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

In this paper we introduce the notion of Hilbert -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 -bimodules and show that tensor products of minimal bimodules are also minimal. For an - bimodule which is compatible with a trace on a unital -algebra , its dimension (square root of Jones index) depends only on its -class. Finally, we show that the dimension map transforms the Kasparov products in to the product of positive real numbers, and determine the subring of generated by the Hilbert -bimodules for a -algebra generated by Jones projections.