𝐾-theory and 𝐾-homology relative to a 𝐼𝐼_{∞}-factor

Abstract

Abstract. Let X be a compact space and M be a factor of type II ", acting on a separable Hubert space. Let KM(X) denote the Grothendieck group generated by the semigroup of isomorphism classes of M-vector bundles over X, and, if A' is also metric, let ExtM(X") denote the group of equivalence classes of extensions of C(X) relative to M. We show that KM(X) is the direct sum of the even-dimensional Cech cohomology groups of X, and that Extw(A') is the direct product of the odd-dimensional Cech homology groups of X.Introduction. Recently Brown, Douglas, and Fillmore [8] have constructed a generalised homology theory called A-homology, which, in a sense made rigorous in [8], is dual to A-theory. Their construction is in terms of extensions of commutative C*-algebras by the ideal of compact operators on a separable Hubert space. Fillmore [12] and Cho [9] have investigated the analogous construction with the compact operators replaced by the closed two-sided ideal generated by the finite projections in a factor of type 11^. They have constructed (see [9]) a generalised homology theory {Ext^} on the category of compact metric spaces, which we shall call A-homology relative to the 11^,-factor M. In [6] Breuer has considered a theory of vector bundles relative to M and introduced a functor KM which has topological properties like those of A-theory. We shall construct a generalised cohomology theory {A^} (A-theory relative to M) from Breuer's functor, identify it in terms of the conventional A-functor and show that KM(X) is the direct sum of the even-dimensional real cohomology of A for any compact space A. Then we shall deduce the corresponding result for Ext^; namely that Extf/(A) is the direct product of the odd-dimensional real homology of A. We mention that the results in this note all follow in standard fashion from the recent literature; our goal is merely to point out some interesting consequences of the work of Breuer [6] and Cho [9]. Along the way we provide a proof of Proposition 2, which has been stated and used by Singer in [18].First we set up some notation. Throughout, all topological spaces will be Hausdorff, and Af will be a factor of type 11^ acting on a separable Hubert space H. We shall denote by Pf(M) the set of finite projections of M and by dim: PAM) ->R+ the Murray-von Neumann dimension function of M. For