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ON THE OUTER AUTOMORPHISM GROUPS OF

arXiv:funct-an/9302003v1 9 Feb 1993ON THE OUTER AUTOMORPHISM GROUPS OFTRIANGULAR ALTERNATION LIMITALGEBRASS. C. PowerDepartment of MathematicsUniversity of LancasterLancaster LA1 4YFEngland.ABSTRACTLet A denote the alternation limit algebra, studied by Hopenwasser and Power, and byPoon, which is the closed direct limit of upper triangular matrix algebras determined byrefinement embeddings of multiplicity rk and standard embeddings of multiplicity sk.

It isshown that the quotient of the isometric automorphism group by the approximately innerautomorphisms is the abelian group ZZd where d is the number of primes that are divisorsof infinitely many terms of each of the sequences (rk) and (sk). This group is also the groupof automorphisms of the fundamental relation of A.

1IntroductionIn Hopenwasser and Power [HP] and in Poon [Po] the alternation limit algebras describedbelow were classified. In this note we determine the quotient group OutisomA = AutisomA/I(A)for these algebras where AutisomA is the group of isometric algebra automorphisms and I(A)is the normal subgroup of AutA of approximately inner automorphisms.

An automorphismα is said to be approximately inner if there exists a sequence (bk) of invertible elementssuch that α(a) = limk bkab−1kfor all a in A.Let (rk), (sk) be sequences of positive integers. Write T(rk, sk) for the Banach algebralimit of the systemIC →Tr1 →Tr1s1 →Tr1s1r2 →.

. .

,where Tn is the algebra of upper triangular n × n complex matrices and where the embed-dings are unital and are alternately of refinement type (ρ(a) = (aij1t), with 1t the t × tidentity and of standard type (σ(a) = a ⊕. .

. ⊕a, t times).Theorem 1 Outisom(T(rk, sk)) = ZZd where d is the number of primes p that are divisorsof infinitely many terms of each of the sequences (rk) and (sk).

(If d = ∞interpret ZZd asthe countably generated free abelian group. )The proof uses the methods of [HP].

A major step is to characterise the automorphismgroup of the fundamental relation, or semigroupoid, which is associated with an alternationalgebra. This order-topological result is of independent interest and is stated and provedseparately below.Let r and s be the generalised integers r1r2 .

. .

, and s1s2 . .

. respectively and suppose thatp is a prime satisfying the condition in the statement of the theorem.

Then p∞divides rand s. Thus we can arrange new formal products r = t1t2 . .

., s = u1u2 . .

., with tk = uk = pfor all odd k. As noted in [HP], because of the commutation of refinement and standardembeddings, we can easily display a commuting zig zag diagram to show that T(rk, sk)and T(tk, uk) are isometrically isomorphic. However, with the new formal product we canconstruct one of the generators of OutisomA.

Consider the automorphism α determined

by the following commuting diagram where the matrix algebras are omitted for notationaleconomy.✲✲✲✲✲✲✲✲✲✲✲✲✲σu1ρt2σu2ρt3σu3ρt1σu1✲ρt2ρt3σu2ρt3σu3ρt1❅❅❅❅❘❅❅❅❅❘✒✒❄. .

.. . .T(tk, uk)T(tk, uk)ICICσt1ρt1σt3It will be shown below that α provides a nonzero coset and that the totality of such cosetsprovides a generating set for the isometric outer automorphism group.2Proof of Theorem 1Let X, or X(rk, sk), be the Cantor spaceX = Π−1k=−∞{1, .

. .

, s−k} × Π∞k=1{1, . .

. , rk},where we have fixed the sequences (rk) and (sk).

Define the equivalence relation ˜R on X toconsist of the pairs (x, y) of points x = (xk), y = (yk) in X with xk = yk for all large enoughand small enough k. ˜R carries a natural locally compact Hausdorfftopology (giving it thestructure of an approximately finite groupoid). Write R, or R(rk, sk), for the antisymmetrictopologised subrelation of ˜R consisting of pairs (x, y) in R for which x preceeds y in thelexicographic order.

