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APPEARED IN BULLETIN OF THE

arXiv:math/9301219v1 [math.OA] 1 Jan 1993RESEARCH ANNOUNCEMENTAPPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 1, January 1993, Pages 75-83FACTORIZATIONS OF INVERTIBLE OPERATORSAND K-THEORY OF C∗-ALGEBRASShuang ZhangAbstract. Let A be a unital C*-algebra.

We describe K-skeleton factorizations ofall invertible operators on a Hilbert C*-module HA, in particular on H = l2, withthe Fredholm index as an invariant. We then outline the isomorphisms K0(A) ∼=π2k([p]0) ∼= π2k(GLpr(A)) and K1(A) ∼= π2k+1([p]0) ∼= π2k+1(GLpr(A)) for k ≥0,where [p]0 denotes the class of all compact perturbations of a projection p in theinfinite Grassmann space Gr∞(A) and GLpr(A) stands for the group of all thoseinvertible operators on HA essentially commuting with p.1.

IntroductionThroughout, we assume that A is any unital C*-algebra. Let HA be the Hilbert(right) A-module consisting of all l2-sequences in A; i.e., HA := {{ai} : P∞i=1 a∗i ai ∈A}, on which an A-valued inner product and a norm are naturally defined by< {ai}, {bi} >= P∞i=1 a∗i bi and ∥{ai}∥= ∥(P∞i=1 a∗i ai)1/2∥.

Let L(HA) stand forthe C*-algebra consisting of all bounded operators on HA whose adjoints exist,and let K(HA) denote the closed linear span of all finite rank operators on HA,respectively. In case A is the algebra C of all complex numbers, HA is the sepa-rable, infinite-dimensional Hilbert space H = l2; correspondingly, L(HA) reducesto the algebra L(H) of all bounded operators on H, and K(HA) reduces to thealgebra K of all compact operators on H. Each element in L(HA) can be identifiedwith an infinite, bounded matrix whose entries are elements in A [Zh4, §1].

Thisidentification can be realized by C*-algebraic techniques and the two important*-isomorphisms L(HA) ∼= M(A ⊗K) and K(HA) ∼= A ⊗K = (lim→Mn(A))−; whereM(A ⊗K) is the multiplier algebra of A ⊗K [Kas]. For more information aboutmultiplier algebras the reader is referred to [APT, Bl, Cu1, El, Br2, Pe1, OP, L,Zh4–5], among others.

The set of projectionsGr∞(A) := {p ∈L(HA) : p = p2 = p∗and p ∼1 ∼1 −p}1991 Mathematics Subject Classification. Primary 46L05, 46M20, 55P10.Received by the editors February 12, 1992.

The main results of this article were presented atthe AMS meeting at Springfield, Missouri, March 27–28, 1992Partially supported by NSFc⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2SHUANG ZHANGis called the infinite Grassmann space associated with A; where ‘q ∼p’ is thewell-known Murray-von Neumann equivalence of two projections; i.e., there existsa partial isometry v ∈L(HA) such that vv∗= p and v∗v = q. If A = C, thenGr∞(A) reduces to the well-known Grassmann space Gr∞(H) consisting of allprojections on H with an infinite dimension and an infinite codimension.2.

Factorizations and K-theoryLet p ∈Gr∞(A). If x is any element in L(HA), with respect to the decompositionp ⊕(1 −p) = 1 one can write x as a 2 × 2 matrix, say ( a bc d ), where a = pxp,b = px(1 −p), c = (1 −p)xp, and d = (1 −p)x(1 −p).

A unitary operator u = ( a bc d )is called a K-skeleton unitary along p, if both b and c are some partial isometriesin A ⊗K. An easy calculation shows that a unitary operator u is a K-skeletonunitary if and only if a is a Fredholm partial isometry on the submodule pHA andd is a Fredholm partial isometry on the submodule (1 −p)HA; in other words, allp −aa∗, p −a∗a, (1 −p) −dd∗, (1 −p) −d∗d are projections in A ⊗K.

