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COMPRESSION LIMIT ALGEBRAS

arXiv:funct-an/9212003v1 18 Dec 1992COMPRESSION LIMIT ALGEBRASAlan HopenwasserCecelia LaurieUniversity of AlabamaI. IntroductionWhile there is now a large and rapidly growing literature on the study of direct limitsof subalgebras of finite dimensional C∗-algebras, the focus in almost every paper on thesubject has been on systems with embeddings which have ∗-extensions to the generatedC∗-algebra.

One notable exception is the paper by Power [P1]. The purpose of this note isto extend the study of systems which are not ∗-extendible and to produce examples whichillustrate some new phenomena.When all the embeddings in a direct system are ∗-extendible, then the limit algebrais, in a natural way, a subalgebra of an AF C∗-algebra.

On the other hand, if the em-beddings are not ∗-extendible, then it is no longer a priori obvious that the limit algebrais an operator algebra. (Initially, the direct limit must be taken within the category ofBanach algebras.) Even if the limit algebra is an operator algebra, its image under somerepresentations may generate a C∗-algebra which is not approximately finite (as happensin the situation studied by Power).

For systems with the type of embeddings which weconsider in this paper, the limit algebra will always be an operator algebra, a fact whichcan easily be seen with the aid of the abstract characterization of operator algebras byBlecher, Ruan, and Sinclair [BRS]. We also produce a representation of the limit algebrawhich is “natural” in the sense that the C∗-envelope of the image algebra (as defined byHamana [H]) is isomorphic to the C∗-algebra generated by the image algebra.II.

Compression EmbeddingsThe key observation behind the choice of embeddings under investigation is that ifA is a CSL-algebra and if p is an interval from the lattice of invariant projections forThe authors would like to thank Vern Paulsen for helpful suggestions on the content of this paper.Typeset by AMS-TEX1

2ALAN HOPENWASSER CECELIA LAURIEA, then the mapping x 7→pxp is an algebra homomorphism.Since we are interestedin direct limits of finite dimensional algebras, we shall assume that every CSL-algebra isfinite dimensional. This implies that, with respect to a suitable choice of matrix units, Ais a subalgebra of some full matrix algebra Mn which contains the algebra Dn of diagonalmatrices.

Such algebras go under a variety of names: poset algebras, incidence algebras,or digraph algebras; henceforth, we use the term digraph algebra.The invariant projections of a digraph algebra are all projections in the diagonal Dn;in particular, they all commute with one another. If e and f are two invariant projectionssuch that e ≤f, then p = f −e is called an interval from the lattice.

(The term semi-invariant projection is also sometimes used. )Since we want our embeddings to be unital, a compression will be the mapping whichtakes x to the restriction of pxp to the range of p, where p is an interval from the lattice.The image algebra under the compression will be viewed as a subalgebra of a full matrixalgebra, usually of rank smaller than the rank of the original containing matrix algebra.Definition.

Let A ⊆Mn be a digraph algebra. A compression embedding is a direct sumof compression mappings on A subject to the proviso that at least one summand is theidentity mapping.The purpose of the assumption that at least one summand is the identity mappingis to ensure that the embedding is completely isometric.

(Compressions are completelycontractive, but not completely isometric except in the trivial case of the identity mapping.)Definition. A compression limit algebra is the limit of a direct systemA1φ1−→A2φ2−→A3φ3−→.

. .

−→Ain which each embedding is a unital compression embedding.A compression embedding of Ak into Ak+1 is a unital map of Ak onto its range; we areassuming then that the unit of Ak+1 is equal to the unit of the subalgebra φk(Ak). (Thisavoids unwanted degeneracies.) The special case in which each summand is compressionto the identity is just the familiar case of a standard embedding.

In this special case,compression embeddings are ∗-extendible; in general they will not be.In this note, we shall focus on a special class of compression algebras: direct limits offull upper triangular matrix algebras,Tn1φ1−→Tn2φ2−→Tn3φ3−→. .