Thus (x, y) ∈R if and only if (x, y) ∈˜R and, either x = y, or for thesmallest k for which xk ̸= yk we have xk < yk.An automorphism of R(rk, sk) is a binary relation isomorphism (implemented by a bijec-tion α of the underlying space X), which is a homeomorphism for the (relative groupoid)topology of R(rk, sk). Necessarily α is a homeomorphism of X.Theorem 2 The group of automorphisms of the topological binary relation R(rk, sk)

is ZZd where d is the number of primes which divide infinitely many terms of each of thesequences (rk) and (sk).Proof:Let O(x) denote the closure of the R-orbit of the point x in X. Here O(x) = {y :(y, x) ∈R}.

Recall from [HP] that the pair of points x, x+ is called a gap pair if x+ ̸∈O(x)andO(x+) = O(x) ∪{x}.Furthermore x, x+ is a gap pair if and only if1) there exists n such that xm = 1 for all m ≤n,2) there exists p such that xq = rq for all q ≥p.Also if p is the smallest integer for which 2) holds (with rp = s−p if p is negative), thenx+ is given by(x+)j =xj if j < p −1xp−1 + 1 if j = p −11 if j ≥pThe usefulness of this for our purpose is that an automorphism α of R necessarily mapsgap pairs to gap pairs and so the coordinate description of these pairs leads ultimately toa coordinate description of α.Let α be an automorphism of R. Consider the (left) gap point x∗= (. .

. , 1, 1, ˆ1, r1, r2, .

. .

)where ˆ1 indicates the coordinate position for s1. Then α(x∗) is necessarily a (left) gap point,thusα(x∗) = (.

. .

, 1, 1, z−t+1, z−t, . .

. , zt−1, rt, rt+1 .

. .

)for some positive integer t. We haveO(x∗) = {x = (. .

. , 1, ˆ1, x1, x2, .

. .) : xk ≤rk for all k},O(α(x∗)) = {y = (.

. .

1, w′, yt, yt+1, . .

. )},where yk ≤rk for all k ≥t and where w′ is any word of length 2t −2 which preceeds (oris equal to) the word w = z−t+1, z−t, .

. .

, zt−1 in the lexicographic order. Restating this, we

have natural homeomorphismsO(x∗) ≈Π∞k=1{1, . .

. , rk}O(α(x∗)) ≈{1, .

. .

n} × Π∞k=t{1, . .

. , rt}where n is the number of words w′.

Moreoever, these identifying homeomorphisms induceisomorphisms between the restrictions R|O(x∗) and R|O(α(x∗)) and the unilateral relationsR1 and R2, respectively, where R1 = R(rk, uk), with uk = 1 for all k, and R2 = R(r′k, uk),with uk as before, r′1 = n, and r′k = rk+t−2 for k = 2, 3, . .

.. Since α induces an isomorphismbetween the restrictions, we obtain an induced isomorphism β between R1 and R2.

It iswell-known that this means that r = r′ where r = r1r2 . .

. and r′ = r′1r′2 .

. .

are generalisedintegers. (See [P2] for example).

Thus we obtain the necessary condition that the integern is a divisor of the generalised integer r.We shall now improve on this necessary condition.The isomorphism between R|O(x∗) and R|O(α(x∗)) is given explicity byα : (. .

. 1, ˆ1, x1, x2, .

. .) →(.

. .

1, w′, yt, yt+1, . .

. )where∥w′∥−1n+∞Xk=1(yt+k−1 −1)nmt+k−1mt−1 =∞Xk=1xk −1mk,(1)where ∥w′∥is the cardinality of the set of points in the order interval from the (2t−2)-tuple(1, 1, .

. .

, 1) to w′, and where mk = r1r2 . .

. rk for k = 1, 2 .

. ..

The identity (1) follows fromthe fact that there are unique canonical R-invariant probability measures on O(x∗) andon O(α(x∗)) and the quantities in (1) are the measures of the subsets O(α(x)) and O(x)respectively.To verify these facts one must recall how the topology of a topological binary relation isdefined. In the case of R1 = R|O(x∗) fix two words(x1, x2, .

. .

, xℓ) ≤(x′1, x′2, . .

. , x′ℓ)

in lexicographic order. Then the set E of pairs((x1, x2, .

