Algebraic orders on K0 and approximately finite

arXiv:funct-an/9302002v1 9 Feb 1993Algebraic orders on K0 and approximately finiteoperator algebras 1Stephen C. PowerDepartment of MathematicsUniversity of LancasterEngland LA1 4YFApproximately finite (AF) C∗-algebras are classified by approximately finite (r-discreteprincipal) groupoids. Certain natural triangular subalgebras of AF C∗-algebras are similarlyclassified by triangular subsemigroupoids of AF groupoids [10].

Putting this in a more intu-itive way, such subalgebras A are classified by the topologised fundamental binary relationR(A) induced on the Gelfand space of the masa AT A∗by the normaliser of AT A∗in A. (This relation R(A) is also determined by any matrix unit system for A affiliatedwith AT A∗.) The fundamental relation R(A) has been useful both in understanding theisomorphism classes of specific algebras and in the general structure theory of triangular andchordal subalgebras of AF C∗-algebras ([7], [15] ,[14],[21], [22]).

Nevertheless it is desirableto have more convenient and computable invariants associated with the K0 group, and webegin such an inquiry in this paper.We introduce the algebraic order and the strong algebraic order on the scale of the K0group of a non-self-adjoint subalgebra of a C∗-algebra. Analogues (and generalisations) ofElliott’s classification of AF C∗-algebras are obtained for limit algebras of direct systemsA1 −→A2 −→...of finite-dimensional CSL algebras (poset algebras) with respect to certain embeddings withC∗-extensions which, in a certain sense, preserve the algebraic order.

We also require thatthe systems have a certain conjugacy property.Despite the restrictions there are manyinteresting applications. For example conjugacy properties prevail for certain embeddingsof finite-dimensional nest algebras (block upper triangular matrix algebras) and for systemsassociated with ordered Bratteli diagrams.1REVISED DECEMBER 19921

Hitherto the study of non-self-adjoint subalgebras of AF C∗-algebras has focused on trian-gular subalgebras, where A T A∗is a certain approximately finite regular maximal abelianself-adjoint algebra ([1], [7], [9], [11], [14], [15]). See also [8] .

However from the point ofview of identifying the algebraically ordered scaled ordered dimension group, the viewpointof this paper, it is the nontriangular subalgebras which are particularly interesting since inthis case K0(A) (which agrees with K0(A T A∗)) can be a ‘small group’, such as ZZ5 orIQ2. In such settings, the algebraic orders can be revealed more explicitly.

For example, inExample 4.5 we have the situation in which A T A∗is a simple C∗-algebra in the simpleC∗-algebra C∗(A), and A is one of only finitely many algebras between AT A∗and C∗(A).The algebraic order of such an algebra corresponds to partial orders on the fibres of thesurjection i∗: K0(A) −→K0(C∗(A)).In section 1 we define the reflexive transitive antisymmetric order S(A) on the scale of theK0 group of a subalgebra of a C∗-algebra, and we recall some basic facts concerning (regular)canonical subalgebras of AF C∗-algebras. In section 2 we discuss various kinds of embeddingsof finite-dimensional algebras of matrices, and we define the strong algebraic order S1(A)associated with a canonical masa.

In section 3 we obtain the main results, Theorems 3.1and 3.2, together with various examples and associated remarks. In particular we considera class of triangular algebras associated with ordered Bratteli diagrams.

In section 4 weconsider examples of non-self-adjoint subalgebras of AF C∗-algebras with small K0 group.In this connection we look at stationary pairs of AF C∗-algebras D ⊆B. This situationincludes the context of the example mentioned above.Most of this paper was completed during a visit to the University of Waterloo in 1989,and the author would like to thank Ken Davidson for some useful discussions.

The presentpreprint replaces an earlier one which contained a mistaken (nontriangular) version of Lemma2.5.1. Algebra orders on K0.We start by recalling some terminology and properties of subalgebras of AF C∗-algebras.A finite-dimensional commutative subspace lattice algebra A, or FDCSL algebra, is anoperator algebra on a finite-dimensional Hilbert space which contains a maximal abelianself-adjoint algebra (masa).