The term‘K-skeleton’ is chosen, since K0(A) is completely described by the homotopy classesof all such unitaries.Let GLpr(A) be the topological group consisting of all those invertible operatorsin L(HA) such that xp−px ∈A⊗K, equipped with the norm topology from L(HA).Let GLp∞(A) stand for the path component of GLpr(A) containing the identity; inthe special case when A = C, we instead use the notation GLpr(H) and GLp∞(H),respectively. Let GL∞(A) and GL0∞(A) denote the group of all invertible elementsin the unitization of A ⊗K and its identity path component, respectively.2.1.K-skeleton factorization theorem [Zh4].

(i) If x ∈GLpr(A), then thereexist an element k ∈A⊗K, an invertible element ( z1 00 z2 ), and a K-skeleton unitary( a bc d ) along p such that 1 + k ∈GL0∞(A) andx = (1 + k)z100z2 abcd.A factorization of x with the form above is called a K-skeleton factorization alongp. (ii) If two K-skeleton factorizations of x along p are given, sayx = x0xpabcd= x′ox′pa′b′c′d′,then [cc∗] −[bb∗] = [c′c′∗] −[b′b′∗] ∈K0(A); in other words, [cc∗] −[bb∗] is aninvariant independent of all (infinitely many) possible K-skeleton factorizations ofx along p.Outline of a proof.

There is a shorter proof solely for this theorem. For the sakeof clarifying some internal relations among π0(GLpr(A)), π0([p]0), and K0(A), weoutline a proof as follows.

First, every element in GLp∞(A) can be written as aproduct of the form x0xp for some invertible x0 ∈GL0∞(A) with x0−1 ∈A⊗K andanother invertible xp with xpp = pxp [Zh4]. Secondly, write the polar decompositionx = (xx∗)1/2u, where (xx∗)1/2 ∈GLp∞(A) and u is a unitary in GLpr(A).

Thenconsider the following subsets of Gr∞(A):[upu∗]r := {wupu∗w∗: w ∈GL0∞(A) with ww∗= w∗w = 1}

FACTORIZATIONS OF INVERTIBLE OPERATORS3and[p]0 := {vpv∗: v ∈GLpr(A)vv∗= v∗v = 1} .Technical arguments show that [upu∗]r is precisely the path component of [p]0containing upu∗. Thirdly, there is a representative in [upu∗]r with the form (p −r1) ⊕r2 for some projections r1, r2 ∈A ⊗K.

It follows that there exists a unitaryu0 ∈GL0∞(A) such thatu∗0upu∗u0 = (p −r1) ⊕r2.Then one obtains a K-skeleton unitary ( a bc d ) such that u = u0( a bc d ), where bb∗= r1and cc∗= r2. Since (xx∗)1/2u0 ∈GLp∞(A), we can rewrite it as a product in thedesired form x0( z1 00 z2 ).

The details are contained in [Zh4].It follows from Theorem 2.1 that x·GLp∞(A) = ( a bc d )·GLp∞(A) (cosets) for eachx ∈GLpr(A). The invariant [cc∗]−[bb∗] associated with the K-skeleton factorizationof x ∈GLpr(A) yields the bijectionabcd· GLp∞(A) ←→[(p −bb∗) ⊕(cc∗)]r.It can be shown that [(p −r1) ⊕r′1]r = [(p −r2) ⊕r′2]r iff[r′1] −[r1] = [r′2] −[r2]in K0(A).

Therefore, we conclude the following theorem whose details are given in[Zh4].2.2.Theorem [Zh4]. The maps defined byabcd· GLp∞(A) 7−→[(p −r1) ⊕r2]r 7−→[r2] −[r1]are two bijections, which induce the following isomorphisms:GLpr(A)/GLp∞(A) ∼= Dh([p]0) ∼= K0(A),where GLpr(A)/GLp∞(A) is the quotient group with the induced multiplication andDh([p]0) = {[upu∗]r : u ∈GLpr(A) with uu∗= u∗u = 1}is the set of all path components of [p]0.

