New York Journal of Mathematics

New York J. Math. 4 (1998) 35–56.

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Homology for Operator Algebras III: Partial Isometry Homotopy and Triangular Algebras

S. C. Power

Abstract. The partial isometry homology groups H_{n} defined in Power [17] and a related chain complex homology CH_{∗} are calculated for various triangular operator algebras, including the disc algebra. These invariants are closely connected with K-theory. Simplicial homotopy reductions are used to identify both H_{n} and CH_{n} for the lexicographic products A(G) ⋆ A with A(G) a digraph algebra and A a triangular subalgebra of the Cuntz algebra O_{m}. Specifically H_{n}(A(G) ⋆ A) = H_{n}(Δ(G)) ⊗_{ℤ} K_{0}(C^{∗}(A)) and CH_{n}(A(G) ⋆ A) is the simplicial homology group H_{n}(Δ(G); K_{0}(C^{∗}(A))) with coefficients in K_{0}(C^{∗}(A)).

Contents

1. The Partial Isometry Homology H_{n}(A; C) 37 2. Vanishing Homology 40 3. The Proof of Theorem 1 43 4. K_{0}-regular Inclusion and Homotopy 44 5. The Cuntz Algebras and T O_{m} 45 6. The Proof of Theorem 2 48 7. The Partial Isometry Chain Complex Homology 51 References 56

Taking the perspective that equivalence classes of projections in the stable algebra of a non-self-adjoint algebra A may be viewed as 0-simplexes one can often identify the resulting homology group H_{0}(A) as K_{0}(C^{∗}(A)). Analogously, viewing partial isometries in the stable algebra as 1-simplexes one can similarly formulate higher order homology group invariants for A. This was done recently in Power [17] with the intention of extending the limit homology groups introduced by Davidson and Power [3], for regular limit algebras, to subalgebras of general C*-algebras.

In the present paper we develop further these homology invariants together with a related chain complex homology CH_{∗} also derived from partial isometries in the stable algebra. Our main purpose is to indicate methods for calculation mainly in

\begin{matrix}c \\ ◯ \end{matrix}1998 State University of New York ISSN 1076-9803/98

Received January 23, 1998.

Mathematics Subject Classification. 47D25, 46K50.

Key words and phrases. operator algebra, homology group, nonselfadjoint, Cuntz algebra.

the setting of triangular algebras, for which A ∩ A^{∗} is abelian. As we see there is a close connection with K-theory for operator algebras, both in terms of the formulation of the invariants and in their identification.

In appropriate contexts induced homology group homomorphisms together with symmetric homology scales seem to provide critical invariants for the position of subalgebras and for the classification of limit algebras. See for example Donsig and Power [5], [6] and Power [17], [18]. In [6] for example, we completely characterise the regular isomorphism classes of the so-called rigid systems of 4-cycle matrix algebras in terms of K_{0}, H_{1} and scales in K_{0} ⊕ H_{1}. This indicates that partial isometry homology may be more accessible and appropriate than Hochschild cohomology at least in the common setting of algebras with a regular diagonal. At the same time it will be of interest to elucidate the connections between H_{∗}(A) and the Hochshild cohomology of operator algebras, as given in Gilfeather and Smith [7], [8], for example.

The partial isometry homology group H_{1}(A) can be viewed as an obstruction to the cancellative triangulability of cycles of partial isometries. (In the sequel we shall restrict attention to homology groups arising from normalising partial isometries, this being appropriate for algebras with a regular maximal abelian self-adjoint subalgebra (masa).) To indicate this idea briefly, consider a partial isometry 2ncycle, by which we mean a 2n-tuple {v_{1}, v_{2}, …, v_{2n}} with v_{2n}^{∗}v_{2n−1}v_{2n−2}^{∗} … v_{2}^{∗}v_{1} = v_{1}^{∗}v_{1}, and with appropriately matching initial and final projections, v_{1}v_{1}^{∗} = v_{2}v_{2}^{∗} etc. Such a cycle is associated with a 2n-sided polygonal directed graph (digraph) with alternating edge directions. It may be that for a particular such cycle in the stable algebra of A that one can add additional partial isometries from the stable algebra so that the totality has a digraph (with compositions of edges respecting operator multiplication) which provides a triangulation of the original 2n-cycle graph. In this case the 2n-cycle gives no contribution to H_{1}(A). Thus, if partial isometry cycles are always triangulable in this way then H_{1}(A) vanishes. This is the case for the disc algebra for example. However there is no general converse assertion because H_{1}(A) may also vanish for reasons of cancellation, as in the case of some of the algebras of Theorem 2.

Theorem 1. Let A(𝔻) be the disc algebra. Then H_{0}(A(𝔻)) = ℤ and H_{n}(A(𝔻)) = 0 for n ≥ 1.

For Theorem 2 below H_{n}(Δ(G)) denotes the integral simpicial homology of the simplicial complex Δ(G) of the digraph algebra A(G). We write A_{1} ⋆ A_{2} for the triangular lexicographic product (see [14], [15]) relative to the natural direct sum decomposition A_{1} = (A_{1} ∩ A_{1}^{∗}) + A_{1}^{0}. Thus A(G) ⋆ A, with A triangular, is simply the algebra

(A(G) ∩ A(G)^{∗}) ⊗ A + A(G)^{0} ⊗ C^{∗}(A),

which is also triangular if A(G) is triangular. The remaining terminology is explained later in the text.

Triangular algebras have a distinguished (maximal abelian self-adjoint) diagonal and an associated family of normalising partial isometries. Accordingly we can define partial isometry homology group invariants based on this family. In general the problem of uniqueness of diagonals must be addressed. On the other hand lexicographic products do give diverse triangular algebras with computable homology.

Theorem 2. Let A(G) be a digraph algebra. Let T O_{m}^{0} and T O_{m} be the refinement nest subalgebras of the algebraic Cuntz algebra O_{m}^{0} and its closure in the Cuntz algebra O_{m}, respectively. Then H_{n}(A(G)⋆T O_{m}^{0}) = H_{n}(A(G)⋆T O_{m}) = H_{n}(Δ(G))⊗ ℤ_{m−1} for all m ≥ 1 and n ≥ 0.

In the important special case of the 4-cycle digraph algebra

A(G) = [ \begin{matrix}ℂ & 0 & ℂ & ℂ \\ 0 & ℂ & ℂ & ℂ \\ 0 & 0 & ℂ & 0 \\ 0 & 0 & 0 & ℂ \end{matrix} ]

the lexicographic product A = A(G) ⋆ T O_{m} may be viewed as

A(G) ⋆ T O_{m} = [ \begin{matrix}T O_{m} & 0 & O_{m} & O_{m} \\ 0 & T O_{m} & O_{m} & O_{m} \\ 0 & 0 & T O_{m} & 0 \\ 0 & 0 & 0 & T O_{m} \end{matrix} ].

Whilst the K_{0} group here is simply K_{0}(A ∩ A^{∗}), which is C(X, ℤ) with X a Cantor space, the homology group H_{1}(A) is ℤ_{m−1}.

There are two stages in the proof of Theorem 2. First we require a key analytical result which is of independent interest, Lemma 5.4, on the structure of normalising partial isometries in O_{m} and its stable algebra. The second stage — Steps 1, 2 and 3 of Section 6 — can be viewed as the identification of H_{n}(A) through simplicial homotopy reductions of general normalising partial isometry complexes to elementary partial isometry complexes. In fact this direct approach is applicable in other contexts for which the normalising partial isometries are well-understood. This is the case for example for crossed products and semicrossed products of C(X) with X a Cantor space [19].

In contrast to direct identifications one can also exploit the established machinery of simplicial homology transferred to partial isometry homology. This theme is taken up in the final section. Here a different but closely related form of partial isometry homology, CH_{∗}, is defined in terms of the homology of a chain complex. This homology is more sensitive to torsion as we see in Theorem 7.2, the analogue of Theorem 2. Algebraic topology techniques are easily imported for CH_{∗} and we illustrate this briefly here with the Mayer Vietoris sequence for regular pairs.

1. The Partial Isometry Homology H_{n}(A; C)

First we define the stable partial isometry homology groups given in Power [17]. It should be borne in mind that the definition given below provides a natural way of recovering the simplicial homology of the digraph of a digraph algebra in purely algebraic terms. Moreover, by doing so in terms of partial isometries (rather than partial orderings of minimal projections for example) we obtain homology groups which give the “correct” limit groups in the case of the fundamental algebras A(G) ⊗ B, with A(G) a digraph algebra and B an approximately finite C*-algebra. Recall that a digraph algebra A(G) is a unital subalgebra of a complex matrix algebra which is spanned by some of the matrix units of a self-adjoint matrix unit system for the matrix algebra. The digraph G for such an algebra has edges (j, i) associated with the matrix units e_{i,j} that belong to the algebra.