We say that a masa C in an AF C∗-algebra B is a canonicalmasa (or, more precisely, a regular canonical masa) if there is a chain of finite-dimensional2

C∗-subalgebras B1 ⊆B2 ⊆..., with dense union, such that the algebras Ck = Bk ∩C aremasas in the algebras Bk, with dense union in C, and such that, for each k, the normaliserof Ck in Bk is contained in the normaliser of Ck+1 in Bk+1. A closed subalgebra A of Bis said to be a (regular) canonical subalgebra if C⊆A ⊆B for some canonical masaC.

In this case A is necessarily the closed union of the FDCSL algebras An = BnT A. (See [16].

)In particular the algebra A is an approximately finite operator algebra andis identifiable with the direct limit Banach algebra lim→An where the embeddings possessstar extensions. Of course the converse is true ; if A1 −→A2 −→... is a direct systemof FDCSL algebras with respect to embeddings, not necessarily unital, which have starextensions C∗(Ak)→C∗(Ak+1), then the Banach algebra A=lim→Ak is completelyisometrically isomorphic to a subalgebra of the AF C∗-algebra B = lim→C∗(Ak).

(Howeversuch a subalgebra need not be a regular canonical subalgebra in the sense above. )The masas above coincide with approximately finite Cartan subalgebras of AF C∗-algebras[18].

A useful discussion of them is given in the notes of Stratila and Voiculescu [20].We now give a definition of K0(A) for a not necessarily self-adjoint subalgebra A of aC∗-algebra. In the unital, or stably unital case, K0(A) coincides with the usual definitionin terms of the stable algebraic equivalence of idempotents.

(See Proposition 5.5.5 of [2]. )Write p →q, or p →A q, or p →v q, for p, q in Proj (A), the set of self-adjoint projections ofA, if there exists a partial isometry v in A with v∗v = p, vv∗= q.

Write p ∼q if v can bechosen in A T A∗. Define K+0 (A) as the set of (Murray von Neumann) equivalence classes[p] of projections in Proj (AN Mn), n = 1, 2, ...., with the usual identifications AN Mn ⊆A N Mn+1, n = 1, 2, ... A semigroup operation is given by [p] + [q] = [p + q] ,where p andq are representatives with pq = 0, and K0(A) is, by definition, the Grothendieck group ofK+0 (A).

For a canonical AF subalgebra K+0 (A) has cancellation and embeds injectively inK0(A). The scale of A in K0(A) is the partially ordered set Σ(A) = {[p] : p ∈Proj(A)}.

Acelebrated theorem of G. Elliott [4] asserts that AF C∗-algebras B1 and B2 are isomorphicif there is a group isomorphism θ : K0(B1) −→K0(B2) with θ (Σ(B1)) = Σ(B2).For a canonical subalgebra A of an AF C∗-algebra note thatK0(A) = lim→K0(An) = lim→K0(An\A∗n) = K0(A\A∗).Define the algebraic order S = S(A) on Σ(A) to be the reflexive transitive antisymmetricrelation such that [p]S[q] if and only if q →v p for some partial isometry v in some algebraA N Mn for some n. For canonical subalgebras we can take p, q, v in A, because K+0 (A) has3

cancellation. The pair (Σ(A), S(A)) does not form a complete invariant for such subalgebrasof AF C∗-algebras, but we shall see that it is complete for certain subclasses.2.

Embeddings and Normalisers.Let C ⊆A ⊆B be as in the second paragraph of section 1. Every self-adjoint projectionin A is equivalent in A T A∗to a projection in AnT A∗n for some n, and so is equivalent toa projection in Cn for some n. Furthermore the algebraic orders can be understood in termsof the partial isometries which normalise C, as we now indicate.Definition 2.1.

The normaliser of C in A is the semigroup NC(A) of partial isometries v inA such that vCv∗⊆C and v∗Cv ⊆C. The strong normaliser of C in A is the subsemigroupNsC(A) of elements v which preserve the relation →A in the sense that if p1 →A p1, withp2 ≤v∗v, and p2 ≤v∗v, then vp1v∗→A vp2v∗, and if p1→A p2, with p1≤v v∗, andp2 ≤v v∗, then v∗p1 v →A v∗p2 v.The normaliser NC(A) has the following important property.

