### Abstract

ABSTRACT. In this paper we introduce the notion of Hubert C*-bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodulcs by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hubert 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 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 Hubert C*-bimodules for a C*-algebra generated by Jones projections.

Original language | English |
---|---|

Pages (from-to) | 3429-3472 |

Number of pages | 44 |

Journal | Transactions of the American Mathematical Society |

Volume | 352 |

Issue number | 8 |

Publication status | Published - Dec 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*352*(8), 3429-3472.

**Jones index theory by hilbert c*-bimodules and k-theory.** / Kajiwara, Tsuyoshi; Watatani, Yasuo.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 352, no. 8, pp. 3429-3472.

}

TY - JOUR

T1 - Jones index theory by hilbert c*-bimodules and k-theory

AU - Kajiwara, Tsuyoshi

AU - Watatani, Yasuo

PY - 2000/12/1

Y1 - 2000/12/1

N2 - ABSTRACT. In this paper we introduce the notion of Hubert C*-bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodulcs by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hubert 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 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 Hubert C*-bimodules for a C*-algebra generated by Jones projections.

AB - ABSTRACT. In this paper we introduce the notion of Hubert C*-bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodulcs by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hubert 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 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 Hubert C*-bimodules for a C*-algebra generated by Jones projections.

UR - http://www.scopus.com/inward/record.url?scp=23044519693&partnerID=8YFLogxK

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M3 - Article

VL - 352

SP - 3429

EP - 3472

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -