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], §10. The deﬁnition of these new ob jects 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 ⊂ 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. |