K-Theory and K-Homology Relative to a II ∞ -Factor

Abstract

Let X be a compact space and M be a factor of type II, acting on a separable Hilbert space. Let KM(X) denote the Grothendieck group generated by the semigroup of isomorphism classes of M-vector bundles over X, and, if X 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 Extm(X) 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 K-homology, which, in a sense made rigorous in [8], is dual to K-theory. Their construction is in terms of extensions of commutative C*-algebras by the ideal of compact operators on a separable Hilbert 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 II,,. They have constructed (see [9]) a generalised homology theory {Ext/} on the category of compact metric spaces, which we shall call K-homology relative to the II.-factor M. In [61 Breuer has considered a theory of vector bundles relative to M and introduced a functor KM which has topological properties like those of K-theory. We shall constructa generalised cohomology theory { KM}) (K-theory relative to M) from Breuer's functor, identify it in terms of the conventional K-functor and show that KM(X) is the direct sum of the even-dimensional real cohomology of X for any compact space X. Then we shall deduce the corresponding result for ExtM; namely that Extm (X) is the direct product of the odd-dimensional real homology of X. 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 M will be a factor of type II acting on a separable Hilbert space H. We shall denote by Pf(M) the set of finite projections of M and by dim: Pf (M) -* R+ the Murray-von Neumann dimension function of M. For Received by the editors November 21, 1977 and, in revised form, January 25, 1978. AMS (MOS) subject classifications (1970). Primary 55B20; Secondary 46L10, 47B30, 47C15. ?d American Mathematical Society 1978