Let B be a unital involutive algebra and let C ⊆ A be unital subalgebras, with C self-adjoint. Usually we take C to be an abelian subalgebra or a maximal abelian subalgebra of B. The stable algebra of A is the algebra M_{∞}(A) of finitely nonzero infinite matrices over A, that is, the natural union of matrix algebras over A. It is immediately clear from the definition below that the homology groups are stable in the sense that

H_{n}(A; C) = H_{n}(M_{∞}(A); M_{∞}(C)).

It is appropriate to consider stable homology since this leads to the natural connections with K-theory. Moreover the stable formulation allows the freedom necessary for the algebraic homotopy aguments in the proof of Theorem 2.

A partial isometry u in M_{∞}(A) is an element for which u^{∗}u is a (self-adjoint) projection and is said to be M_{∞}(C)-normalising, or simply normalising if the context is clear, if ucu^{∗} and u^{∗}cu belong to M_{∞}(C) whenever c does. In particular, if p is a projection in M_{∞}(C) and pu is a partial isometry, then pu is also normalising.

Let D ⊆ M_{∞}(B) be star isomorphic to the matrix algebra M_{r}(ℂ), with full matrix unit system {u_{ij} : 1 ≤ i, j ≤ r} consisting of normalising partial isometries. In particular each projection u_{ii} belongs to M_{∞}(A) and it follows that the subalgebra

A_{D} = D ∩ M_{∞}(A)

is spanned by the matrix units u_{ij} in M_{∞}(A). The associated pairs (j, i) form the edges of a (transitive relation) digraph. In particular A_{D} is (completely) isomorphic to a digraph algebra, and we refer to A_{D} as a digraph algebra of A. More generally define the digraph algebras A_{D} when D is star isomorphic to a direct sum of full matrix algebras. Also it is convenient to refer to unital subdigraph algebras of A_{D} (those unital subalgebras given by a subsemigroup of the matrix units) also as digraph algebras of A. Note that the partial matrix unit systems of these subalgebras must not only satisfy the obvious multiplicative relations but must also generate a complete matrix unit system for a finite dimensional C*-algebra.

It is through such algebras, or partial isometry complexes, together with associated regular inclusions and direct sums, that we define the partial isometry homology H_{n}(A; C). At least, this is appropriate in the case of unital and sigma unital algebras.

Two digraph subalgebras A_{1} = A_{D_{1}} and A_{2} = A_{D_{2}} are said to be equivalent if there is a unitary element v in M_{∞}(A ∩ A^{∗}) (more precisely, in some sufficiently large matrix algebra over (A ∩ A^{∗}), which is normalising, such that vA_{1}v^{∗} = A_{2}.

To the resulting equivalence class [A_{D}] of A_{D} there is a well-defined digraph G and simplicial complex Δ(G). This complex is obtained from the undirected graph \overline{G} of G by including 0-simplexes ⟨v_{i}⟩ for the vertices v_{i} of \overline{G} and t-simplexes for each complete subgraph of \overline{G} with t + 1 vertices.

Define the simplicial homology group H_{n}([A_{D}]) to be the usual simplicial homology group of Δ(G) with coefficients in ℤ. The group H_{n}(A; C) is defined as the quotient

(∑_{[A_{D}]} ⊕H_{n}(([A_{D}]))/J_{n}

where J_{n} is a subgroup of the (restricted) direct sum associated with inclusions and splittings of the subalgebras A_{D}. Explicitly, J_{n} is generated by elements

−g ⊕ θ(g)

and

−h ⊕ θ_{1}(h) ⊕ θ_{2}(h),

where g ∈ H_{n}([A_{D_{1}}]) and θ : H_{n}([A_{D_{1}}]) → H_{n}([A_{D_{2}}]) is induced by an inclusion of matrix unit systems, and where h ∈ H_{n}([A_{D}]) and

θ_{1} + θ_{2} : H_{n}([A_{D}]) → H_{n}([A_{D_{1}}]) ⊕ H_{n}([A_{D_{2}}])

is the mapping induced by a splitting u_{ij} = u_{ij}^{1} + u_{ij}^{2}. By a splitting, we mean that there is a projection p in M_{∞}(C), dominated by the initial projection of u_{1,1} such that u_{ij}^{1} = u_{i1}^{1}pu_{1j}^{1} for all appropriate i, j. In view of the assumed normalising property of the u_{ij} the new systems {u_{ij}^{1}} and {u_{ij}^{2}} obtained in this way are also normalising.

One can also express H_{n}(A; C) as a universal object amongst groups G with families of embeddings H_{n}(Δ[A_{D}]) → G respecting the splitting and inclusion induced maps.

In the case when A is a triangular algebra in the sense that A ∩ A^{∗} is abelian it is convenient to define H_{n}(A) to be the group H_{n}(A; A ∩ A^{∗}). This definition is particularly appropriate for operator algebras A in which C is a regular subalgebra in the sense that the normalising partial isometries in A generate A.

The following basic result is in Section 2 of [17]. See also the parallel Theorem 7.1 below. The partial isometry homology H_{n}(A(G)) is defined to be H_{n}(A(G); C) where C is any maximal abelian self-adjoint subalgebra of A(G). This is well-defined since all such diagonal algebras C are unitarily equivalent. A convenient feature of triangular algebras is that we can employ the definition of the last paragraph and avoid problems of uniqueness of diagonals. In this regard there are already complications in the case of diagonals of approximately finite algebras. (See [4].)

Theorem 3. The partial isometry homology of a digraph algebra A(G) is naturally isomorphic to the simplicial homology of the simplicial complex Δ(G) of the digraph G.

The homology scale

Let A(G) ⊆ M_{r}(ℂ) be a digraph algebra with diagonal subalgebra D_{r} and with homology groups H_{n}(A(G); D_{r}) where r = |G|. Identify these groups with the simplicial homology groups H_{n}(Δ(G)). We may define the scale of H_{n}(A(G)) as a subset determined by n-cycles which, in the following sense, lie in the complex Δ(G) for A(G). For simplicity we take n = 1.

A 1-cycle is said to belong to the scale Σ_{1}(A(G)) of H_{1}(A(G)) if it has the form

∑_{i=1}^{m_{1}} δ_{1,i} + ⋯ + ∑_{i=1}^{m_{k}} δ_{k,i}

where each pair δ_{k,i}, δ_{j,l} is disjoint if k ≠ j, and each partial sum

∑_{i=1}^{m_{1}} δ_{k,i}

is a 1-cycle for which the 1-simplexes δ_{k,i} are essentially disjoint.

For a general pair (A, C) define the scale of H_{n}(A; C) to be the images of the scales for all the inclusions

H_{n}(A) → H_{n}(A; C)

arising from digraph subalgebras A contained in A (rather than the stable algebra of A).

As an illustration consider the digraph algebra

A = [ \begin{matrix}M_{2,2} & M_{2,3} & 0 & M_{2,2} \\ 0 & M_{3,3} & 0 & 0 \\ 0 & M_{3,2} & M_{2,2} & M_{2,2} \\ 0 & 0 & 0 & M_{2,2} \end{matrix} ]

where M_{2,3} is the space of 2 × 3 complex matrices. In this case H_{1}(A) = ℤ and the scale is the subset {−2, −1, 0, 1, 2}. (The reduced digraph of A is a 4-cycle and so, in the terminology of [5], A is a 4-cycle algebra.)

For the algebras of Theorem 2 the scales of the higher homology groups are improper in that they coincide with the groups themselves.

For another elementary example consider the tensor product algebra A ⊗ C(X) with A the digraph algebra above and X a Cantor space. Then

H_{1}(A ⊗ C(X)) = H_{1}(A) ⊗ K_{0}(C(X)) = ℤ ⊗ C(X, ℤ) = C(X, ℤ)

(see [17]) and the scale can be identified with the subset C(X, {−2, −1, 0, 1, 2}).

The scale is a symmetric subset of the abelian group H_{n}(A; C). That is, if g is an element then so too is −g. In many approximately finite contexts it is a generating subset. Also, as with the K_{0} scale, the homology scales provide isomorphism invariants. In particular we remark that the scaled first homology group plays a key role in the regular classification of limits of cycle algebras. (See [5], [6].)

2. Vanishing Homology

Cancellation

Let (A_{1}, C_{1}), (A_{2}, C_{2}) be pairs as in Section 1. A regular homomorphism between such pairs is an algebra homomorphism φ : A_{1} → A_{2} such that φ(C_{1}) ⊆ C_{2} and φ maps the normaliser of C_{1} in A_{1} into the normaliser of C_{2} in A_{2}. A star-extendible homomorphism φ : A_{1} → A_{2} is one which is a restriction of a star homomorphism between the generated star algebras. In particular, such a map maps a partial matrix unit system in M_{∞}(A_{1}) to one in M_{∞}(A_{2}). Accordingly it is the starextendible regular maps that induce natural group homomorphisms

H_{n}φ : H_{n}(A_{1}; C_{1}) → H_{n}(A_{2}; C_{2}).

