  Local Hochschild, cyclic Homology and K-theory were introduced by N. Teleman
in [10] with the purpose of unifying different settings of the index theorem.
This paper is one of the research topics announced in [10], {\S}10. The
definition of these new objects inserts the Alexander-Spanier idea for defining
the co-homology [8] into the corresponding constructions. This is done by
allowing only chains which have smal l support about the diagonal. This
definition, applicable at least in the case of the Banach sub-algebras of the
algebra of bounded operators on the Hilbert space of L2-sections in vector
bundles, differs from various constructions due to A. Connes [1], A. Connes,
H.Moscovici [2], M. Puschnigg [7], J. Cuntz [4]. In this paper we prove that
the local Hochschild homology of the Banach algebra of Hilbert-Schmidt
operators on any countable, locally finite homogeneous simplicial complex X is
naturally isomorphic the Alexander-Spanier homology of the space X, Theorem 1.
This result may be used to compute the local periodic cyclic homology of the
algebra of Hilbert-Schmidt operators on such spaces X. The same result should
hold in the case of the algebra of trace class operators L1 as well as in the
case of smoothing operators s \subset L 1. In addition, the tools we introduce
in this paper should apply also for computing the local Hochschild and periodic
cyclic homology of the Schatten class ideals Lp, at least for the other values
1 < p < 2. Parts of what is presented here were stated in author's lecture at
the International Alexandroff Reading Conference, Moscow, 21-25 May 2012.