. −→A.The non-∗-extendible embeddings studied by Power [P1] require non-trivial cohomologyfor the digraph algebras in the system; hence nest algebras can never appear as finite

COMPRESSION LIMIT ALGEBRAS3dimensional approximants. The systems studied here have one other new feature: the onlyconstraint on nk and nk+1 is that nk < nk+1; no divisibility is required.For each k, one of the summands of φk is the identity compression; let qk denote thesupport interval for this summand from the nest of invariant projections for Tnk+1.

Also,let ψk denote the compression of Tnk+1 onto qk. We may, in a natural way, view ψk as analgebra homomorphism from Tnk+1 onto Tnk.

It is clear that ψk ◦φk = id, for all k. Thuscompression embedding systems are structured in the sense introduced by Larson [L]; inparticular, compression limit algebras are all structured Banach algebras.Compression embeddings satisfy one additional property: they map matrix units tosums of matrix units. This implies that they are regular embeddings, in the sense used byPower [P2, section 4.9].Remark.

Compression embeddings can be placed into a broader context. If C = [ cij ] isan n × n matrix, the mapping φC : Mn →Mn given by φC[ aij ] = [ cijaij ] is known as aSchur mapping.

It is easy to check that φC is an algebra homomorphism if, and only if, Csatisfies the cocycle condition: cik = cijcjk. Furthermore, φC is unital if, and only if, allcii = 1.

By taking a direct sum of algebra homomorphisms each of which is a compressioncomposed with a Schur mapping, we can obtain embeddings more general than compressionembeddings. (Compression embeddings arise by insisting that appropriate cij = 1.) Wedefer the study of these more general systems to another time.III.

RepresentationsLetTn1φ1−→Tn2φ2−→Tn3φ3−→. .

. −→Abe a direct system with compression embeddings.Since each φk is a unital completeisometry, we may identify each Tnk with a unital subalgebra of A; the subalgebras soobtained are nested and the closure of their union is A.

Each subalgebra carries a matricialnorm and the sequence of norms is compatible with the nesting; consequently, A has amatricial norm. It is easy to see that this matricial norm satisfies the axioms of Blecher,Ruan and Sinclair [BRS], so A is an (abstract) operator algebra, i.e., there is a Hilbertspace H and a completely isometric unital algebra homomorphism ρ mapping A into B(H).This argument remains valid for systems of digraph algebras and for systems with the moregeneral Schur embeddings, provided that the embeddings are complete isometries.Our primary interest, however, is to obtain an explicit representation which is, in anappropriate sense, canonical.

To that end, let H denote a Hilbert space with basis {en}.This index set will depend on the specific direct system, but will always be either Z or{ n: n ≤b } for some integer b ≥0 or { n: n ≥a } for some integer a ≤0. For each k,let pk denote the projection on the closed linear span of { en : n ≤k } and let N denotethe nest consisting of the projections pk together with 0 and I. Alg (N ) denotes the nest

4ALAN HOPENWASSER CECELIA LAURIEalgebra associated with N . The representation which we will construct will map A to aweakly dense subalgebra of Alg (N ).For each k, we may write φk = ψka ⊕· · · ⊕ψkb , where each ψkj is a compression map.At least one of these compression maps must be the identity; in order to keep track of aselection of identity compressions, arrange the indexing so that ψk0 = id for all k. Thus, wewill always have that the first index a is non-positive and the last index b is non-negative.Now suppose that φ: Tnk →Tnk+1 is a compression embedding and that ψ is a singlecompression to an interval from the invariant projection lattice of Tnk+1.

It is easy to checkthat ψ ◦φ is a sum of compressions to intervals from the lattice of Tnk. Note that theintervals associated with ψ ◦φ may include some which were not associated with φ itself.A consequence of the observation above is that a composition of two compressionembeddings is again a compression embedding.

Since we need to consider multiple compo-sitions, let φjk denote the composition φk ◦· · · ◦φj. Then φjk can be written in the formbXa⊕ηi, where each ηi is a simple compression.

Furthermore, since ψn0 = id for each n, wemay arrange the indexing so that η0 = id. Thus, a ≤0 and b ≥0.

The indices a and bdepend, of course, on j and k, but that dependence is suppressed in the notation.Observe also that the string of η’s in the summation for φjk appears as a substringin the expression for φj,k+1. The reason, of course, is that ψk+10= id.

