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 - 2000
Externally publishedYes

Fingerprint

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

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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, 2000, p. 3429-3472.

Research output: Contribution to journalArticle

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