Each v in NC(A) has theform cw with c a partial isometry in C and w an element of NCk(Ak) for some k. (Also,every operator of this form is in NC(A).) We use this below without further explanation.For details see [14] or [16].Lemma 2.2 Let p and q be projections in A.

Then [p] S(A) [q] if and only if there existprojections p′ and q′ in C and a partial isometry v in NC(A) such that p ∼p′ , q ∼q′and v∗v = q , vv∗= p.Proof: Suppose that [p] S(A) [q]. For some large k there are projections in Ak which are closeto p and q , and so it follows that there exist projections p′ and q′ in Ck with p ∼p′ , q ∼q′.By the hypothesis it follows that there is a partial isometry w in A with w∗w= q′ andww∗= p′.

Increasing k if necessary, choose an operator x in Ak, close to w, with x = p′xq′,such that x is invertible when viewed as an operator from q′H to p′H where H is the finite-dimensional Hilbert space underlying Ak. Let p′ = p′1 + ... + p′r , q′ = q′1 + ... + q′r4

be the decomposition into minimal projections of Ck. By the invertibility of x it followsthat there is a permutation π of1, ..., rsuch that p′i x q′π(i) is nonzero for each i. Bythe minimality of the p′i and q′j it follows that there is a partial isometry vi with initialprojection qπ(i) and final projection pi.

The partial isometry v = v1 + ... + vr satisfiesv∗v = q , vv∗= p, and since v belongs to NCk(Ak) it follows that v belongs to NC(A). ✷In the next section we consider embeddings affiliated with maximal abelian subalgebrasand the following terminology will be useful.Definition 2.3.

Let A, A ′ be FDCSL algebras containing masas C, C ′ respectively and letα : A −→A ′ be an injective algebraic homomorphism with C −→C ′. The embedding issaid to be(i) star-extendible, if there is an extension C∗(A) −→C∗(A ′),(ii) regular (with respect to C, C ′) if NC(A) −→NC ′(A ′),(iii) strongly regular (with respect to C, C ′) if NsC(A) −→NsC ′(A ′),If A ⊆Mn, A ′ ⊆MnN Mm, identified with Mn(Mm), and if A N ICI ⊆A ′, then werefer to the natural embedding ρ : A −→A ′, given by ρ(a) = a N 1, as a refinementembedding.

In particular, viewing Tnm as the upper triangular matrix subalgebra of Mn(Mm)we have the refinement embedding ρ : Tn −→Tnm. In contrast, if ICI N A ⊆A ′, then werefer to the embedding σ : A −→A ′, given by σ(a) = 1 N a = a ⊕... ⊕a (n times) asa standard embedding.

In particular, with the same identification Tnm ⊆Mn(Mm), we havethe standard embeddings σ : Tm −→Tnm. Standard embeddings, refinement embeddings,and many other hybrid embeddings are strongly regular.

In contrast the embedding T2 −→T4 given byabc→a00bab0c0cis a regular star-extendible embedding which is not strongly regular.5

It is straightforward to check that a strongly regular star-extendible embedding Tn −→Tmis determined, up to conjugation by a unitary in the diagonal algebra Dm = TmT(Tm)∗,by its restriction to the diagonal algebra Dn. Recall that a finite-dimensional nest algebraA is an FDCSL algebra whose lattice of invariant projections is totally ordered.

Similarlyit can be shown that a strongly regular embedding α : A −→A ′ between direct sums ofsuch algebras is determined, up to conjugacy by a unitary in A ′ T(A ′)∗, by its restriction toAT A∗.Definition 2.4 Let C ⊆A ⊆B be as in the second paragraph of Section 1. The strongalgebraic order S1(A) is the subrelation of the algebraic order S(A) such that [p]S1(A) [q] ifand only if there are representatives p, q in Proj C and a partial isometry v in NsC(A) withq →v p.In general S1(A) is a proper subrelation of S(A).

This can be seen for elementary FDCSLalgebras. On the other hand these relations agree in the case of triangular nest algebras.Note that S1(A) depends on an implicit choice of canonical masa, so it is not clear, a priori,whether S1(A) is even an invariant for isometric isomorphism.