Furthermore, thissubstring includes the η0 = id term. Thus, as we increase k, we change the summationfor φjk by adding terms at one or both ends of the sum.

By increasing k indefinitely, weobtain a mapping ρj =X ⊕ηi. The index set for this sum will either be Z or a set of theform { i: i ≥a } or of the form { i: i ≤b } for some a ≤0 or b ≥0.An index set consisting of integers greater than or equal to some a will occur preciselywhen all but finitely many of the embeddings φk have φk0 = id as the first term; the othersingly infinite index set arises when the distinguished identity summand occurs as the lastterm in all but finitely many embeddings.

In either case, we can change the presentationby deleting finitely many embeddings to arrange that the index set for the η’s is eitherthe non-negative integers or the non-positive integers, as appropriate. In either of thesecases, we may now view ρj as a representation on the Hilbert space H with respect tothe appropriate basis.

The doubly infinite case still presents an ambiguity: we need to“anchor” the representation ρj with respect to the basis { en : n ∈Z }.This “anchoring” may be done as follows. Let the index set for the matrices in Tn1be { 0, 1, .

. ., b }, so that f 100 is the matrix unit which has a 1 in the upper left corner andzeros elsewhere.

Identify ψ10 with the mapping 0 ⊕· · · ⊕0 ⊕ψ10 ⊕0 ⊕· · · ⊕0, a mappingof Tn1 into Tn2. (In other words, replace all terms in φ1 except for the distinguished copyof the identity by a 0 mapping of appropriate dimension.) The image of f 100 under ψ10is now a matrix unit in Tn2; arrange the index set for Tn2 so that this matrix unit is

COMPRESSION LIMIT ALGEBRAS5f 200. By iterating this procedure, we can index all the Tnk so that the successive imagesof the matrix unit f 100 under the distingushed identity compressions will be f k00.

In thedoubly infinite case, the indexing of the Tnk’s will begin with negative integers and endwith positive integers each of arbitrarily large magnitude for sufficiently large k.It is now clear how to define the representation ρk: the image of the distinguishedmatrix unit f k00 in the component η0 should be the one dimensional projection onto thespan of the basis vector e0. For each k, ρk is a completely isometric representation ofTnk into Alg (N ) acting on the Hilbert space H.Furthermore, the following diagramcommutes:Tnkφk−−−−→Tnk+1yρkyρk+1Alg (N )id−−−−→Alg (N )This system of representations induces a completely isometric representation ρ from thedirect limit A into Alg (N ).Remark.

Observe that the image ρ(A) is weakly (or σ-weakly) dense in Alg (N ). To seethis it is enough to note that, given a ≤0 and b ≥0, the image contains matrices witharbitrarily specified entries in the locations ij for a ≤i ≤j ≤b.

To accomplish this,choose k sufficiently large, select an appropriate matrix in Tnk, and inspect the image ofthat matrix under the representation ρ.In the special case in which each φk is a direct sum of identity mappings, i.e., φk isa standard embedding, the representation which we have constructed is just the represen-tation of the standard limit algebra introduced by Roger Smith. For a description of thisrepresentation, see [OP].

Depending on the locations of the distinguished copy of the iden-tity, the representation will act on a Hilbert space with a basis indexed by the integers, bythe non-negative integers, or by the non-positive integers. The choice here is arbitrary, butsince the representations in this special case are all ∗-extendible, the C∗-algebras generatedby the images are all isomorphic.

(Indeed, they are all the UHF algebra associated withthe appropriate supernatural number. )In the general case, it is possible to have two completely isometric representations,ρ and σ, of A with C∗(ρ(A)) not isomorphic to C∗(σ(A)).

It is desirable, then, to finda representation for which the C∗-algebra generated by the image is the C∗-envelope inthe sense of Hamana. We shall show below that ρ, as constructed above, has just thisproperty.

It is in this sense that ρ is canonical.First, we provide a brief review of the idea of a C∗-envelope. The appropriate settingfor this is actually the category of unital operator spaces with unital complete order injec-tions, but we only need to deal with operator algebras so we will restrict the discussion tothat domain.

