### 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 - 2000 |

Externally published | Yes |

### Fingerprint

### 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.

**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

Y1 - 2000

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.

KW - Hubert c-bimodule

KW - Jones index

KW - K-theory

KW - Subfactor

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

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

M3 - Article

AN - SCOPUS:23044519693

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 -