The group operation on Dh([p]0) is definedby[(p −r1) ⊕r′1]r + [(p −r2) ⊕r′2]r = [(p −r1 −s2) ⊕(r′1 ⊕s′2)]rfor some projections s2 ∈p(A ⊗K)p and s′2 ∈(1 −p)(A ⊗K)(1 −p) such thats2 ∼r2, s2r1 = 0, s′2 ∼r′2, and s′2r′1 = 0.

4SHUANG ZHANG2.3.Theorem. Let the base point of [p]0 be p and the base point of GLpr(A) bethe identity.

Thenπ2k+1([p]0) ∼= π2k+1(GLpr(A)) ∼= K1(A),andπ2k+2([p]0) ∼= π2k+2(GLpr(A)) ∼= K0(A)∀k ≥0.Outline of a proof. Let U∞(A) be the unitary group of the unitization of A⊗K, andlet Up(A) be the subgroup of U∞(A) consisting of all those unitaries commutingwith p. First, the map ψp : U∞(A) −→[p]r defined by ψp(u) = upu∗is a Serre(weak) fibration with a standard fiber Up(A) [Zh6, §2].

Secondly, the long exactsequence of homotopy groups associated with this fibration breaks into short exactsequences [Zh6, 2.5, 2.8]:0 −→πk+1([p]r) −→πk(Up(A)) −→πk(U∞(A)) −→0(k ≥0).Thirdly, by an analysis on this short exact sequence one concludesπ2k+2([p]0) ∼= K0(A)andπ2k+1([p]0) ∼= K1(A)(k ≥0).It is well known that the subgroup U pr (A) consisting of all unitary elements inGLpr(A) is homotopy equivalent to GLpr(A). We consider the maps U pr (A) −→[p]0defined by φp(u) = upu∗.It can be shown that φp is a weak fibration with astandard fiber U p(A), where U p(A) is the group consisting of all those unitariesin U pr (A) commuting with p. An argument similar to that above applies to thisfibration.

One can show that π2k+1(U pr (A)) ∼= K1(A) and π2k+2(U pr (A)) ∼= K0(A)for k ≥0. The details are given in [Zh6, §4].2.4.Special case A = C(X).

In particular, if A is taken to be the commutativeC*-algebra C(X) consisting of all complex-valued continuous functions on a com-pact Hausdorffspace X, then each element in L(HC(X)) can be identified with anorm-bounded, *-strong continuous map from X to L(H) [APT]. Here L(H) ⊃{xλ}converges to x in the *-strong operator topology iff∥(xλ −x)k∥+ ∥k(xλ −x)∥→0for any k ∈K.Obviously, L(HC(X)) contains the C*-tensor product L(H)⊗C(X) consisting of allnorm-continuous maps from X to L(H) as a C*-subalgebra.

Then Theorems 2.1and 2.2 in this special case are interpreted as follows.2.5.Corollary. Let GL∞(H) be the group of all invertible operators in L(H).

(i) If f : X −→GL∞(H) is a norm-bounded, *-strong continuous map and p is aprojection in the infinite Grassmann space Gr∞(H) such that pf −fp ∈K⊗C(X),then f can be factored as the following product of three invertible mapsf(.) =1 + k11(.)k12(.)k21(.

)1 + k22(.) g1(.)00g2(.) a(.)b(.)c(.)d(.

);

FACTORIZATIONS OF INVERTIBLE OPERATORS5where kij(. )’s are norm-continuous maps from X to K, g1(.) ⊕g2(.) is a norm-bounded, *-strong continuous map from X to GL∞(H), a(.