6ALAN HOPENWASSER CECELIA LAURIEIf A is an operator algebra, then a C∗-extension of A is a C∗-algebra B together witha unital complete order injection ρ of A into B such that C∗(ρ(A)) = B. A C∗-extensionB is a C∗-envelope of A provided that, given any operator system, C, and any unitalcompletely positive map τ : B →C, τ is a complete order injection whenever τ ◦ρ is.Hamana [H] proves the existence and uniqueness (up to a suitable notion of equivalence)of C∗-envelopes; further, he shows that the C∗-envelope of A is a minimal C∗-extension inthe family of all C∗-extensions of A.After proving the existence of C∗-envelopes, Hamana then uses this to prove theexistence of a ˇSilov boundary for A.

The ˇSilov boundary is a generalization to operatorspaces of the usual notion of ˇSilov boundary from function spaces; it was first developedby Arveson in [A1]. Here is the appropriate definition.

Let B be a C∗-algebra and let Abe a unital subalgebra such that B = C∗(A). An ideal J in B is called a boundary idealfor A if the canonical quotient map B →B/J is completely isometric on A.

A boundaryideal which contains every other boundary ideal is called the ˇSilov boundary for A.Hamana shows that if B is a C∗-extension for A, then the C∗-envelope of A is isomor-phic to B/J, where J is the ˇSilov boundary for A. In particular, B is the C∗-envelope forA if, and only if, the ˇSilov boundary is 0.

This last fact is the one which we shall use toshow that C∗(ρ(A)) is the C∗-envelope of the image ρ(A) for the representation ρ definedabove.The first step needed to accomplish this goal is to compute the C∗-algebra generatedby each ρk(Tnk). The following simple lemma is helpful:Lemma.

Let Tn be the n × n upper triangular matrices acting on Cn and let p and q bedistinct intervals from the nest of invariant projections for Tn. ThenC∗({ pap ⊕qaq: a ∈Tn }) = B(pCn) ⊕B(qCn).Proof.

Here, we view pap and qaq as being restricted to the ranges pCn and qCn of pand q respectively.Since p and q are distinct, there is an element a of Tn such thatone of pap and qaq, say pap, is non-zero while the other is 0. So we have an element ofthe form b ⊕0 in C∗({ pap ⊕qaq: a ∈Tn }).

From this and the fact that the C∗-algebragenerated by Tn is Mn, it follows that B(pCn)⊕0 ⊆C∗({ pap⊕qaq: a ∈Tn }. This in turnimplies that 0 ⊕B(qCn) ⊆C∗({ pap ⊕qaq: a ∈Tn } and hence that B(pCn) ⊕B(qCn) ⊆C∗({ pap ⊕qaq: a ∈Tn }.

The reverse containment is evident.By using the obvious extension of this lemma to multiple direct sums (including count-able direct sums), we can describe the C∗-algebra, C∗(ρk(Tnk)), both as a subalgebra ofB(H) and as an abstract (finite-dimensional) C∗-algebra. Indeed, write ρk = P⊕ηi and,for each i, let qi be the interval in the nest for Tnk to which ηi is a compression.

ThenC∗(ρk(Tnk)) = { (bj) ∈X ⊕B(qjCnk): bi = bj whenever qi = qj }.

COMPRESSION LIMIT ALGEBRAS7If r1, . .

., rm is the list of distinct intervals to which the ηi are compressions, thenC∗(ρk(Tnk)) ∼=mXj=1⊕B(rjCnk).We shall need both the isomorphism class of C∗(ρk(Tnk)) and its actual expression as anoperator algebra acting on H. Also, since C∗(ρ(A)) is the closure of the union of theC∗(ρk(Tnk)), we have proven the following:Proposition. Let A be the direct limit of a system of full upper triangular matrix algebraswith compression embeddings and let ρ be the representation of A defined above.

ThenC∗(ρ(A)) is an AF C∗-algebra.Our main result is the following theorem.Theorem. Let A be the direct limit of a system of full upper triangular matrix algebraswith compression embeddings and let ρ be the representation of A defined above.