Suppose that α : A → A is a regular star-extendible automorphism with respect to a regular masa C of A. Let us write, simply, id ⊕ α for the maps M_{2^{k}} ⊗ A → M_{2^{k+1}} ⊗ A given by a → a ⊕ (id_{M_{2^{k}}} ⊗ α)(a) for k = 0, 1, 2, . . .. These maps are regular homomorphisms, with respect to the diagonal masas D_{2^{k}} ⊗ C, and we may form the algebraic direct limit

(Ã; C̃) = lim_{→}((M_{2^{k}} ⊗ A; D_{2^{k}} ⊗ C), id ⊕ α).

It can be verified that

H_{n}(Ã; C̃) = lim_{→}(H_{n}(M_{2^{k}} ⊗ A; D_{2^{k}} ⊗ C), H_{n}(id ⊕ α))
= lim_{→}(H_{n}(A; C), H_{n}(id ⊕ α)).

In particular if H_{n}α : H_{n}(A; C) → H_{n}(A; C) is the map g → −g then (id ⊕ α)_{∗} is the zero map and H_{n}(Ã; C̃) = 0. These limit algebras illustrate how homology groups may vanish through cancellation.

For a concrete example we may take A ⊆ M_{4}(ℂ) to be the fundamental 4-cycle matrix algebra (spanned by the diagonal matrix units e_{ii} and e_{13}, e_{14}, e_{24}, e_{23}) and let α be a reflection automorphism. If one considers the 4-cycle {v_{1}, v_{2}, v_{3}, v_{4}} = {e_{13}, e_{14}, e_{24}, e_{23}} in the first algebra ℂ ⊗ A then no image in a subsequent superalgebra M_{2^{k}} ⊗ A is triangulable (in the sense mentioned in the introduction and below). Nevertheless the 4-cycle provides a generator for H_{1}(ℂ⊗A; ℂ⊗A∩A^{∗}) = ℤ, and its image in M_{2} ⊗ A can be split, in our sense, as a direct sum of two 4-cycles of opposite orientation.

We remark that one can consider, more generally, direct limits lim_{→}(M_{n_{k}} ⊗ A; D_{n_{k}} ⊗ C) where each embedding is a direct sum of automorphisms of M_{n_{k}} ⊗ A coming from automorphisms of a fixed digraph algebra A. In this way one obtains a very wide family of subalgebras of C*-algebras with computable normalising partial isometry homology.

In the examples above cancellation is built in at the outset in the presentation of the algebras. The following simple example illustrates how one might have to be more creative in seeking homology cancellation or reduction.

Let A ⊆ M_{2} ⊗ L(H) be spanned by e_{1,1} ⊗ (ℂ + K), e_{2,2} ⊗ (ℂ + K), e_{1,2} ⊗ L(H), that is

A = [ \begin{matrix}ℂ + K & L(H) \\ 0 & ℂ + K \end{matrix} ]

where K ⊆ L(H) is the ideal of compact operators and L(H) is the algebra of all operators on H, a separable Hilbert space. Consider the 4-cycle {u_{1}, …, u_{4}} where u_{i} = e_{1,2} ⊗ w_{i}, 1 ≤ i ≤ 4, and where w_{1}^{∗}w_{1} = w_{4}^{∗}w_{4} = I, w_{1}w_{1}^{∗} = w_{2}w_{2}^{∗} has defect 1, w_{2}^{∗}w_{2} = w_{3}^{∗}w_{3} has defect 2, and w_{3}w_{3}^{∗} = w_{4}w_{4}^{∗} has defect 3. We can indicate this 4-cycle data with the array

[ \begin{matrix}1 & 0 \\ 3 & 2 \end{matrix} ]

and the diagram

[figure: 2; 3; 1; 0]

The 4-cycle admits no trivial triangulation, in the sense that none of the operators u_{3}^{∗}u_{2}, u_{2}^{∗}u_{3}, u_{3}^{∗}u_{4}, u_{4}^{∗}u_{3} belong to A, as partial isometries in ℂ + K have zero Fredholm index. For similar reasons, if A_{1} is a digraph algebra associated with {u_{1}, …, u_{4}} there is no multiplicity one inclusion i : A_{1} → A_{D} for which i_{∗} = 0.

However if A_{2} is associated with a 4-cycle {v_{1}, …, v_{4}} with defect array

[ \begin{matrix}1 & 2 \\ 1 & 0 \end{matrix} ]

then the direct sum {u_{1} ⊕ v_{1}, …, u_{4} ⊕ v_{4}} is associated with the defect array

[ \begin{matrix}2 & 2 \\ 4 & 2 \end{matrix} ] .

Both these arrays have a constant column and so A_{2} and the algebra A_{3}, for the direct sum cycle, have trivially triangulable 4-cycle graphs. In particular the inclusions H_{n}(A_{2}) → H_{n}(A; C), and H_{n}(A_{3}) → H_{n}(A; C) are the zero maps for n > 1. A simple elaboration of this argument leads to H_{n}(A) = H_{n}(A; C) = 0.

Triangulation

We have observed how H_{n}(A; C) may vanish by virtue of cancellation giving rise to induced zero maps i_{∗} : H_{n}(A_{D_{1}}) → H_{n}(A_{D_{2}}). Here the inclusion i necessarily is of multiplicity greater than one. It can also happen that H_{n}(A; C) vanishes for more geometric reasons in the following sense. Suppose that for every digraph algebra A_{D_{1}} for the pair (A, C) there is a containing digraph algebra A_{D_{2}}, with multiplicity one star-extendible regular inclusion i : A_{D_{1}} → A_{D_{2}}, such that H_{n}i = 0. For the case n = 1 we can view the digraph G(A_{D_{2}}) as providing triangulations of the cycles in G(A_{D_{1}}). In this case H_{n}(A; C) = 0.

We give two illustrations.

Let (A, C) = (T_{m}, D_{m}) where T_{m} ⊆ M_{m}(ℂ) is the upper triangular subalgebra. Let A_{D} ⊆ M_{N} ⊗ T_{m} be an M_{N} ⊗ D_{m} normalising digraph algebra for A, with connected digraph G and partial matrix unit system {u_{ij} : (i, j) ∈ E(G)}. The partial matrix unit system can be decomposed as a direct sum u_{ij} = u_{ij}^{(1)} +⋯+u_{ij}^{(r)} (in the appropriate splitting sense) where each u_{ij}^{(k)} has rank one. We show that for each associated digraph algebra A_{D^{(k)}} there is a regular multiplicity one inclusion inducing a zero map on H_{n} for each n ≥ 1. We may as well assume already that rank u_{ij} = 1 for all i, j. Note that since each u_{ij} is normalising it follows that

(I_{N} ⊗ e_{kk})u_{ij}(I_{N} ⊗ e_{ll}) = u_{ij}

for precisely one pair k, l with 1 ≤ k, l ≤ m. It now follows that {u_{ij} : (i, j) ∈ E(G)} is unitarily equivalent to a subset of the standard matrix unit system for M_{N} ⊗ T_{m}. In particular the inclusion i : A_{D} → M_{N} ⊗ T_{m} is of the admissible kind specified in the definition of H_{n}(A; C). Plainly H_{n}i = 0 for n ≥ 1, since H_{n}([M_{n} ⊗ T_{m}]) = 0, and so H_{n}(T_{m}; D_{m}) = 0 for n ≥ 1.

For the second illustration, consider the disc algebra A(𝔻), viewed in the usual way, as a function algebra on the unit circle S^{1}. Let {u_{1}, …, u_{2n}} be a 2n-cycle of partial isometries in M_{2n} ⊗ A(𝔻) of the form

u_{k} = v_{k} ⊗ w_{k}

where {v_{1}, …, v_{2n}} is the standard 2n-cycle in M_{2n} given by v_{1} = e_{12}, v_{2} = e_{32}, v_{3} = e_{34}, …, v_{2n−1} = e_{2n−1,2n}, v_{2n} = e_{1,2n}, and where w_{1}, … w_{2n} are inner functions in A(𝔻) with w_{1}w_{3} … w_{2n−1} = w_{2}w_{4} … w_{2n}. The 2n-cycle {u_{1}, …, u_{2n}} together with the diagonal projections e_{ii} ⊗ 1 spans a subalgebra, A(w_{1}, …, w_{2n}) say, which is completely isometrically isomorphic to the digraph algebra in M_{2n}(ℂ) associated with the standard 2n-cycle. Furthermore, A(w_{1}, …, w_{2n}) is a digraph algebra of A(𝔻) according to our definition.

The algebra A(w_{1}, …, w_{2n}), can be triangulated as follows.