), d(.) are *-strong con-tinuous maps from X to the set of Fredholm partial isometries on pH and (1−p)H,respectively, and c(.

), b(.) are norm-continuous maps from X to the set of partialisometries in K.

Furthermore,[c(.)c(. )∗] −[b(.)b(.

)∗] ∈K0(C(X)) (∼= K0(X))is an invariant independent of all possible factorization with the above form. (ii)The groups [X, GLpr(H)], [X, [p]0], and K0(C(X)) are isomorphic, where[X, .] is the set of homotopy classes of norm-bounded, *-strong continuous mapsfrom X to (.

).2.6.Invertible dilations of a Fredholm operator. Let us illustrate a K-skeleton factorization of any invertible dilation of a Fredholm operator x ∈L(HA).There are of course infinitely many invertible 2 × 2 matrices with the formD2(x) :=xy1y2z∈M2(L(HA)).Each such 2 × 2 invertible matrix is called an invertible dilation of x.Specificconstructions of such a dilation were given by P. Halmos [Ho, 222] and A. Connes[Co].

For each invertible dilation of x it follows from the K-skeleton FactorizationTheorem 2.1 thatxy1y2z=1 + a11a12a211 + a22 z100z2 v1 −vv∗1 −v∗v−v∗,where aij’s are some elements in A⊗K, z1, z2 ∈GL∞(A), and the above matrix onthe right, say w, is a familiar unitary matrix occurring in the index map in K-theory[Bl, 8.3.2] in which v is a Fredholm partial isometry in L(HA). Set p = diag(1, 0).It is well known that[1 −v∗v] −[1 −vv∗] ∈K0(A)is precisely the Fredholm index Ind(v) = Ind(pxp) (on pHA).

It follows from The-orem 2.1(ii) that those K-skeleton unitaries associated with all possible invertibledilations of x in M2(L(HA)) only differ from w by a factor in GLp∞(A).3. Factorizations of invertible operators with integer indicesNow we consider some special cases such that K0(A) ∼= Z (the group of allintegers); for example, A = C, or A = C(S2n+1) where Sm is the standard m-sphere, or A = O∞, the Cuntz algebra generated by isometries {si}∞i=1 ⊂L(H)such that P∞i=1 sis∗i ≤1.Let p be any projection in Gr∞(H) ⊂Gr∞(A) [the inclusion holds becauseH ⊂HA and L(H) ⊂L(HA)].Let {ξi}+∞i=0 be any orthonormal basis of thesubspace pH and {ξi}i=−1−∞be any orthonormal basis of the subspace (1−p)H. Then{ξi}+∞−∞is an orthonormal basis of both H and HA.

Let u0 denote the bilateral shiftassociated with the basis {ξi}+∞−∞of H, defined by u0(ξi) = ξi+1 for all i ∈Z. Clearly,u0 is a K-skeleton unitary of L(HA) along p. Applying the K-skeleton FactorizationTheorem 2.1 to the above special cases, we have the following factorizations ofinvertible operators orientated by the integer-valued Fredholm index:

6SHUANG ZHANG3.1.Corollary. Suppose that K0(A) ∼= Z is generated by [1] where 1 is the iden-tity of A.

If x is an invertible operator on HA such that px −xp ∈A ⊗K, thenx = (1+k)xpu−n0 , where k ∈A⊗K, xp is an invertible operator commuting with p,and the integer n is the Fredholm index of pxp on the submodule pHA, say Ind(pxp),which is independent of the choice of {ξi}+∞i=0 , {ξi}−1−∞and all possible factorizationsalong p with the same form above.Outline of a proof. It is obvious that Ind(pun0p) = −n.

Let G be the group {un0 :n ∈Z} in which every element is a K-skeleton unitary along p. As a special caseof Theorem 2.1 one can show that the map from G to GLpr(A)/GLp∞(A) definedby un0 7−→un0 · GLp∞(A) is a group isomorphism. It follows that π0(GLpr(A)) ={un0 · GLp∞(A) : n ∈Z}.