ThenC∗(ρ(A)) is the C∗-envelope of ρ(A).Proof. In order to prove that C∗(ρ(A)) is the C∗-envelope of ρ(A), it is sufficient to showthat the ˇSilov boundary of ρ(A) is 0.

Since the ˇSilov boundary is the largest boundaryideal, we need merely show that if J is a non-zero ideal then J is not a boundary ideal.This requires proving that the canonical quotient map C∗(ρ(A)) →C∗(ρ(A))/J is notcompletely isometric when restricted to ρ(A). We shall, in fact, prove that ρ(A) ∩J ̸= ∅;the quotient map is not even isometric.For convenience, let Bk = C∗(ρk(Tnk)) and B = C∗(ρ(A)).

Each Bk is isomorphicto a direct sum of full matrix algebras, one for each compression map which appears inthe expression for ρk. Since the identity must appear, one summand must be Mnk.

Thissummand is the largest rank summand and it appears one time only. Mnk−1 appears atmost two times, reflecting the fact that the nest for Tnk has exactly two interval projectionsof rank nk −1.

More generally, Mnk−j appears at most j + 1 times in the expression forBk.Thus, the isomorphism class of Bk is Mnk ⊕Mt1 ⊕· · ·⊕Mts, where t1, . .

.ts are integersless than nk. In the Bratteli diagram for B, the kth level has a node for each summandin the isomorphism class of Bk.

Next, we need the following observation: given k anda summand Md in the isomorphism class of Bk, there is an integer j > k such that Mdpartially embeds into the summand Mnj in the isomorphism class of Bj.The observation is verified by inspecting the chain of finite dimensional C∗-algebrasacting on H. Now, ρk(Tnk) is an infinite sum whose terms are selected from finitely manycompressions of Tnk. Consequently, there exist integers a ≤0 and b ≥0 such that if p isthe projection onto the linear span of { ei : a ≤i ≤b }, then p is reducing for C∗(ρk(Tnk))

8ALAN HOPENWASSER CECELIA LAURIEand every compression, and in particular the one corresponding to Md, appears in therestriction of C∗(ρk(Tnk)) to p. Then, by the way in which the representations are con-structed, there is an integer j > k so that the η0 = id term in ρj acts on a subspace ofH which includes the range of p. (This was the reason for “anchoring” the ρj so that theη0 terms act on the vector e0 and that the support space for the η0 terms “grows away”from e0.) This shows that in the abstract Bratteli diagram, each node eventually partiallyembeds into a node corresponding to the identity summand (the Mnj node).Now let J be a non-zero closed two-sided ideal in B.

By a result in [B], J is the closureof the union of the J ∩Bk. In particular, there is a positive integer k such that J ∩Bk ̸= 0.Since Bk is isomorphic to a finite direct sum of full matrix algebras, J ∩Bk is isomorphicto a direct sum of some of those algebras, with 0’s as the remaining summands.

Let Mdbe one of the non-zero summands appearing in J ∩Bk. Let j > k be an integer such thatthe Md term partially embeds into Mnj.

It now follows that Mnj is one of the non-zerosummands for J ∩Bj. By utilizing the support projection for the Mnj term (a reducingprojection for C∗(ρj(Tnj))), we see that the subalgebra of C∗(ρj(Tnj)) isomorphic to Mnjis contained in J.

This subalgebra consists of all sequences (bi) in C∗(ρj(Tnj)) for whichbi = 0 whenever the corresponding interval qi is not the identity and the remaining bi areall equal.The proof is now completed by observing that there is an element of this subalgebrawhich lies in ρj(Tnj). Indeed, let v be the matrix unit in Tnj which has a 1 in the extremeupper right corner and zeros elsewhere.

Then the compression of v to any interval q otherthan the identity is zero. Thus, ρj(v) ∈J ∩ρ(A), and the proof is complete.Remark.

We have shown that the C∗-envelope of a direct limit of full upper triangularmatrix algebras with compression embeddings is an AF C∗-algebra. If, on the other hand,A is the direct limit of full upper triangular matrix algebras with ∗-extendible embeddings,then A can be represented as a generating subalgebra of a UHF algebra.Since UHFalgebras are simple, it is immediate that the generated C∗-algebra is the C∗-envelope.More is true.