Consider the natural inclusion

A(w_{1}, … w_{2n}) → A(w_{1}, …, w_{2n}) ⊕ ℂe_{2n+1,2n+1}.

Let w = w_{1}w_{3} … w_{2n−1} and define

x_{1} = e_{1,2n+1} ⊗ w,

x_{2} = e_{2,2n+1} ⊗ \frac{w}{w_{1}} ,

x_{3} = e_{3,2n+1} ⊗ \frac{ww_{2}}{w_{1}} ,

⋮

x_{2n} = e_{2n,2n+1} ⊗ \frac{ww_{2}}{w_{1}} .\frac{w_{4}}{w_{3}} … \frac{w_{2n−2}}{w_{2n−1}}

Note that in view of the equality w_{1}w_{3} … w_{2n−1} = w_{2}w_{4} … w_{2n} it follows that A(w_{1}, … w_{2n}) together with x_{1}, …, x_{2n} and e_{2n+1,2n+1} span a digraph algebra, A^{+} say. The digraph of A^{+} is the cone over the digraph of A and so the inclusion A(w_{1}, …, w_{2n}) → A^{+} induces the zero map on H_{1}.

3. The Proof of Theorem 1

We now turn to the proof that H_{n}(A(𝔻)) = 0 for n ≥ 1. The essential idea is the triangulation argument above, although this time we must consider matrix functions.

Let D ⊆ M_{N} ⊗C(S^{1}) be a finite-dimensional C*-algebra with matrix unit system {u_{ij}} normalising M_{N} ⊗ ℂ. Without loss of generality assume that D is isomorphic to M_{r} and that A_{D} is spanned by {u_{ij} : (i, j) ∈ E(G)} where G is a connected digraph with r vertices. We claim that there is a multiplicity one inclusion D → D^{+} where D^{+} has matrix unit system {u_{ij} : 1 ≤ i, j ≤ r+1}, where D^{+} ⊆ M_{2N} ⊗C(S^{1}) and where A_{D^{+}} = {u_{ij} : (i, j) ∈ E(H)} where H is a digraph containing G and all edges from the new vertex r + 1 to vertices of G (labelled 1, …, r). That is H contains the cone over G.

This step will complete the proof since the simplicial complex of the cone has trivial higher order homology.

Define v_{ij} ∈ M_{2} ⊗ M_{N} ⊗ C(S^{1}) by v_{ij} = e_{11} ⊗ u_{ij}, for 1 ≤ i, j ≤ r. Let v_{r+1,r+1} = e_{22}⊗u_{r,r}, v_{r,r+1} = e_{12}⊗u_{r,r}, and consider the full matrix unit system for M_{r+1} which is generated by these matrix units and denoted {v_{ij} : 1 ≤ i, j ≤ r + 1}. Consider the partial isometries v_{i,r+1}, for 1 ≤ i ≤ r, which correspond to edges from vertex r + 1 to the vertices of G. Each such partial isometry is a word in the set

{e_{11} ⊗ u_{ij}, e_{11} ⊗ u_{ij}^{∗} : 1 ≤ i, j ≤ r} ∪ {v_{r,r+1}}.

Moreover we can choose words of length at most r, in which no partial isometries are repeated. Thus, each v_{i,r+1} has the form

e_{12} ⊗ u_{1} … u_{k}

where k ≤ r and for each i with 1 ≤ i ≤ r either u_{i} or u_{i}^{∗} belongs to M_{N} ⊗ A(𝔻). A partial isometry w in M_{N} ⊗ A(𝔻) has the form

w = U_{1}D_{1}U_{2}D_{2} … U_{t}D_{t}U_{t+1}

where each U_{k} is a scalar partial isometry and where each D_{k} is a diagonal matrix of functions whose entries consist of a single Blaschke factor ϕ_{k} and constant functions. (This may be deduced from the well-known corresponding assertion for inner functions in M_{N} ⊗ A(𝔻).) It follows that if Φ(z) is the inner function Φ(z) = I ⊗ (ϕ_{1} … ϕ_{t}) then

Aw^{∗}BΦ(z)

is a partial isometry in M_{N} ⊗A(𝔻) for all partial isometries A, B in M_{N} ⊗A(D) for which Aw^{∗}B is a partial isometry. Let Ψ(z) be the product of the Blaschke factors associated with all the partial isometries u_{ij} in A_{D}. Define

v′_{ij} = [ \begin{matrix}I & 0 \\ 0 & Ψ(z) \end{matrix} ]^{∗} v_{ij} [ \begin{matrix}I & 0 \\ 0 & Ψ(z) \end{matrix} ]

for 1 ≤ i, j ≤ r + 1. This is a matrix unit system for D^{+} = M_{r+1} with the desired properties.

4. K_{0}-regular Inclusion and Homotopy

We begin by considering H_{0}(A; C) and the following idea (see also [17]) will be useful.

The inclusion C → A is said to be K_{0}-regular if

(i) the induced map K_{0}C → K_{0}(C^{∗}A) is a surjection, and (ii) whenever p, q are projections in M_{N} ⊗C which are unitarily equivalent in M_{N} ⊗

C^{∗}A, then there is a digraph subalgebra for A with connected graph which contains projections p′ and q′ as minimal projections, where the K_{0}(C^{∗}(A)) classes [p], [p′], [q], [q′] all agree.

Note, for example, that D_{m} → T_{m}, the diagonal algebra inclusion, is K_{0}-regular. Also, ℂ → A(𝔻) is K_{0}-regular. We shall see that the masas for the triangular algebras of Theorem 2 have K_{0}-regular inclusions.

Proposition 4.1. If C → A is a K_{0}-regular inclusion then H_{0}(A; C) = K_{0}(C^{∗}(A)).

Proof. By the hypotheses

∑_{[A_{D}]} ⊕ H_{0}([A_{D}])

contains the subgroup

∑_{[p]∈(K_{0}C^{∗}(A))_{+}} ⊕ ℤ

arising from the degenerate digraph algebras ℂp associated with projections p in M_{N} ⊗ C for some N. Moreover, in view of the inclusion and splitting relations used in the definition of J_{0} we see that

H_{0}(A; C) = (∑_{[A_{D}]} ⊕H_{0}([A_{D}]))/J_{0}
= (  ∑_{[p]∈(K_{0}C^{∗}(A))_{+}} ⊕ℤ)/J_{0}
= K_{0}C^{∗}(A).

The second equality here is a consequence of the inclusion relations in J_{0}. The last equality holds since the splitting relations correspond to the semigroup relations for K_{0}(C^{∗}(A))_{+}, and for any semigroup S the quotient (∑_{s∈S} ⊕ℤ)/R, arising from the semigroup relations R, is the Grothendieck group of S. □

The assertions concerning H_{0}A for the triangular algebras of Theorem 2 follow from this proposition and the K_{0}-regularity of the diagonal inclusions discussed below.

We now consider a simple retraction procedure which will be useful for identifying the partial isometry homology of lexicographic products.

Let A_{1}, A_{2} be digraph algebras for the pair (A, C) for which there is a containing digraph algebra A with the following properties:

(i) The digraph G(A) has vertices {v, w} ∪ {v_{1}, …, v_{r}} and A_{1} (respectively A_{2}) is the subalgebra of A determined by the full subgraph of G(A) for the vertices {v} ∪ {v_{1}, …, v_{r}} (respectively {w} ∪ {v_{1}, …, v_{r}}).

(ii) (v, v_{i}) ∈ E(G(A)) if and only if (w, v_{i}) ∈ E(G(A)) and (v_{i}, v) ∈ E(G(A)) if and only if (v_{i}, w) ∈ E(G(A)), and at least one of the edges (v, w) or (w, v) belongs to E(G(A)).

In this case we say that there is an elementary homotopy between A_{1} and A_{2} (and between A_{2} and A_{1}). Since Δ(G(A_{1})) and Δ(G(A_{2})) are simplicial retractions of Δ(G(A)) it follows that H_{n}(Δ(G(A_{1}))) = H_{n}(Δ(G(A_{2}))) = H_{n}(Δ(G(A))) for all n ≥ 0. Moreover, since the inclusions A_{1} → A, A_{2} → A induce simplicial homology isomorphisms it follows, in the notation of Section 1, that

(H_{n}([A_{1}]) ⊕ H_{n}([A_{2}]) ⊕ H_{n}([A]))/J_{n} = H_{n}([A_{i}])/J_{n}

for i = 1 or 2. In this way we will be able to obtain reductions through inclusion relations corresponding to homotopy. More generally this reduction will also hold if A_{1} and A_{2} are homotopic, by which we mean that there is a finite chain of elementary homotopies connecting A_{1} to A_{2}. Plainly there is a more general notion of homotopy, allowing for retractions, but the present usage suffices for the proof below.

