Jones index theory by Hilbert C*-bimodules and K-theory

Abstract

In this paper we introduce the notion of Hilbert C ∗ {\mathrm {C}}^{*} -bimodules, replacing the associativity condition of two-sided inner products in Rieffel’s imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur’s Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert C ∗ {\mathrm {C}}^{*} -bimodules and show that tensor products of minimal bimodules are also minimal. For an A A - A A bimodule which is compatible with a trace on a unital C ∗ {\mathrm {C}}^{*} -algebra A A , its dimension (square root of Jones index) depends only on its K K KK -class. Finally, we show that the dimension map transforms the Kasparov products in K K ( A , A ) KK(A,A) to the product of positive real numbers, and determine the subring of K K ( A , A ) KK(A,A) generated by the Hilbert C ∗ {\mathrm {C}}^{*} -bimodules for a C ∗ {\mathrm {C}}^{*} -algebra generated by Jones projections.