It is easy to show that if A is a generating subalgebra of an AF C∗-algebraB and if A contains a Stratila-Voiculescu masa, then B is the C∗-envelope of A. Thus, ifA is the direct limit of a system of digraph algebras with ∗-extendible embeddings, thenthe C∗-envelope of A is an AF algebra.The simple proof of this last fact does not apply to compression limits.

Examples inthe next section will show that the image of the limit algebra under the representation ρneed not contain a masa in the generated C∗-algebra. (The natural diagonal in ρ(A) neednot be a masa in C∗(ρ(A)).) The following problem is suggested.Problem.

Is the C∗-envelope of a direct limit of digraph algebras with compression embed-dings an AF C∗-algebra?Remark. In [P1], Power studies a direct limit of digraph algebras with embeddings which

COMPRESSION LIMIT ALGEBRAS9are neither ∗-extendible nor compression embeddings. For systems of tri-diagonal algebraswith certain natural non-∗-extendible embeddings, Power shows that the limit algebra iscompletely isometrically isomorphic to a generating subalgebra of an appropriate Bunce-Deddens algebra.

Since this latter algebra is simple, it is the C∗-envelope of the limitalgebra for the tri-diagonal system. In particular, this provides an example of a system ofdigraph algebras whose limit algebra has a C∗-envelope which is not AF.Remark.

Since ρ(A) is weakly dense in Alg (N ), C∗(ρ(A)) is an irreducible C∗-algebra.If it happens that ρ(A) contains a non-zero compact operator, then Arveson’s boundarytheorem [A2] immediately implies that the Silov boundary for ρ(A) is 0. The boundarytheorem is a deep theorem, so the argument above could be considered more elementary.In most of the examples in the next section, ρ(A) contains non-zero compact oper-ators.Here is how to determine in general if ρ(A) contains compact operators.Eachrepresentation ρk of Tnk can be written as an infinite direct sum P⊕ηi, where the ηi’s arecompressions.

Let zk be the number of times that ηi = id in the expression for ρk. Thenzk ∈{ 1, 2, .

. ., ∞} and the sequence zk is decreasing.

Thus there are two possibilities: allzk = ∞, or there is a finite integer y such that zk = y for all large k. In the first case ρ(A)will contain no non-zero compact operators. In the second case, ρ(A) contains finite rankoperators and C∗(ρ(A)) contains all compact operators.For the first assertion, for any a ∈Tnk, ∥a∥= ∥ρk(a)∥= ∥ρk(a)∥ess.

It follows fromthe density of S ρk(Tnk) that ∥ρ(a)∥= ∥ρ(a)∥ess for all a ∈A. Thus, ρ(A) contains nonon-zero compact operators.

As for the second assertion, choose k so that zk is finite.Let v be the matrix unit in Tnk which has a 1 in the extreme upper right hand cornerand zeros elsewhere. Then ρk(v) has finite rank and lies in ρ(A).

Since ρ(A) is weaklydense in Alg (N ), it has no non-trivial reducing subspaces. Consequently, C∗(ρ(A)) isirreducible and contains a non-zero compact operator, which implies that it contains allcompact operators.IV.

Some ExamplesA. Fix an integer i between 1 and n and consider the system:Tnφ1−→Tn+k1φ2−→Tn+k1+k2φ3−→.

. .

−→Ai.Each embedding φn is given bya 7→a ⊕aiiIkn,

10ALAN HOPENWASSER CECELIA LAURIEwhere Ik is the k × k identity matrix. For example,a 7→aaiiaii7→aaiiaiiaiiaiiaii7→The representation ρ of the limit algebra Ai will act on a Hilbert space H withothornormal basis { ej } indexed by N ∪{0}.

Let pk denote the projection on the linearspan of { e0, . .

., ek } and N the nest consisting of the pk’s. Also, let KN be the algebra ofcompact operators which leave N invariant.It is easy to check that for each j, the image of any matrix under ρj is the sum of afinite rank matrix and a scalar multiple of the identity I. Consequently, ρ(Ai) ⊆KN +CI.The containment will, in fact, be proper.For each k, let fk = pk −pk−1.