5. The Cuntz Algebras and T O_{m}

The Cuntz algebra O_{m} is the universal C*-algebra generated by m isometries S_{1}, …, S_{m} with S_{1}S_{1}^{∗} +⋯+S_{m}S_{m}^{∗} = 1. In fact, any C*-algebra generated by a set of isometries satisfying this relation is isomorphic to O_{m}. It will be convenient to consider the specific representation on L^{2}[0, 1] generated by the natural isometries S_{1}, …, S_{m} where S_{i}S_{i}^{∗} is the orthogonal projection onto L^{2}[(i − 1)/m, i/m], for 1 ≤ i ≤ m. Specifically, for f ∈ L^{2}[0, 1],

(S_{i}f)(x) = \{ \begin{matrix}\sqrt{m}f(mx − (i − 1)) & , & for i − 1 ≤ mx ≤ i \\ 0 & , & otherwise. \end{matrix}

The (unclosed) star algebra generated by S_{1}, …, S_{m} is also uniquely determined by any realisation, and we denote this algebra by O_{m}^{0}.

Recall the following basic facts from Cuntz [1]. Let W_{k} denote the words of length k in the letters 1, 2, …, n. If μ = μ_{1} … μ_{k} ∈ W_{k} then write S_{μ} = S_{μ_{1}} … S_{μ_{k}}, l(μ) = k and let

d(μ) = \frac{μ_{1} − 1}{m} + ⋯ + \frac{μ_{k} − 1}{m^{k}}.

Every word in the operators S_{1}, …, S_{m} and their adjoints can be reduced to the form S_{μ}S_{ω}^{∗} for uniquely determined words μ, ω. For words of the same length, we have

Lemma 5.1. Let μ, ω ∈ W_{k}. Then S_{μ}S_{ω}^{∗} is the natural partial isometry with initial space L^{2}[d(ω), d(ω) + m^{−k}] and final space L^{2}[d(μ), d(μ) + m^{−k}].

Thus {S_{μ}S_{ω}^{∗} : μ, ω ∈ W_{k}} is a set of matrix units for a copy of the matrix algebra M_{m^{k}}. The union of these algebras will be denoted F^{0}, and the closed union, which is a UHF C*-algebra of type m^{∞}, will be denoted F. Also write C for the masa in F generated by {S_{μ}S_{μ}^{∗} : μ ∈ W_{k}}.

We now define the triangular algebra T O_{m}. Let N be the nest of projections in F^{0} corresponding to the subspaces L^{2}[0, i/m^{k}], for 1 ≤ i ≤ m^{k}, k = 1, 2, . . .. Define

T F = {a ∈ F : (1 − p)ap = 0, ∀ p ∈ N },
T O_{m} = {a ∈ O_{m} : (1 − p)ap = 0, ∀ p ∈ N },

and define T F^{0} and T O_{m}^{0} similarly. Then T F is a copy of the refinement algebra lim_{→}(T_{m^{k}}, ρ) determined by the so-called refinement embeddings. (See [13].) For this reason we refer to T O_{m} as the refinement subalgebra of O_{m}. We can also think of the algebras T F and T O_{m} as the Volterra nest subalgebras of the realisations of F and O_{m}. Alternatively, the algebras T O_{m} and T O_{m}^{0} can be described in purely intrinsic terms, as follows. This description will not be needed below, and we refer the reader to [11] for a proof.

Proposition 5.2. If l(μ) ≤ l(ω) then S_{μ}S_{ω}^{∗} ∈ T O_{m} if and only if d(μ) ≤ d(ω). If l(μ) < l(ω) then S_{μ}S_{ω}^{∗} ∈ T O_{m} if and only if

d(μ) + m^{−l(μ)} ≤ d(ω) + m^{−l(ω)}

Furthermore, T O_{m} is generated as an operator algebra by the operators S_{μ}S_{ω}^{∗} in T O_{m}.

In particular, it follows that T O_{m} is generated by a subsemigroup of an inverse semigroup of normalising partial isometries. The same is true for the algebras A(G) ⋆ T O_{m}.

The next two lemmas provide the purely C*-algebraic technical results that we need to understand normalising partial isometries and the normalising finitedimensional C*-algebras associated with O_{m}. In brief they allow for a reduction to the case of standard matrix unit systems with matrix units that are orthogonal sums of the standard partial isometries S_{μ}S_{ω}^{∗}.

Lemma 5.3 (Cuntz [1].). Each operator a in the star algebra O_{m}^{0} generated by S_{1}, …, S_{m} has a unique representation

a = ∑_{i=1}^{N}(S_{1}^{∗})^{i}a_{−i} + a_{0} + ∑_{i=1}^{N} a_{i}S_{1}^{i}

where a_{i} ∈ F for each i. Moreover the linear maps E_{i} given by E_{i}(a) = a_{i} extend to continuous contractive linear maps from O_{m} to F.

Lemma 5.4. (i) If v is a C-normalising partial isometry in O_{m} then there is a partial isometry w in O_{m}^{0} such that v = cw where c is a partial isometry in C and w has the form

w = ∑_{i=1}^{N}(S_{1}^{∗})^{i}v_{−i} + v_{0} + ∑_{i=1}^{N} v_{i}S_{1}^{i}

where the sum is an orthogonal sum of partial isometries with each v_{j} equal to an orthogonal finite sum of partial isometries in {S_{μ}S_{ω}^{∗} : μ, ω ∈ W_{j}}.

(ii) If v is a M_{n} ⊗ C-normalising partial isometry in M_{n} ⊗ O_{m} then there is a partial isometry w′ such that v = cw′ where c is a partial isometry in M_{n} ⊗ C and where w′ = (w_{ij})_{i,j=1}^{n} with each w_{ij} a partial isometry as in (i).

Let us say that two standard partial isometries v, w in O_{m}, by which we mean those of the form S_{μ}S_{ω}^{∗}, are disjoint if for all projections p, q in C the equality pvq = pwq implies pvq = pwq = 0. This is equivalent to the graphs of the partial homeomorphisms inducing v and w (as composition operators) having at most one point in common.

Proof of Lemma 5.4. Note that if the index l(μ) − l(ω) for S_{μ}S_{ω}^{∗} differs from the index for S_{ρ}S_{δ}^{∗} then these partial isometries are disjoint. Let v ∈ N_{C}(O_{m}). It follows from Lemma 5.3 that there is a finite complex combination w = α_{1}u_{1} + ⋯ + α_{n}u_{n} of disjoint standard partial isometries such that ∥v − w∥ < \frac{1}{4}. Here the coefficients α_{i} are nonzero complex numbers.

We claim that there is a subset of {u_{i}}, which we may relabel as u_{1}, …, u_{l}, consisting of partial isometries with orthogonal initial projections and orthogonal final projections such that v = c(u_{1} + ⋯ + u_{l}) for some partial isometry c in C.

By disjointness there are projections p, q in C such that pvq ≠ 0 and pwq = α_{i}pu_{i}q for some i. Relabel to arrange i = 1. Thus ∥pvq − α_{1}pu_{1}q∥ < \frac{1}{4}. Also |1 − |α_{1}|| < \frac{1}{4} and so ∥v_{1} − t∥ < \frac{1}{2} where v_{1} = pvq and t = pu_{1}q. In particular tt^{∗} is a projection in C and ∥tt^{∗} − v_{1}t^{∗}∥ < \frac{1}{2}. Thus ∥P v_{1}t^{∗}P^{⊥}∥ < \frac{1}{2} for all projections P in C. Since v_{1}t^{∗} is normalising it follows that P v_{1}t^{∗}P^{⊥} = 0 for all such P. Since C is a masa in O_{m} it follows that v_{1}t^{∗} is a partial isometry, d_{1} say, in C. Thus v_{1} = d_{1}t with t a standard partial isometry and d_{1} a partial isometry in C.

The partial isometry u_{1} can be expressed as a strong operator topology sum ∑_{i=1}^{∞} p_{i}u_{1}q_{i} of orthogonal partial isometries of the form above. In this way we deduce that v(u_{1}u_{1}^{∗}) = cu_{1} where c = ∑_{i=1}^{∞} d_{i}. Furthermore, since v = cu_{1} + (v − cu_{1}) is necessarily an orthogonal sum of two partial isometries it follows that c = E_{0}(vu_{1}^{∗}) and hence that c belongs to C.

Repeating the argument above obtain an orthogonal decomposition

v = v′ + v′′

where v′ = c_{1}u_{1} + ⋯ + c_{l}u_{l} with c_{i} ∈ C for each l, where, after relabelling, |1 − |α_{i}∥ < \frac{1}{4} precisely for 1 ≤ i ≤ l. But now consider v′′ = v − v′. This is a normalising partial isometry. Suppose that v′′ ≠ 0. Then (by disjointness again) there are projections p, q in C such that pvq = pv′′q ≠ 0 and pwq = α_{j}pu_{j}q. Of necessity j > l and |1−|α_{j}|| ≥ \frac{1}{4}. Since ∥w−v∥ < \frac{1}{4} we obtain ∥α_{j}pu_{j}q−pv′′q∥ < \frac{1}{4} which, given the inequality for α_{j}, leads to the contradiction ∥pv′′q∥ ≠ 1

The proof of (ii) is similar. □

Although in this paper we focus on T O_{m}, we note that O_{m} has many other natural maximal triangular subalgebras.