However in the followingtriangular context we have :Lemma 2.5 Let A be the limit of the system A1→A2→... consisting of directsums of triangular finite-dimensional nest algebras and strongly regular embeddings. ThenS1(A) = S(A), where S1(A) is the strong algebraic order of A.Proof:If Ak is a triangular finite-dimensional nest algebra then S1(Ak) = S(Ak).

Indeed,if p, q are projections in a masa Ck of Ak, and q→p, then we can order the minimalsubprojections (in Ck) of q and p and obtain q→vp where v∈NCk(Ak) is a partialisometry which matches these subprojections in order. Since v preserves the partial orderingon minimal projections (induced by Ak) v belongs to NsCk (Ak).If [e] S(A) [f], and if C is the limit of the subsystem C1 →C2 →..., choose p, q inProjCk for some large k with [e] = [p], [f] = [q] and with q →Ak p. Choose v as abovein NsCk (Ak).

The embeddings are strongly regular and so it follows that v is in NsC(A) . ✷6

3. Classifying limit algebras.Let A1 −→A2 −→... be a direct system of FDCSL algebras with star-extendible injectiveembeddings.

We say that the system has the conjugacy property if whenever α : An −→An+k is a star-extendible embedding whose restriction α| Cn to a masa in An is equal to thegiven injection i : Cn −→An+k, then there is a unitary operator u in An+kT A∗n+k suchthat α = (Ad u) ◦i, where i is the given injection of An. Similarly we say that the systemhas the conjugacy property for strongly regular maps if the same conclusion holds just formaps α : An −→An+k which are additionally strongly regular relative to two masas.

Thisis the appropriate concept for direct systems with strongly regular embeddings.We noted above that a strongly regular direct system A1 −→A2 −→..., with each Aka direct sum of triangular finite-dimensional nest algebras, has the conjugacy property forstrongly regular maps. On the other hand it can be shown that the natural direct systemA1 −→A1 ⊗A2 −→A1 ⊗A2 ⊗A3 −→...,where each Ak is an FDCSL algebra, has the ordinary conjugacy property.In the theorem below we obtain a generalisation of Elliott’s classification of AF C∗-algebras, and the proof is modelled on the self-adjoint case.However, in our generalityit is necessary to do extra work to lift relations on the scale of K0(A) to normalising partialisometries in such a way that we obtain star-extendible embeddings.

(We remark that evenstrongly regular embeddings of FDCSL algebras need not be star-extendible. )Let A be a canonical subalgebra with canonical masa C as before, and let R ⊆S(A) be aconnected transitive reflexive finite subrelation.

We view this as a binary relation on the set{ 1, ..., n }. It follows from Lemma 2.2 that we can find orthogonal projections p1, ..., pn inM∞(C) and partial isometries vij in M∞(NC(A)), with pj →vijpi, whenever (i, j) ∈R.We say that S(A) has the star realisation property if for every such subrelation there is achoice with {vij : (i, j) ∈R} a subset of a complete matrix unit system.

By this wemean that the natural map A(R) →M∞(A) given by (aij) →(aijvij) is a star-extendibleinjection from the FDCSL algebra A(R) associated with R. The star realisation propertyis rather restricted, as we observe below in Remark 3.3. Nevertheless it holds in severalcontexts of interest and allows for the statement of a general theorem.In an exactly analogous way one can define when S1(A) has the star realisation property,7

and there is a corresponding variant of the following theorem for strongly regular systems.Theorem 3.1.Let A and A′ be the limits of the systems A1 −→A2 −→... andA′1 −→A′2 −→... consisting of FDCSL algebras and injective star-extendible regular embed-dings. Suppose further that the systems have the conjugacy property and that the algebraicorders S(A) and S(A′) have the star realisation property.

Then A and A ′ are isometricallyisomorphic if and only if there is a scaled order group isomorphism θ : K0(A) −→K0(A ′)which gives an isomorphism of the algebraic order.Proof:Assume that θ exists.Thus it is assumed that θ gives a bijection between thealgebraic orders S(A), S(A′) associated with canonical masas C, C′, respectively, affiliatedwith the given direct systems. It will be enough to construct a system of embeddingsA1 →φ1 A ′n1 →ψ1 Am1 →φ2 ...which commute with the given embeddings A1 −→Am1, A ′n1 −→A ′n2, ... .