LetBi = { s ∈KN + CI :limk→∞fksfk = fisfi }.The image of each ρk is clearly contained in Bi, whence ρ(Ai) ⊆Bi. To see the reversecontainment, let b ∈Bi.

There is a compact operator c in KN and a scalar α such thatb = c + αI. Observe thatbii = fibfi = limk→∞fkbfk = limk→∞(ckk + α) = α.

(Strictly speaking, this is abuse of notation, since, for example, fibfi is not a scalar butan infinite matrix with a single non-zero entry, viz., bii in the ith position on the diagonal.In cases like this we identify the two and the meaning should be clear. )Now, fix an integer k > i such that bkk is close to α and pkcpk is close to c. Thena = pkbpk + αp⊥k is in ρ(Ai) and is close to b.

Thus, ρ(Ai) is dense in, and hence equal toBi.Fact. The family of algebras Ai are pairwise non-isomorphic.Indeed, the identity and the pk are the only projections in KN + CI (or in B(H),for that matter) which are invariant under ρ(Ai).

But pk ∈ρ(Ai) if, and only if, k < i.Thus, keeping in mind that the indexing starts with 0, there are exactly i + 1 invariantprojections in Ai. This establishes the claim.

Note also that the C∗-envelope for each Aiis K + CI. The “linking condition” that the ith diagonal element is equal to the limit ofthe diagonal elements disappears in the passage to the generated C∗-algebra.

This is anexample in which the diagonal of ρ(A) is not a masa in C∗(ρ(A)).

COMPRESSION LIMIT ALGEBRAS11B. This time consider the systemTnφ1−→Tn+k1φ2−→Tn+k1+k2φ3−→.

. .

−→Awith the embeddings φj defined byφj(a) = a ⊕appIkj,where p = n + k1 + · · · + kj−1, i.e., app is the last entry on the diagonal of the matrix a.The Hilbert space H and the nest N are as before.This time, however, ρ(A) =KN +CI; the arguments are much the same as in the previous example. The limit algebraA now has infinitely many invariant projections, and so is not isomorphic to any of the Aifrom the previous example.

Also, the C∗-envelope of A is evidently once again K + CI.C. For this example, it is most convenient if the indices for each matrix algebra Tm aredrawn from the set { −m + 1, .

. ., −1, 0 }.

The basis for H is indexed by the non-positiveintegers and the pk will be the obvious infinite rank projections. Fix an integer i such that−n + 1 ≤i ≤0.

The embeddings in the systemTnφ1−→Tn+k1φ2−→Tn+k1+k2φ3−→. .

. −→Biare given byaφj7−→aiiIkj ⊕a.Arguments analogous to the one in example A show thatρ(Bi) = { s ∈KN + CI :limk→−∞fksfk = fisfi }.Once again, Bi ∼= Bj if, and only if i = j, and the C∗-envelope of each Bi is K + CI.

Thisis another example in which the diagonal of ρ(A) is not a masa in the C∗-envelope.The limit algebras in examples A and C are all non-isomorphic. All that needs to bechecked is that Bi ̸∼= Ai for each i, since when i ̸= j, Bi and Aj have a different number ofinvariant projections.

Observe that if p is a projection in ρ(Ai) ∩ρ(Ai)∗which is invariantunder ρ(Ai) and is not the identity, then for any sequence xj in Ai, the set { xjp } is linearlydependent. On the other hand, if p ̸= 0 is an invariant projection in ρ(Bi) ∩ρ(Bi)∗, thenthere exist operators x1, x2, .

. .

∈ρ(Bi) such that the set { xjp } is linearly independent.D. In this example, H is a Hilbert space with basis indexed by N ∪{0}, N = { pk } andKN are as before, and D denotes the algebra of diagonal matrices.

Also, d will denotethe conditional expectation from Tn onto Dn; in other words, if a is an upper triangularmatrix, d(a) is the diagonal part of a. Note that d is a direct sum of rank one compressions.

12ALAN HOPENWASSER CECELIA LAURIENow consider the stationary systemT2φ1−→T4φ2−→T8φ3−→. .

