Homology for operator algebras. I: Spectral homology for reflexive algebras (Q1908147)

A reflexive operator algebra \({\mathcal A}\) for which the lattice \({\mathcal L}=\text{Lat }{\mathcal A}\) of invariant projections is a set of commuting projections is called a CSL algebra. A CSL algebra for which \({\mathcal L}\) is completely distributive as an abstract lattice is called a CDCSL algebra. \textit{S. C. Power} [Proc. Lond. Math. Soc., III. Ser. 61, No. 3, 571-614 (1990; Zbl 0783.47060)] and \textit{J. L. Orr} and \textit{S. C. Power} [Indiana Univ. Math. J. 40, No. 2, 617-638 (1991; Zbl 0776.47020)] have given the spectral representation theorem for CDCSL algebras using the partially ordered space \(M({\mathcal L})\) of meet-irreducible elements of \({\mathcal L}\) with partial order \(L_1\leq L_2\) defined by \(L_1L_2=L_1\). In this article the spectral homology of \({\mathcal A}\), \(H_{*}^{sp}({\mathcal A})\) is defined and its properties are studied. This homology group is more tractable than the Hochschild cohomology for Banach algebras. A Kunneth formula for the spectral homology of spatial tensor products, together with a natural suspension formula is also given. This article deals with the connection between the spectral homology and the Hochschild cohomology. It is shown that if \(H_1^{sp}({\mathcal A})\) is trivial then every Schur automorphism \(\alpha\) relative to a fixed masa of \({\mathcal A}\) is quasispatial in the sense of Gilfeather and Moore.