### 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 |
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Pages (from-to) | 3429-3472 |

Number of pages | 44 |

Journal | Transactions of the American Mathematical Society |

Volume | 352 |

Issue number | 8 |

Publication status | Published - 2000 |

Externally published | Yes |

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### Keywords

- Hubert c*-bimodule
- Jones index
- K-theory
- Subfactor

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*352*(8), 3429-3472.