. .

, xℓ, zℓ+1, zℓ+2, . .

. ), (x′1, x′2, .

. .

, x′ℓ, zℓ+1, zℓ+2, . .

. ))is, by definition, a basic open and closed subset for the topology.Notice that for thisset, the left and right coordinate projection maps, πℓ: E →O(x∗), πr :→O(x∗), areinjective.

In the language of groupoids, E is a G-set. If λ is a Borel measure such thatλ(πℓ(E)) = λ(πr(E)) for all closed and open G-sets E, then λ is said to be R-invariant.

Itis easy to see that this requirement forces λ to be the product measure λ1 × λ2 × . .

. whereλk is the uniformly distributed probability measure on {1, .

. .

, rk}. (One can also bear inmind that R-invariant measures are also ˜R-invariant, where ˜R is the topological equivalencerelation (i.e.

groupoid) generated by R, and that the ˜R-invariant measures correspond totraces on the C*-algebra of ˜R. In our context C∗( ˜R) is UHF, and the R-invariant measurecorresponds to the unique trace.

)Let ν(x) denote the right hand quantity of (1).Then the coordinates for α(x) arecalculated from the identity (1), bearing in mind that the ambiguity arising from theequality ν(x) = ν(x+), for a gap pair x, x+, is resolved by the known correspondenceof left and right gap points.Note that if x is in O(x∗), and α(x) = y = (yk), and ∥w′∥= 1 (so that y−t+1, y−t, . .

. , ytare all equal to 1), then, by (1),ν(α(x)) =∞Xk=1yk −1mk=∞Xk=1yt+k−1 −1mt+k−1= nν(x)mt−1.We have obtained the identity ν(α(x)) = cν(x), with c = n/mt−1, for all points x in O(x∗)for which ν(x) is small.

In fact, because of the R-invariance of the measures on O(x∗) andO(α(x∗)), which we shall call λ1 and λ2 respectively, it follows that ν(α(x)) = cν(x) forall points x for which α(x) ∈O(x∗). To be more precise about this, consider the left gappointsg=(.

. .

1, ˆ1, 1, . .

. , 1, rℓ+1, .

. .

),x=(. .

. 1, ˆ1, w, rℓ, rℓ+1, .

. .

),x′=(. .

. 1, ˆ1, w, rℓ−1, rℓ+1, .

. .

),

where w is some word w1, w2, . .

. , wℓ−1.

Note that the setE = {((. .

. 1, ˆ1, w, rℓ, zℓ+1, zℓ+2, .

. .

), (. .

. 1, ˆ1, .

. .

, 1, zℓ+1, zℓ+2, . .

.)) : zj ≤rj}has πℓ(E) = O(x) \ O(x′) and πr(E) = O(g), and so ν(g) = ν(x) −ν(x′).

Since α preservesorbits and G-sets we also deduce thatν(α(g))=λ1(O(α(g))) = λ1(πr((α × α)(E)))=λ1(πℓ((α × α)(E))) = λ1(O(α(x)) \ O(α(x′)))=ν(α(x)) −ν(α(x′)).Thus, if we choose ℓlarge, so that we know that ν(α(g)) = cν(g), we deduce thatν(α(x)) −ν(α(x′)) = ν(α(g)) = cν(g) = c(ν(x) −ν(x′)),from which it follows that ν(α(x)) = c(ν(x)) for general points x with α(x) in O(x′).We can similarly extend this identity to points in the setX0 = {(yk) ∈X : ∃k0 such that yk = 1 for all k ≤k0}and the extension of ν given byν(y) =∞Xk=1(y−k −1)s0s1 . .

. sk−1 +∞Xk=1yk −1mkfor y in X0, where s0 = 1.

The range of ν on the gap points of X0 is the additive cone ofrationals of the form ℓ/mk for some k = 1, 2, . .

. and some natural number ℓ.

The identityν(α(x)) = cν(x) for x in X0 shows that multiplication by c is a bijection of the cone. Fromthis we obtain the necessary condition that c has the formc = pa11 .

. .

paddwhere ai ∈ZZ, 1 ≤i ≤d, and where p1, . .

