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Homology for operator algebras. III: Partial isometry homotopy and triangular algebras. (English) Zbl 0906.46055

Summary: The partial isometry homology groups \(H_n\) defined in part II [the author, J. Funct. Anal. 135, No. 1, 233-269 (1996; Zbl 0859.46047)] 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 \({\mathcal O}_m\). Specifically, \(H_n(A(G)*A)= H_n(\Delta(G))\otimes_{\mathbb{Z}} K_0(C^*(A))\) and \(CH_n(A(G)* A)\) is the simplicial homology group \(H_n(\Delta(G);K_0(C^*(A)))\) with coefficients in \(K_0(C^*(A))\).

MSC:

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
47L30 Abstract operator algebras on Hilbert spaces
46K50 Nonselfadjoint (sub)algebras in algebras with involution
18G50 Nonabelian homological algebra (category-theoretic aspects)
46L55 Noncommutative dynamical systems

Citations:

Zbl 0859.46047
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