In particularthe constructed isomorphism mapsS An ontoS A ′n. This isomorphism φ = lim→φk willalso implement the given isomorphism θ.The algebra A is a regular canonical subalgebra of the AF C∗-algebra C∗(A) = lim→C∗(An),and from this it follows that we can choose systems {enij} of matrix units for C∗(An), forn = 1, 2, ..., such that each enij is a sum of matrix units in {en+1ij} , and such that {enii}spans the masa Cn in An.

(See [16] for example.) Let{enij : (i, j) ∈Ωn } be the set ofmatrix units in An.

Similarly choose the matrix unit system {f nij} for C∗(A ′n), with {f nii}spanning C ′n, and let {f nij : (i, j) ∈Ω′n } be the set A ′nT {f nij} .Choose n1 large enough so that there are orthogonal projections gii in C ′n1 such thatθ([e1ii]) = [gii] for all i. (We need not be precise about the range of i.) Let (i, j) ∈Ω1.Since θ preserves the algebraic order, and since S(A′) has the star realisation property, thereis a choice of partial isometries vij in M∞(NC′(A′)), for (i, j) ∈Ω1, such that gjj →vij gii,and such that the induced map φ1 : A1 →M∞(A′) is star-extendible.

Since the orthogonalprojections gii lie in C′n1, the range of φ1 is actually in A′. In view of the remarks preceedingLemma 2.2, φ1 has the formφ1XΩ1aij e1ij=XΩ1aij cij wij8

where, for some large enough k , wij is a partial isometry which is a sum of some of thematrix units in {f kij : (i, j) ∈Ω′k} , and where cij ∈C′ for all (i, j) ∈Ω1. However,by the star-extendibility of φ1 the set { cij wij : (i, j) ∈Ω1} is a subset of a completematrix unit system.

From this it follows that { wij : (i, j) ∈Ω1} is necessarily a subsetof a complete matrix unit system. It can now be shown that cijwij=cwijc∗for somepartial isometry c in C′.

So, replacing φ1 by (Adc∗) ◦φ1, and replacing n1 by k, we obtainthe desired map φ1. It is regular because the images of the matrix units of A1 lie in thenormaliser of a masa.We have obtained a regular star-extendible embedding φ1 : A1 −→A ′n1 such that[φ1(e1ii)]S(A ′)[φ1(e1jj)] for each matrix unit e1ij in A1.

We now construct the desired mapψ1 : A ′n1 −→Am1. Choose orthogonal projections hii in Cm1, for suitably large m1, so that[hii] = θ−1([gn1ii ]) for all i.

We can do this in such a way so that, for each i, if gii =PJi f n1iithen PJi hii coincides with i(e1ii). In other words, the choice of the projections hii determinean injection ω : C ′n1 −→Cm1 and we can arrange this so that ω ◦φ1 agrees with thegiven injection i : C1 −→Cm1.

By our earlier arguments, increasing m1 if necessary, thereis a regular star-extendible embedding ˆω : A ′n1 −→Am1 which extends ω. Because ofthe hypothesised conjugacy property there is a unitary element u in Am1T A∗m1 so thatq1 = (Adu) ◦ˆω is the desired injection from A ′n1 to Am1, with q1 ◦φ1 = i.

Continue toobtain the desired system.✷It will be noticed that the star realisation property is much stronger than is necessary forthe proof of Theorem 3.1. The essential point is that any finite transitive reflexive subrelationof S(A′) (respectively S(A)) which is isomorphic to the relation for Ak (respectively A′k) forsome k, is star realisable.Theorem 3.2.

Let A and A ′ be limits of direct sums of triangular finite-dimensional nestalgebras with respect to injective strongly regular star-extendible embeddings associatedwith ordered Bratteli diagrams (as in 3.8). Then A and A ′ are isometrically isomorphic ifand only if there is a scaled group isomorphism θ : K0(A) −→K0(A ′) which preserves thealgebraic order.Proof:The proof above applies with simplifications.

Firstly, note that the maps φ1, ψ1, . .

.are easily defined by specifying images for the superdiagonal matrix units (the matrix units9

of the first superdiagonal). Secondly, observe that ordered Bratteli diagram systems have theconjugacy property.