. pd are primes which divide infinitely many termsof the sequence (rk).We now improve further on this condition by considering the fact that α is a homeomor-phism of X and is determined by its restriction to X0.

Suppose, by way of contradiction, that a1 ̸= 0 and that p1 does not divide infinitely manyterms of the sequence (sk). Note that c only depends on α, thus, replacing α by its inverseif necessary, we may assume that a1 > 0.

By relabelling we may also assume that p1 dividesno terms of the sequence. Without loss of generality assume that s1 > 1 and consider theproper clopen subset E of points y = (yk) in X with y−1 = 1.

We show that α(E) is dense,which is the desired contradiction. Observe first that the range of ν on E ∩X0 is the unionof the intervals [ks1, ks1 + 1] for k = 0, 1, 2, .

. .

Pick x in X0 arbitrarily, pick j large, andconsider the countable setFj(x) = {x′ ∈X0 : x′ = (x′k) and x′k = xk for all k ≥−j}.The range of ν on Fj(x) is an arithmetic progression of period s1s2 . .

. sj.In view ofthe identity ν(α(y)) = cν(y), the range of ν on α(E) ∩X0 is the union of the intervals[cks1, cks1 +c], which is an arithmetic progression of intervals of period cs1.

It follows fromour hypothesis on p1 that one of these intervals contains a point in ν(Fj(x)), and so α(E)meets Fj(x). Since the intersection of the sets F1(x), F2(x), .

. .

is the singleton x, it followsthat x lies in the closure of α(E). Since X0 is dense it follows that α(E) is dense as desired.We have now shown that if α is an automorphism of R = R(rk, sk), then ν(α(x)) = cν(x)for all x in X0 where c has the form c = pa11 pa22 .

. .

paddwhere a1, . .

. , ad are integers andwhere p1, .

. .

pd are primes which divide infinitely many terms of (rk) and of (sk). It is alsoclear from the above that for each such c there is at most one automorphism α satisfyingthe identity ν(α(x)) = cν(x).

It follows that the mapα →(a1, . .

. , ad)is an injective group homomorphism from AutR to ZZd.

(d may be infinite.) It remains toshow that this map is surjective.

One way to do this is to start with c of the required formabove and to show that the bijection of X0 induced by multiplication by c (that is, thebijection α satisfying ν(α(x)) = cν(x)) does extend to an order preserving homeomorphismof X which defines an automorphism of R. Another way, which we now follow, is to makethe connection between R(rk, sk) and T(rk, sk), and to determine generators of AutR interms of commuting diagrams, as we indicated after the statement of Theorem 1.

Consider the diagramICρr1→Mr1σs1→Ms1 ⊗Mr1ρr2→Ms1 ⊗Mr1 ⊗Mr2σs2→. .

.B↑↑↑↑↑ICρr1→Tr1σs1→Ts1r1ρr2→Ts1r1r2σs2→. .

.AThe vertical maps are inclusions, where Ts1r1r2, for example, is realised in terms of thelexicographic order on the indices (i, j, k) of the minimal projections eii ⊗ejj ⊗ekk inMs1 ⊗Mr1 ⊗Mr2. (For more detail concerning this discussion, read the introduction of[HP].) The maximal ideal space of the diagonal C*-algebra A ∩A∗is naturally identifiedwith the space X.

Indeed, x = (xk) in X corresponds to the point in the intersection ofthe Gelfand supports of the projectionse(x, N) = ex−N,−N ⊗. .

. ⊗ex−1,−1 ⊗ex1,1 ⊗.

. .

⊗exN,Nfor N = 1, 2, . .

.. Furthermore, (x, y) belongs to R = R(rk, sk) if and only if for all largeN there is a matrix unit in the appropriate upper triangular matrix algebra with initialprojection e(y, N) and final projection e(x, N).

(In fact R is the fundamental relation ofthe limit algebra A. )Suppose now that rk = sk = p for all odd k and let α be the isometric automorphismof T(rk, sk) determined by the diagram given in the introduction.

Let α also denote theinduced automorphism of R. We prove that ν(α(x)) = p−1ν(x), completing the proof ofthe theorem.Let us calculate α(e(x, N)), where N is even, x = (. .