Let A ⊆ F be a maximal triangular subalgebra which contains the masa C. Let A ⊆ O_{m} be the set of operators a for which E_{0}(a) ∈ A and E_{i}(a) = 0 for i < 0. Both A and its superalgebra A + F have, roughly speaking, the nature of an analytic subalgebra or semicrossed product in the sense of Muhly and Solel [9] and Peters [10], for example.

In general one expects a maximal triangular subalgebra to have trivial partial isometry homology groups for n > 1 and this is so for T F and T O_{m} by simple triangulation argument in the spirit of the next section. The analytic algebras A above (“bianalytic” is a more accurate designation) also have trivial higher homology. Nevertheless we now indicate a maximal triangular subalgebra A of M_{4}⊗C(X) for which H_{1}(A) = ℤ, and this can be used in the construction of more elaborate examples, with trivial centre for example, also with nonzero H_{1}.

Let X be a Cantor space and let U, V be open subsets with dense union and with X/(U ∪ V) = {x} where x is a point of closure of U and of V. Write C(U) and C(V) for the subalgebras of C(X) supported on U and V and define

A = [ \begin{matrix}C(X) & C(V) & C(X) & C(X) \\ C(U) & C(X) & C(X) & C(X) \\ 0 & 0 & C(X) & C(V) \\ 0 & 0 & C(U) & C(X) \end{matrix} ] .

This is a maximal triangular algebra. Over the point x in the maximal ideal space of the centre of A, the local algebra for x is isomorphic to the 4-cycle algebra A(D_{4}). This in turn leads to the fact that H_{1}(A) = ℤ.

6. The Proof of Theorem 2

Let (A, C) = (A(G) ⋆ T O_{m}, ℂ^{|G|} ⊗ C). Clearly the K_{0}(O_{m}) classes [1] and [S_{i}S_{i}^{∗}] coincide for i = 1, …, m and so m[1] = [S_{1}S_{1}^{∗}] + ⋯ + [S_{m}S_{m}^{∗}] = [1]. Thus (m−1)[1] = 0. Moreover Cuntz [2] has shown that for m > 1 K_{0}(O_{m}) = ℤ/(m−1), with [1] as generator. In particular the inclusion C → O_{m} induces a K_{0} group surjection. In fact the inclusion C → A is K_{0}-regular as we now show.

We may assume that G is connected. Let us say that a projection p in M_{N} ⊗ ℂ^{|G|} ⊗ C is A-connected to a projection p′ if there is a digraph subalgebra for A, with connected digraph, which contains p, p′ as minimal projections. In particular, if p ≠ p′ then these projections are orthogonal. Note first that each projection p in M_{N} ⊗ ℂ^{|G|} ⊗ C is A-connected to a projection p′ in M_{N} ⊗ e_{1,1} ⊗ C. Accordingly, it will be enough to show that two orthogonal projections p′, p′′ in M_{N} ⊗ e_{1,1} ⊗ C, if unitarily equivalent in M_{N} ⊗ e_{1,1} ⊗ O_{m}, are A-connnected. But M_{n} ⊗ A contains M_{N} ⊗ e_{1,1} ⊗ T O_{m}, and there exist partial isometries u, v in M_{N} ⊗ e_{1,1} ⊗ T O_{m} with uu^{∗} = vv^{∗}, u^{∗}u = p′ and v^{∗}v = p′′, and so p′ and p′′ are A-connnected.

Since the standard partial isometries S_{μ}S_{ω}^{∗} are normalising the K_{0}-regularity of the inclusion C → A follows.

Step 1. Reduction to standard digraph subalgebras.

Let D_{1} be a finite-dimensional C*-algebra in the stable algebra M_{∞}(C^{∗}(A)) with a normalising matrix unit system for M_{∞}(C). Then, for some integer N,

D_{1} ⊆ M_{N} ⊗ C^{∗}(A(G)) ⊗ O_{m} = M_{N} ⊗ C^{∗}(A),

with matrix unit system normalising M_{N} ⊗ ℂ^{|G|} ⊗ C. Using Lemma 5.4 (ii) it follows from routine C*-algebra theory that D_{1} is unitarily equivalent, by a unitary operator in M_{N} ⊗ ℂ^{|G|} ⊗ C to a finite-dimensional C*-algebra D_{2} with a matrix unit system which is decomposable, by splitting, as a direct sum of matrix unit systems each with matrix units v of the standard form f_{kl} ⊗ e_{ij} ⊗ S_{μ}S_{ω}^{∗}, where (f_{kl}) is a standard system for M_{N}, {e_{ij}} the standard system for C^{∗}(A(G)) and with k, l, i, j, μ, ω depending on v. We say A_{D_{2}} is a standard digraph algebra for A. Thus, if D is the sub-collection of digraph algebra classes [A_{D}] associated with the elementary tensor systems then we have the first reduction

H_{n}(A; C) = (∑_{D} ⊕H_{n}([A_{D}]))/I_{n}

where I_{n} is the ideal in the direct sum (of the right hand side above) generated by the splitting and inclusion relations for the standard algebras A_{D}.

In fact this reduction follows in two stages. First note, as we did above, that A_{D_{1}} is equivalent to a subalgebra A_{D_{2}} of a standardised algebra A_{D} where the subalgebra inclusion map A_{D_{2}} → A_{D} is regular. Secondly, note that, for similar reasons,

J_{n} ∩ (∑_{D} ⊕H_{n}([A_{D}]) = I_{n}

since a generator for the ideal J_{n}, arising from a splitting, or an inclusion, can be written as a sum of elements in I_{n}.

Step 2. Homotopies to equalise traces of diagonal matrix units.

Assume now that D is a standardised finite-dimensional C*-algebra matrix summand, with associated digraph subalgebra A_{D} = D ∩ (M_{N} ⊗ A) with a matrix unit system of elementary tensors. Further reduction will be obtained by non-self-adjoint homotopy in the sense described earlier.

Claim: Suppose that the digraph of A_{D} is connected so that D is a full matrix algebra. Then A_{D} is homotopic to A_{D′} where D′ is also a full matrix subalgebra but with matrix units v of the form f_{kl} ⊗ e_{ij} ⊗ S_{μ}S_{ω}^{∗} where l(μ) and l(ω) are equal to a fixed integer (which is independent of v).

Consider first the subalgebras of A_{D} of the form

A_{1} = (f_{kk} ⊗ e_{ll} ⊗ 1)A_{D}(f_{kk} ⊗ e_{ll} ⊗ 1)

for some self-adjoint matrix unit v = f_{kk} ⊗ e_{ll} ⊗ S_{λ}S_{λ}^{∗} for D. Then A_{1} is a digraph subalgebra of f_{kk} ⊗ e_{ll} ⊗ T O_{m} with matrix units of the form f_{kk} ⊗ e_{ll} ⊗ w_{ij} where w_{ij} has the form S_{μ}S_{ω}^{∗} for various μ, ω.

In particular each w_{ii} is an interval projection for the projection nest in T O_{m}. First we consider homotopies to equalise the traces of these matrix units.

Since D is a full matrix algebra we know that these diagonal projections are equivalent in O_{m}. Order the projections w_{ii} as a family {w_{1}, w_{2}, …, w_{r}} with the usual ordering of intervals of a projection nest. Also, relabel the w_{ij} for consistency so that w_{ij} = w_{i}w_{ij}w_{j}. Let u_{1} be an interval projection of small trace which is the final projection of a partial isometry z = S_{μ}S_{ω}^{∗} in T O_{m} with initial projections w_{1}. We then see that A_{1} is homotopic (and isomorphic) to the digraph algebra Â_{1} in which w_{1} is replaced by u_{1} and each w_{1j} is replaced by zw_{1j}. Strictly speaking, according to our definition, this is not quite a homotopy since u_{1} may not be orthogonal to w_{1} and in this case we cannot consider the containing digraph algebra for the pair A_{1}, Â_{1}. However we are only interested in equivalence classes and it is a simple matter to replace Â_{1} by an equivalent algebra in M_{2N} ⊗A(G)⊗O_{m} for which there does exist the necessary containing algebra.

Similarly we can, through an elementary homotopy, replace w_{2} by an interval projection u_{2} of the same trace as u_{1}, as long as the trace of u_{1} is small enough. Repeating, obtain a homotopy between A_{1} and (an isomorphic) digraph algebra A′_{1} with diagonal matrix units of the form f_{kk} ⊗ e_{ll} ⊗ u_{i} with the projections u_{i} of the same trace.