Indeed, if φ : E →F is an ordered Bratteli diagram embedding betweentriangular elementary algebras, and if e is a matrix unit in E, then in each summand of Fthe minimal subprojections of e∗e interlace those of ee∗. It follows that the partial isometriesv in F with e∗e →v ee∗agree modulo a multiplier of C.✷Remark 3.3 In general the algebraic order of a FDCSL algebra may not have the starrealisation property.

In fact more is true. There are FDCSL algebras A1 , A2 and a scaledordered group injection θ:K0(A1)→K0(A2) with θ(2)(R(A1))⊆R(A2) which isnot induced by any regular injective embedding.

To see this consider the subalgebra A2 ofmatrices (aij) in M4 ⊗M2 of the form(aij) =a110a130a150a17a18a220a240a26a27a28a33000a370a44000a48a5500a58a66a670a770a88Let A1=T2N T2 and consider the injection φ:A1T A∗1→A2T A∗2 whichis given by c→cN I2. This in turn induces a mapθ:Σ (A1)→Σ (A2)with θ(2) (R(A1))⊆R(A2).

Examination shows that there is no unital regular injectionA1→A2, star-extendible or otherwise, which induces θ. We remark that in the caseof infinite tensor products of proper finite-dimensional nest algebras the algebraic ordercontains arbitrary subrelations, and in particular subrelations isomorphic to θ(2) (R(A1)).Thus it does not seem that the methods of Theorem 3.1 are immediately applicable in theclassification of infinite tensor products.Remark 3.4.

If A and A ′ are isomorphic as Banach algebras then it can be shown thatthere is a scaled ordered group isomorphism from K0(A) to K0(A ′) which preserves thealgebraic order. As a consequence the algebraically ordered scaled ordered group K0(A) is a10

complete invariant for bicontinuous isomorphism within the classes considered in Theorems3.1 and 3.2.It seems plausible that any two canonical subalgebras of an AF C∗-algebra are isometricallyisomorphic if they are algebraically isomorphic. Settling this problem will be a good test ofthe effectiveness of any future methods in the study of subalgebras of AF C∗-algebras.Remark 3.5.There exist triangular canonical subalgebras A and A′, with S(A)=S1(A) , withS(A′)=S1(A′), and with an algebraic order preserving isomorphismθ : K0(A) →K0(A′), which are nevertheless not isometrically isomorphic.

To see this letB = lim→(M2k, ρk), where ρk : M2k →M2k+1 , k = 1, 2, ... are refinement embeddings,and let B′ be the subspace lim→(M′2k, ρk) where M′2k is the subspace of matrices with zerodiagonal. Furthermore, let D = lim→(D2k, ρk) be the canonical diagonal subalgebra of B.Adopting a little notational distortion, defineA=" DOBD#,A′=" DOB′D#.These canonical subalgebras of M2N B are not isometrically isomorphic.

This can bededuced from the fact that A and A′ do not have topologically isomorphic fundamentalrelations. See [15].

We leave the verification of the other assertions as a simple exercise.Remark 3.6. Theorem 3.2 is not true if the embedding condition is relaxed.

Let A=lim→(T2n, ρ) be the limit of upper triangular matrix algebras with respect to refinement em-beddings, and let A ′ = lim→(T2n, θn) be the limit algebra where θn(enij) = ρ(enij) if j < 2nor (i, j) = (2n, 2n), and θn(eni,2n) = en+12i,2n+1−1 + en+12i−1,2n+1, otherwise. These embeddings,in which the final column of matrix units is embedded with twisted orientation, are notstrongly regular.

Despite the fact that the binary relations (Σ(A), S(A)), (Σ(A ′), S(A ′))are naturally isomorphic, the algebras A and A ′ are not isomorphic. This observation isessentially due to Peters, Poon and Wagner [9].

See also [15].In [9] a partial order

In [11] related invariants are exploited inthe study of nest subalgebras. In particular it is shown that there are uncountably many11

nonisomorphic triangular nest algebras A in any given UHF algebra, all having the sametrace invariant {trace(p) : p ∈Lat A} . Here LatA is the projection nest in C determiningA.Remark 3.7.

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