. , 1, ˆ1, 2, 1, .

. .

), and where weabuse notation somewhat and write e(x, N) for the image of e(x, N) in the limit algebra.Let d(N) = sN . .

. s1r1 .

. .

rN, and let e(x, N) occupy position a(N) in the lexicographicordering of the d(N) matrix units. Consider the following part of the diagram defining α.TsN..rNρrp→TsN..rN+1i→Aσp↓TsN..rN+1i→A

Thenρp(e(x, N)) =pXk=1e(x, N) ⊗ekk.On the other hand σp(e(x, N)) is the summation of the diagonal matrix units in positionsa(N), a(N)+d(N), . .

. , a(N)+(p−1)d(N) in the lexicographic order.

Let these projectionscorrespond to the matrix unit tensors with subscripts z(i) = (z(i)−N, . .

. z(i)N+1) for 1 ≤i ≤p, and denote the projections themselves by f1, .

. .

fp, respectively. It follows (from thepartial diagram above) that the homeomorphism α : X →X maps the support of e(x, N)onto the union of the supports of f1, .

. .

, fp. Denote these supports by E(x, N), F1, .

. .

, Fprespectively. Since X0 is invariant for α,α(E(x, N) ∩X0) =p[k=1Fk ∩X0.Notice that x is the unique point in E(x, N)∩X0 with the property that if y ∈E(x, N)∩X0and (x, y) ∈˜R then (x, y) ∈R.

The point in the union of F1 ∩X0, . .

. , Fp ∩X0 with thisminimum property is the pointu = (.

. .

1 1 z(1)−N, . .

. , z(1)N+1, 1, 1, .

. .

)and so α(x) = u. Finally one can verify that ν(x) = p−1 and ν(u) = p−2, as desired.✷Recall that the fundamental relation R(A) of a canonical triangular subalgebra A ofan AF C*-algebra B is the topological binary relation on the Gelfand space M(A ∩A∗)induced by the partial isometries of A which normalise A ∩A∗.

(See [P2].) In [HP] weidentified R(A), for A = T(rk, sk), with R(rk, sk).

(This identification is also effected inthe proof above by virtue of the fact that a matrix unit system determines R(A).) Let βbe an isometric automorphism of A.

Then β induces an automorphism of R(A) (becauseβ(A ∩A∗) = A ∩A∗and β maps the normaliser onto itself).Thus β determines anautomorphism of R(rk, sk) and so by the last theorem there is an isometric automorphismα of A such that γ = α−1 ◦β induces the trivial automorphism of R(rk, sk). This meansthat γ is an isometric automorphism with γ equal to the identity map on A ∩A∗.Lemma Let γ be an automorphism of T(rk, sk) which is the identity on the diagonalsubalgebra (and which is not necessarily isometric).

Then γ is approximately inner.

Proof:Let A = T(rk, sk) and let A1 →A2 →. .

. be the direct system defining A. Thehypothesis is that γ(c) = c for all c in C = A ∩A∗.

This ensures that γ( ˜An) = ˜An where˜An is the subalgebra generated by An and C. To see this,recall from Lemma 1.2 of [P1]that there are contractive maps Pn : A →˜An which are defined in terms of limits of sumsof compressions by projections in C, and so, for a in ˜An, γ(a) = γ(Pn(a)) = Pn(γ(a)).The restriction automorphism γ| ˜An is necessarily inner. Indeed identify ˜An with Tr ⊗D,for appropriate r, where D is an abelian approximately finite C*-algebra and let ui ∈D,1 ≤i ≤r −1, be the invertible elements such that γ(ei,i+1) = ei,i+1 ⊗ui.

Also set u0 = 1.Then it follows that γ(a) = u−1au, whereu =rXi=1ei,i ⊗u0u1 . .

. ur−1Furthermore, since γ(e1,r) = e1,r ⊗u0u1 .

. .

ur−1, it follows that ∥u∥≤∥γ∥.Similarly∥u−1∥≤∥γ−1∥. The inner automorphisms Adu−1, for varying n, thus form a uniformlybounded sequence which converge pointwise on each An, and so determine an approximatelyinnder automorphism.✷It follows from Lemma 1 and the preceeding discussion thatAutisomA/I(A) = AutR(A) = ZZd.Remark 1.