Since A is the lexicographic product A(G) ⋆ T O_{m} there is in fact no obstruction to extending these homotopies to homotopies between A_{D} and A_{D′} for some other standard matrix algebra D′. Indeed at the first stage if w is any matrix unit of the form

f_{k′,k} ⊗ e_{l′,l} ⊗ u

where l′ ≠ l, e_{l,l′} ∈ A(G) and where u = S_{μ}S_{ν}^{∗} in O_{m} (but perhaps not in T O_{m}) and has final projection w_{1}, then simply replace u by zu. Similarly, if

f_{k′,k} ⊗ e_{l′,l} ⊗ v

is a matrix unit of A_{D}, where l′ ≠ l and v has initial projection w_{1}, then we may replace v by vz^{∗}. In this way extend the first elementary homotopy and it is clear that the subsequent homotopies can be extended in the same way.

Similarly we can continue with homotopies to equalise the traces of all the diagonal matrix units of A_{D} and this establishes the claim, and Step 2.

Note that the algebra A_{D′} is necessarily a (multiplicity one) subalgebra of M_{N} ⊗ A(G)⋆T_{m^{r}} for some r (equal to the common values of l(μ)) and some matrix algebra M_{N}. Furthermore note that the algebras M_{N} ⊗ A(G) ⋆ T_{m^{r}}, r = 0, 1, 2, …, are all homotopic. In particular it follows that H_{n}(A; C) is finitely generated since it coincides with the subgroup H_{n}([A(G) ⋆ T_{m}])/I_{n}.

In fact, in view of these homotopies, this subgroup is necessarily a quotient of H_{n}(Δ(G)) ⊗ ℤ_{m−1}.

Step 3. The Isomorphism.

We have shown that given the special digraph subalgebra A_{D} with matrix unit system consisting of elementary tensors, and with connected digraph, there is a similar digraph subalgebra for A of the form

A_{D^{+}} = [ \begin{matrix}A_{D′} & ? \\ 0 & A_{D} \end{matrix} ]

where A_{D′} is a subalgebra of M_{N} ⊗ A(G) ⋆ T_{m^{r}} spanned by elementary tensors. We refer to this as a homotopy construction for A_{D}. Although the class [A_{D′}] is not determined explicitly in the construction — there is freedom in the choice of r and in the choice of small interval projections — there is however a well-defined surjective map

ν_{A_{D}} : H_{n}([A_{D}]) → H_{n}(Δ(G)) ⊗ ℤ_{m−1}

determined from the composition

H_{n}([A_{D}]) → H_{n}([A_{D′}]) → H_{n}([M_{N} ⊗ A(G) ⋆ T_{m^{r}}])
→ H_{n}(Δ(G)) → H_{n}(Δ(G)) ⊗ ℤ_{m−1}

where the first map is the isomorphism induced by the inclusion in A_{D^{+}}, the second is induced by inclusion and the third is the natural indentification isomorphism.

A splitting A_{D} → A_{D_{1}} ⊕ A_{D_{2}} of A_{D} need not yield digraph algebras A_{D_{i}} of the same type, with elementary tensor matrix unit systems (since, for example, I − S_{1}S_{1}^{∗} is not of the form S_{μ}S_{μ}^{∗} if m ≥ 3). Nevertheless it is clear that one also has a homotopy construction for digraph algebras A_{E} of this type (with connected digraph) and well-defined maps ν_{A_{E}} as before.

We now note that the family {ν_{A_{E}}} (we can restrict to the case of connected digraphs for simplicity, although this is not necessary) respects splittings and inclusions in the following sense. If A_{E_{1}} → A_{E_{2}} is an inclusion of standard digraph algebras for A then the following diagram commutes

H_{n}([A_{E_{1}}])  →^{i_{∗}}  H_{n}([A_{E_{2}}])
ν_{A_{E_{1}}} ↓  ν_{A_{E_{2}}} ↓
H_{n}(Δ(G)) ⊗ ℤ_{m−1} →^{id} H_{n}(Δ(G)) ⊗ ℤ_{m−1}

and if θ : A_{E} → A_{E_{1}} ⊕ A_{E_{2}} is a splitting, with induced map

θ_{∗} : H_{n}([A_{E}]) → H_{n}([A_{E_{1}}]) ⊕ H_{n}([A_{E_{2}}])

then the following diagram commutes

H_{n}([A_{E}])  →^{θ_{∗}} H_{n}([A_{E_{1}}]) ⊕ H_{n}([A_{E_{2}}])
ν_{A_{E}} ↓  ν_{A_{E_{1}}} + ν_{A_{E_{2}}} ↓
H_{n}(Δ(G)) ⊗ ℤ_{m−1} →^{id} H_{n}(Δ(G)) ⊗ ℤ_{m−1}

By Step 1, H_{n}(A; C) is the quotient group associated with the groups H_{n}([A_{E}]) and the family of splitting maps θ_{∗} and inclusion maps i_{∗}. Thus, in view of the above there is a surjection

H_{n}(A; C) → H_{n}(Δ(G)) ⊗ ℤ_{m−1}

In view of the remarks at the end of Step 2, for example, this surjection must be an isomorphism. □

7. The Partial Isometry Chain Complex Homology

Whilst the definition of the groups H_{n}(A; C) is suitable for the purpose of direct identifications for particular algebras it has the drawback that it does not present normalising partial isometry homology as the homology of a chain complex. With such a presentation one can more fully exploit the standard techniques of algebraic topology and we illustrate this below.

For a pair A, C, as before, we form the chain complex (C_{n}(A; C), ∂) obtained from the quotients C_{n}(A; C) in the short exact sequences

0 → Q_{n} → ∑_{[A_{D}]} ⊕C_{n}([A_{D}]) → C_{n}(A; C) → 0

where Q_{n} is the subgroup of the full n-chain group ∑_{[A_{D}]} ⊕C_{n}([A_{D}]) (restricted direct sum) associated with inclusions and splittings of the digraph spaces D. Explicitly, Q_{n} is generated by elements

−g ⊕ θ(g)

and

−h ⊕ θ_{1}(h) ⊕ θ_{2}(h),

where g ∈ C_{n}([A_{D_{1}}]) and θ : C_{n}([A_{D_{1}}]) → C_{n}([A_{D_{2}}]) is induced by an inclusion of matrix unit systems, and where h ∈ C_{n}([A_{D}]) and

θ_{1} + θ_{2} : C_{n}([A_{D}]) → C_{n}([A_{D_{1}}]) ⊕ C_{n}([A_{D_{2}}])

is the mapping induced by a splitting u_{ij} = u_{ij}^{1} + u_{ij}^{2}.

The boundary operators ∂ : C_{n}([A_{D}]) → C_{n−1}([A_{D}]) respect inclusions and splittings and so induce group homomorphisms Q_{n} → Q_{n−1} and boundary operator ∂_{n} : C_{n}(A; C) → C_{n−1}(A; C). We define the (C−normalising)-partial isometry chain complex homology of the pair (A, C) to be the homology CH_{n}(A; C) of the chain complex (C_{n}(A; C), ∂_{n}), where ∂_{0} is the zero map.

Put another way, (Q_{n}, ∂) is a subcomplex of the direct sum of the chain complexes (C_{n}([A_{D}]), ∂), the complex (C_{n}(A; C), ∂) is defined to be the associated quotient complex, and the homology CH_{∗} is defined to be the homology of this quotient.

The following theorem, and the analogous Theorem 1, show that H_{∗} and CH_{∗} coincide for digraph algebras.

For a directed graph G of a digraph algebra define the reduced graph G_{r} to be the undirected graph obtained from G through an identification of the vertices v, w of G for which both of the edges (v, w) and (w, v) belong to G.

Theorem 7.1. If C is a masa in a digraph algebra A(G) then CH_{n}(A(G); C) is naturally isomorphic to H_{n}(Δ(G)) for all n.

Proof. Let A ⊆ M_{k} ⊗ A(G) be a normalising digraph algebra for A(G). The unitary equivalence class [A] has a representative, which we may take to be A, with partial matrix unit system {u_{i,j}} such that each u_{i,j} is an orthogonal sum of standard matrix units f_{l,m} ⊗ e_{p,q} in M_{k} ⊗ A(G). Such choices of A give inclusion induced maps

i_{A} : C_{n}([A]) → C_{n}([M_{k(A)} ⊗ A(G)]).

Let

q_{k} : C_{n}([M_{k} ⊗ A(G)]) → C_{n}(Δ(G_{r}))

be the natural surjections determined by the identification of equivalent 0-simplexes of Δ([M_{k} ⊗ A(G)]). Then the homomorphism

i = ∑_{[A]} ⊕q_{k(A)} ∘ i_{A} : ∑_{[A]} ⊕C_{n}([A]) → C_{n}(Δ(G_{r}))

maps Q_{n} to zero and so induces a surjective group homomorphism

θ : C_{n}(A(G); C) → C_{n}(Δ(G_{r})).

