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

Tsuyoshi Kajiwara, Yasuo Watatani

Research output: Contribution to journalArticle

24 Citations (Scopus)

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 languageEnglish
Pages (from-to)3429-3472
Number of pages44
JournalTransactions of the American Mathematical Society
Volume352
Issue number8
Publication statusPublished - Dec 1 2000

Fingerprint

Index Theory
Bimodule
K-theory
Algebra
Hilbert
Tensors
C*-algebra
Associativity
Subring
Minimality
Reciprocity
Frobenius
Unital
Square root
Scalar, inner or dot product
Tensor Product
Lemma
Trace
Projection
Transform

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Transactions of the American Mathematical Society, Vol. 352, No. 8, 01.12.2000, p. 3429-3472.

Research output: Contribution to journalArticle

Kajiwara, Tsuyoshi ; Watatani, Yasuo. / Jones index theory by hilbert c*-bimodules and k-theory. In: Transactions of the American Mathematical Society. 2000 ; Vol. 352, No. 8. pp. 3429-3472.
@article{dd7a38bfe0974ff6aa0184e941b46c2c,
title = "Jones index theory by hilbert c*-bimodules and k-theory",
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.",
author = "Tsuyoshi Kajiwara and Yasuo Watatani",
year = "2000",
month = "12",
day = "1",
language = "English",
volume = "352",
pages = "3429--3472",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "8",

}

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

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

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 -