Then the factorization follows. The reader may want toconsider the extreme case A = C and then generalize the conclusion to a largerclass of C*-algebras.A similar proof yields the following alternative factorization of x as a product ofthree invertibles under the same assumptions as of Corollary 3.1:x =(1 + k1)x1if Ind(pxp) = 0,(1 + k2)x2(u1 ⊕u2 ⊕· · · ⊕u−n ⊕w1)if Ind(pxp) = n < 0,(1 + k3)x3(u∗1 ⊕u∗2 ⊕· · · ⊕u∗n ⊕w2)if Ind(pxp) = n > 0,where ui is a bilateral shift on a subspace Hi of H for 1 ≤i ≤n, wj’s are unitaryoperators on (Lni=1 Hi)⊥, kj ∈A⊗K, and xj’s are invertible operators commutingwith p.3.2.Corollary.

Suppose that K0(A) ∼= Z is generated by [1]. If x is an arbitraryelement L(HA) and p ∈Gr∞(H) (as above) such that px −xp ∈A ⊗K, then thereexists a unique norm-continuous map x(λ) from C \ σ(x) to GLp∞(A), where σ(x)is the spectrum of x, such that x −λ = x(λ)u−ni0, where ni = Ind(p(x −λi)p) andλi is any complex number in the ith path component Oi of C \σ(x).

An alternativeK-skeleton factorization of x −λ for λ ∈Oi is as follows (when ni ̸= 0):x −λ = yi(λ)(u1 ⊕u2 ⊕· · · ⊕u|ni| ⊕wi)if Ind(p(x −λi)p) = ni < 0,y′i(λ)(u∗1 ⊕u∗2 ⊕· · · ⊕u∗ni ⊕vi)if Ind(p(x −λi)p) = ni > 0,where ui’s are bilateral shifts on mutually orthogonal closed subspaces Hi’s of H,wi, vi’s are unitary operators on the subspace (L|ni|i=1 Hi)⊥, and yi(λ), y′i(λ) arenorm-continuous maps from Oi to GLp∞(A).3.3.Winding numbers of invertible operators. Using the first factorizationin Corollary 3.2, we assign an integer ni to each path component Oi of C \ σ(x),which is precisely the minus winding number of u−ni0as a continuous map fromS1 to S1 (via the Gel’fand transformation).

We call ni the winding number of xalong p over Oi. As a particular case, if x is an operator whose essential spectrum,the spectrum of π(x) in the generalized Calkin algebra L(HA)/K(HA), does notseparate the plane, then all winding numbers of x along any p ∈Gr∞(A) are zeroas long as px −xp ∈A ⊗K.

There is another way to describe the integer ni.3.4.Corollary. Let Gi(x) denote the subgroup of GLpr(A) generated by GLp∞(A)and x−λi where λi ∈Oi.

Then Gi(x)/GLp∞(A) ∼= niZ, and hence GLpr(A)/Gi(x) ∼=Zni, the finite cyclic group of order ni.

FACTORIZATIONS OF INVERTIBLE OPERATORS7In particular, one can apply the above factorizations to an invertible dilationof a pseudodifferential operator of order zero on a compact manifold and classicalmultiplication operators. Let us spend few lines to look at the following familiarexamples.3.5.Multiplication operators.

Let Mf be the invertible multiplication opera-tor with symbol f in L∞(S1), where S1 is the unit circle; i.e., Mf(g) = fg for anyg ∈L2(S1). If p is a projection on L2(S1) such that dim(p) = codim(1 −p) = ∞and pMf −Mfp is a compact operator, then it follows from Corollary 3.1 thatMf = (1 + k)xpu−n0 , where n = Ind(pMfp), k is a compact operator on L2(S1), xpis an invertible operator on L2(S1) commuting with p, and u0 is a bilateral shiftoperator associated with a fixed orthonormal basis of L2(S1).

