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

Kajiwara, Tsuyoshi; Watatani, Yasuo

doi:10.1090/S0002-9947-00-02392-8

Jones index theory by Hilbert C${}^{*}$-bimodules and K-theory
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by Tsuyoshi Kajiwara and Yasuo Watatani PDF

Trans. Amer. Math. Soc. 352 (2000), 3429-3472 Request permission

Abstract:

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

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