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Jones index theory by Hilbert C${}^{*}$-bimodules and K-theory

Author(s): Tsuyoshi Kajiwara; Yasuo Watatani
Journal: Trans. Amer. Math. Soc. 352 (2000), 3429-3472.
MSC (2000): Primary 46L08, 46L80
Posted: April 14, 2000
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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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Additional Information:

Tsuyoshi Kajiwara
Affiliation: Department of Environmental and Mathematical Sciences, Okayama Unniversity, Tsushima, Okayama 700, Japan
Email: kajiwara@math.ems.okayama-u.ac.jp

Yasuo Watatani
Affiliation: Graduate School of Mathematics, Kyushu University, Ropponmatsu, Fukuoka, 810 Japan
Email: watatani@math.kyushu-u.ac.jp

DOI: 10.1090/S0002-9947-00-02392-8
PII: S 0002-9947(00)02392-8
Keywords: Hilbert C${}^{*}$-bimodule, K-theory, Jones index, subfactor
Received by editor(s): April 28, 1995
Received by editor(s) in revised form: March 18, 1998
Posted: April 14, 2000
Additional Notes: Partially supported by the Grants--in--aid for Scientific Research, The Ministery of Education
Dedicated: Dedicated to the 60th birthday of Professor Takeshi Hirai
Copyright of article: Copyright 2000, American Mathematical Society

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