It is well known thatpMfp is a familiar Toeplitz operator on the subspace pL2(S1).3.6.Restricted loop group along p ∈Gr∞(H). Consider the following re-stricted loop group along p consisting of all norm-bounded, *-strong continuousmaps from S1 to GLpr(H), denoted by Map(S1, GLpr(H))β.

Since K0(C(S1)) = Z,each f ∈Map(S1, GLpr(H))β can be factored as f = (1 + f0)f1u−n0 , where n =Ind(pfp), f0 is a norm-continuous map from S1 to K, f1 is a *-strong continuousmap from S1 to GL∞(H) such that f1(z)p = pf1(z) for any z ∈S1, and u0 is a bi-lateral shift with respect to a fixed orthonormal basis of H. If f is norm-continuous,then f1 is also norm continuous. Furthermore, [S1, GLpr(H)] ∼= [X, [p]0] ∼= Z. Thesame conclusions also hold, if S1 is replaced by S2n+1 for any n ≥1.3.7.Remarks.

(i) Theorems 2.1–2.3 still hold, if A is any stably unital C*-algebra;i.e., A ⊗K has an approximate identity consisting of a sequence of projections [Bl,5.5.4; Zh4]. (ii) Let Index(x, p) denote the invariant [cc∗]−[bb∗] ∈K0(A) in Theorem 2.1(ii).If p is fixed, then Index(x, p) is precisely the Fredholm index of pxp as an operatoron pHA and fits into the established theory of the K0(A)-valued Fredholm index.However, some new results do arise from invariants of Index(x, p) as the variablep runs in {p ∈Gr∞(A) : xp −px ∈A ⊗K} or as x and p jointly change [Zh7].As a matter of fact, Index(x, p) is an invariant under homotopy and perturbationby elements in A ⊗K with respect to both variables x and p. For example, bythe combination of the K-skeleton Factorization Theorem and certain invariants ofIndex(x, p), we proved [Zh7] the following:Theorem.π0(GL(Mn(C)′e)) ∼= {k ∈K0(A) : n · k = 0}for any n ≥2;where GL(Mn(C)′e) denotes the group of all invertibles in the essential commutantMn(C)′e of Mn(C) which is naturally embedded in Mn(L(HA)).

(iii) The reader may want to compare (3.1)–(3.3) and the famous BDF theory[BDF1,2] to see their obvious relations; we work with invertibles on HA, while theBDF theory dealt with Fredholm operators. (iv) In [PS] Pressley and Segal have studied the restricted general linear groupGLres(H) := {x ∈GL∞(H) : xp −px is Hilbert-Schmidt }and given some applications to the Kdv equations.

It is a hope that our results willshed some light in the same direction.

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I, Springer-Verlag, Berlin, Heidelberg, and NewYork, 1979.[Zh1]S. Zhang, Certain C∗-algebras with real rank zero and their corona and multiplier alge-bras, Part II, K-theory (to appear).

[Zh2], On the homotopy type of the unitary group and the Grassmann space of purelyinfinite simple C*-algebras, K-Theory (to appear).

FACTORIZATIONS OF INVERTIBLE OPERATORS9[Zh3], Exponential rank and exponential length of operators on Hilbert C*-module, Ann.of Math. (2) (to appear).

[Zh4], K-theory, K-skeleton factorizations and bi-variable index Index(x, p), Part I,Part II, Part III, preprints. [Zh5], K-theory and bi-variable index Index(x, [p]e): properties, invariants and appli-cations, Part I, Part II, Part III, preprints.

[Zh6], K-theory and homotopy of certain groups and infinite Grassmann spaces asso-ciated with C*-algebra, preprint. [Zh7], Torsion of K-theory, bi-variable index and certain invariants of the essentialcommutant of Mn(C).

I, II, preprints.Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio45221-0025E-mail address: szhang@ucbeh.san.uc.edu

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