construction of the Kasparov product map KK(A,,B,OD)xKK(A,OD,B,)~KK(A,OA,,B,OB?): the operator N, together with its “complement” M= (1 -N) appear as weights in the averaging of two operators, and the theorem asserts that these weights can be chosen as to produce an average with certain desirable properties (which, for example, make it amenable to study from the point of view of index theory). For details, see Kasparov’s papers (see [33; 35, especially Remark 3, p. 7731 for a motivation of the construction). Our uses of Theorem 1.1.11 will on the whole be more algebraic in nature. We will appeal to it in Section 1.3 when we discuss exact sequences related to mul- tiplier algebras. In Section 3.5 we will use it to prove various excision properties of extension groups, and in Section 5.2 we will use it to deduce the existence of local units in certain C*-algebra ideals (see Theorem 5.2.1). Finaly, it makes an appearance in a technical result in Section 6.1. 1.2. Tensor Products Let A and B be C*-algebras, and denote by A 0 B the (algebraic) tensor product of A and B. We put a C*-norm on A 0 B as follows. Pick faithful representations pa and pB of A and B, on Hilbert spaces Y& and ~8’~. Using and pe we embed A 0 B in B’(yic,) 0 a(~&), and since PA B(yic4) 0 .B(&$) embeds in an obvious way into a(%” @ %B), we obtain a faithful representation of A Q B in !B(;ci”A0 XB): the operator norm on a(~?~ @ ZB) then gives the C*-norm on A 0 B. It is not hard to show that