Suppose that δ ∈AutA. Then δ determines a scaled group homomorphismδ∗: K0(A) →K0(A) which preserves the algebraic order on the scale Σ(A) of K0(A).

Thus,by the main theorem of [P3], (which can also be found in [P4]) there is an isometric algebraautomorphism of A, φ say, with φ∗= δ∗. In particular ψ = φ−1 ◦δ has ψ∗trivial.

Thismeans that if P : A →A ∩A∗is the diagonal expectation, then P(ψ(e)) = e for eachprojection e in A ∩A∗. Thus to show that AutA/I(A) = ZZd it remains only to show thatsuch automorphisms ψ are approximately inner.Remark 2.

There are approximately inner automorphisms of alternation algebras whichare not inner. To see this, consider the standard limit algebra A = lim→(T2n, σ).Let λ be a unimodular complex number and let dn = λe1,1 +λ2e2,2 +.

. .+λ2ne2n,2n.

Then

dnad−1n = dmad−1m if a ∈T2n and m > n, from which it follows that α(a) = limn(dnad−1n ) isan isometric approximately inner automorphism.Suppose now that α is inner, and α(a) = gag−1 for some invertible g in A. Since α(c) = cfor all c in the masa C it follows that g ∈C.

In particular ∥α −β∥≤14 for some innerautomorphism β of the form β(a) = hah−1 where, for some large enough n, h ∈T2n ∩(T2n)∗.However, in T2m, for large m, the diagonal element h has matrix entries which are periodicwith period 2n. One can now verify that if λ is chosen so that no power of order 2k is unitythen for large enough m there exist matrix units e ∈T2m such that ∥λe −heh−1∥> 14, acontradiction.Remark 3.

Let (x, y) be a point in R(C∗(A(rk, sk))) with x = (. .

. , x−2, x−1, x1, x2, .

. .

),y = (. .

. , y−2, y−1, y1, y2, .

. .

). Then, although ν(x) and ν(y) may be infinite, we may defined(x, y) as the sum∞Xk=1(y−k −x−k)s0s1 .

. .

sk−1 +∞Xk=1yk −xkr1r2 . .

. rkbecause only finitely many terms are nonzero.

Since d(x, y) = d(x, z)+d(z, y), and (x, y) ∈R(rk, sk) if and only if d(x, y) ≥0, it follows that d(x, y) is a continuous real valued cocyledetermining A(rk, sk) as an analytic subalgebra of C∗(A(rk, sk)).See [V], where somespecial cases are discussed as well as some general aspects of analyticity.Added Dec 1992 : Unfortunately the proof of the classification of alternation algebrasgiven in [HP] and [P4] appears to be incomplete. (It is not clear, in [P4], whether q can bechosen with the desired properties.) However the present paper is independent of [HP] andthe arithmetic progression argument above can be adapted, to the case of an isomorphismα between two alternation algebras, to show that the supernatural numbers for the standardmultiplicities are finitely equivalent.References[HP]A. Hopenwasser and S.C. Power, Classification of limits of triangular matrixalgebras, Proc.

Edinburgh Math. Soc., to appear.

[P1]S.C. Power, On ideals of nest subalgebras of C*-algebras, Proc. London Math.Soc., 50 (1985), 314-332.[P2]S.C.

Power, The classification of triangular subalgebras of AF C*-algebras, Bull.London Math. Soc., 22 (1990), 269-272.

[P3]S.C. Power, Algebraic orders on K0 and approximately finite operator algebras,preprint 1989, to appear in J. Operator Th.[P4]S.C.

Power, Limit Algebras : An Introduction to Subalgebras of C*-algebras,Pitman Research Notes in Mathematic Series, No. 278, Longman, 1992.[Po]Y.T.

Poon, A complete isomorphism invariant for a class of triangular UHF alge-bras, preprint 1990.[V]B.A. Ventura, Strongly maximal triangular AF algebras, International J.

Math., 2(1991), 567–598.

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