. −→Ain which φk : T2k →T2k+1 is given by φk(a) = a ⊕d(a).The representations ρk : T2k →KN + D are given by ρk(a) = a ⊕d(a) ⊕d(a) ⊕· · · =a ⊕P⊕d(a).

The image ρ(A) of the limit algebra under the canonical representation isthus a subalgebra of KN + D. Once again, the image will be a proper subalgebra.Definition. We say that an element b ∈D is periodic with period p if bm+p = bm, for allm.

Here, bm denotes the matrix entry bmm. We say b is dyadic periodic is the period p is apower of 2.Let DAP (2∞) denote the closure in the norm topology of the dyadic periodicelements of D.Let K0N = { a ∈KN : d(a) = 0 }.

Clearly, ρk(T2k) ⊆K0N + DAP (2∞), for each k.Thus, ρ(A) ⊆K0N + DAP (2∞). Furthermore, an argument analogous to the one used inexample A shows that we have equality: ρ(A) = K0N + DAP (2∞).

The C∗-envelope of Ais therefore K + DAP (2∞).E. The notation is the same as in the previous example, but the embedding is changed toa 7−→a ⊕dlh(a) ⊕dlh(a)where dlh(a) is the diagonal last half of a.

(When a is a 2k × 2k matrix, dlh(a) is a2k−1 × 2k−1 matrix. )If a is a matrix in T2k, then the first 2k−1 diagonal terms in ρk(a) bear no relationto the remaining diagonal terms in ρk(a).

This difference in behaviour compared with thelast example results in a slightly different image for the limit algebra under the canonicalrepresentation: ρ(A) = KN + DAP (2∞). The C∗-envelope is unchanged.F.

Consider the stationary systemT2φ1−→T4φ2−→T8φ3−→. .

. −→Ain which φk : T2k →T2k+1 is given by φk(a) = a ⊕lh(a) ⊕lh(a).Here lh(a) denotescompression to the last half of a, resulting in a 2k−1 × 2k−1 matrix when a is 2k × 2k.Let S(2∞) denote the norm closure of all the dyadic periodic matrices.

(A matrix on His periodic with period p if it has the form a ⊕a ⊕· · · , where a is a p × p matrix.) Theintersection of S(2∞) and Alg (N ) will be denoted by SN (2∞).

With this notation, theimage of the limit algebra A under the canonical representation ρ is KN + SN (2∞) andthe C∗-envelope is K + S(2∞).

COMPRESSION LIMIT ALGEBRAS13G. We conclude by mentioning an old example, the systemT2φ1−→T4φ2−→T8φ3−→.

. .

−→Awith the standard embeddings of multiplicity 2: φk(a) = a ⊕a. The canonical representa-tion is the Smith representation described in [OP]; ρ(A) = SN (2∞) and the C∗-envelopeis the UHF algebra S(2∞).References[A1]W. Arveson, Subalgebras of C∗-algebras, Acta Math.

123 (1969), 141–224. [A2], Subalgebras of C∗-algebras II, Acta Math.

128 (1972), 271–308. [BRS] D. Blecher, Z. Ruan, and A. Sinclair, A characterization of operator algebras, J. Functional Analysis89 (1990), 188–201.[B]O.

Bratteli, Inductive limits of finite dimensional C∗-algebras, Trans. Amer.

Math. Soc.

171 (1972),195–234.[H]M. Hamana, Injective Envelopes of Operator Systems, Publ.

RIMS, Kyoto Univ. 15 (1979), 773–785.[L]D.

Larson, Structured Limit Algebras, This journal.[OP]J. Orr and J. Peters, Some representations of TAF algebras, Preprint.[P1]S.

C. Power, Non-self-adjoint operator algebras and inverse systems of simplicial complexes, J.Reine Angewandte Mathematik 421 (1991), 43–61.[P2]S. C. Power, Limit algebras; an introduction to subalgebras of C∗-algebras, Pitman Research Notes,Longman, 1992.Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487E-mail address: ahopenwa@ua1vm.ua.edu and claurie@ua1vm.ua.edu

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