On the other hand any natural inclusion Δ(G_{r}) → Δ(\overline{G}) = Δ([A(G)], with \overline{G} the undirected graph, which gives a cross-section for q_{1}, induces a cross-section for θ. By the first paragraph above this cross-section is surjective and so θ is an isomorphism. Moreover θ determines a chain complex isomorphism and so the theorem follows. □

The chain complex homology need not coincide with H_{n}(A; C). Perhaps the simplest way to see this is to consider a digraph algebra for which Tor(H_{n}(A(G)), ℤ_{2}) is nonzero. Then H_{n}(A(G) ⊗ O_{3}; D_{|G|} ⊗ C) = H_{n}(Δ(G)) ⊗ ℤ_{2}, by Theorem 2.1 of [17]. On the other hand CH_{n}(A(G) ⊗ O_{3}; D_{|G|} ⊗ C) is computable, by direct methods similar to the proof below, (or by a Kunneth formula), as the homology of the chain complex (C_{n}(A(G) ⊗ K_{0}(O_{3})), ∂ ⊗ id). Accordingly, by the universal coefficient theorem, the chain complex homology has the extra torsion term. Such a difference also appears in the next theorem.

Theorem 7.2. Let A(G) be a digraph algebra. Let T O_{m}^{0} and T O_{m} be the refinement nest subalgebras of the algebraic Cuntz algebra O_{m}^{0} and its closure in the Cuntz algebra O_{m}, respectively. Then for all m ≥ 1 and n ≥ 0 the groups CH_{n}(A(G) ⋆ T O_{m}^{0}) and CH_{n}(A(G) ⋆ T O_{m}) coincide with the simplicial homology group H_{n}(Δ(G); ℤ_{m−1}) with coefficients in ℤ_{m−1}. In particular

CH_{n}(A(G) ⋆ T O_{m}) = (H_{n}(Δ(G)) ⊗ ℤ_{m−1})) ⊕ (T or(H_{n−1}(Δ(G)), ℤ_{m−1})).

Sketch of proof. Let A = A(G)∗T O_{m}. The case for T O_{m}^{o} is essentially the same. As in step one of the proof of Theorem 2 obtain the reduction

C_{n}(A; C) = (∑_{D} ⊕C_{n}([A_{D}]))/IQ_{n}

where the sum is taken over classes [A_{D}] associated with elementary tensor systems and where IQ_{n} is the ideal in this direct sum generated by splitting and inclusions.

For each A_{D} (with connected digraph) the existence of a homotopy with a digraph subalgebra of M_{N} ⊗ A(G) ⋆ T_{m^{r}}, for some r, leads to a group homomorphism

i_{D} : C_{n}([A_{D}]) → C_{n}([M_{N} ⊗ A(G) ⋆ T_{m^{r}}]).

This homomorphism depends on the particular homotopy and on N and r. But as before the natural induced maps

ĩ_{D} : C_{n}([A_{D}]) → C_{n}(Δ(G_{r})) ⊗ ℤ_{m−1}

are well defined and are chain maps. The sum

i = ∑_{D} ⊕ ĩ_{D} : ∑_{D} ⊕C_{n}([A_{D}]) → C_{n}(Δ(G_{r})) ⊗ ℤ_{m−1}

respects splittings and inclusions and so gives a chain complex surjection

i : (C_{n}(A; C), ∂) → (C_{n}(Δ(G_{r})) ⊗ ℤ_{m−1}, ∂ ⊗ id).

On the other hand the homotopy reductions show that the cross section of i induced by a natural injection Δ(G_{r}) → Δ(G) is surjective. Thus i is a chain isomorphism. □

Mayer Vietoris Sequence

Let C ⊆ A_{1} ⊆ B, C ⊆ A_{2} ⊆ B where A_{1} and A_{2} are subalgebras of the star algebra B. We say that the pair A_{1}, A_{2} is a regular pair if every M_{n} ⊗C-normalising partial isometry in M_{N} ⊗ (A_{1} + A_{2}) can be decomposed as an orthogonal sum of normalising partial isometries in M_{N} ⊗ A_{1} and M_{N} ⊗ A_{2}. There are many contexts providing regular pairs. We shall not go into the details of this but nevertheless we remark that Lemma 5.4 is useful for identifying regular pairs. Furthermore all pairs of subalgebras of an AF C*-algebra B which contain a fixed regular canonical masa are regular pairs. (See Chapter 4 of [13].)

Theorem 7.3. If C ⊆ A_{i} ⊆ B where A_{1} and A_{2} is a regular pair with sum A = A_{1} + A_{2} then we have the Mayer Vietoris sequence

.. → CH_{n}(A_{1} ∩ A_{2}; C) → CH_{n}(A_{1}; C) ⊕ CH_{n}(A_{2}; C) →
CH_{n}(A; C) → CH_{n−1}(A_{1} ∩ A_{2}; C) → ..

The theorem follows immediately from the excision lemma below and standard algebraic topology. (See Chapter 6 of [20] for example.)

If C ⊆ A_{1} ⊆ A_{2}, with (C, A_{i}) as before, then we define the relative homology CH_{n}(A_{2}, A_{1}; C) as the homology of the quotient chain complex

(C_{n}(A_{2}; C)/C_{n}(A_{1}; C), ∂).

In particular one has the long exact sequence for the connecting homomorphisms

d_{n} : CH_{n}(A_{2}, A_{1}; C) → CH_{n−1}(A_{1}; C).

Lemma 7.4 (Excision). For a regular pair A_{1}, A_{2} as above,

CH_{n}(A_{1}, A_{1} ∩ A_{2}; C) = CH_{n}(A_{1} + A_{2}, A_{2}; C).

Proof. It is sufficient to show that the inclusion induced chain complex map

(C_{n}(A_{1}; C) + C_{n}(A_{2}; C), ∂) → (C_{n}(A_{1} + A_{2},; C), ∂)

induces an isomorphism of homology. However if A is a normalising digraph subalgebra of A_{1} + A_{2} then, since A_{1}, A_{2} is a regular pair it follows, from a simple finiteness argument, that A has a splitting A → A_{1} ⊕ ⋯ ⊕ A_{r} where each A_{i} is a digraph subalgebra of A_{1} or A_{2}. Thus the inclusion induced map above is already an isomorphism. □

By way of a simple application of the Mayer-Vietoris sequence in our context consider the nontriangular operator algebra

A = [ \begin{matrix}F & 0 & O_{m} & O_{m} \\ 0 & F & O_{m} & O_{m} \\ 0 & 0 & F & 0 \\ 0 & 0 & 0 & F \end{matrix} ]

contained in M_{4} ⊗ O_{m}, which can be viewed as the intersection A_{1} ∩ A_{2} where A_{1} = A(D_{4}) ⊗ O_{m} and

A_{2} = [ \begin{matrix}F & O_{m} & O_{m} & O_{m} \\ 0 & F & O_{m} & O_{m} \\ 0 & 0 & F & O_{m} \\ 0 & 0 & 0 & F \end{matrix} ] ,

and let C ⊆ A be the diagonal ℂ^{4} ⊗ C. It can be shown that, for reasons of cancellation, CH_{1}(A_{2}; C) = 0. The argument for this, which we leave to the reader, uses defect symmetrisation as in Section 2.

The sum A_{1} +A_{2} is T_{4} ⊗O_{m} and so in particular H_{n}(A_{1} +A_{2}; C) = 0, for n > 0, either by direct calculation or by a natural Kunneth formula. The pair A_{1}, A_{2} is a regular pair and, with C suppressed, the Mayer-Vietoris sequence gives

CH_{2}(A_{1} + A_{2}) → CH_{1}(A_{1} ∩ A_{2}) → CH_{1}(A_{1}) ⊕ CH_{1}(A_{2}) → H_{1}(A_{1} + A_{2})

which is

0 → CH_{1}(A_{1} ∩ A_{2}) → ℤ_{m−1} ⊕ 0 → 0

Thus, in view of exactness we get, as expected, CH_{1}(A; C) = ℤ_{m−1}.

Modifying A_{2}, with T F replacing F, similar arguments show that the triangular algebra

E = [ \begin{matrix}T F & 0 & O_{m} & O_{m} \\ 0 & T F & O_{m} & O_{m} \\ 0 & 0 & T F & 0 \\ 0 & 0 & 0 & T F \end{matrix} ]

also has CH_{1}(E; C) = ℤ_{m−1}.

56 S. C. Power

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Department of Mathematics and Statistics, Lancaster University, England s.power@lancaster.ac.uk http://www.maths.lancs.ac.uk/˜ power/

This paper is available via http://nyjm.albany.edu:8000/j/1998/4-4.html.