Yasuo Watatani

Jones index theory by Hilbert C*-bimodules and K-theory

In this paper we introduce the notion of Hilbert C∗-bimodules, replacing the associativity condition of two-sided inner products in Rieffel’s imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur’s Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert 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 Hilbert C∗-bimodules for a C∗-algebra generated by Jones projections. Introduction Strong Morita equivalence for C∗-algebras A and B was introduced by M. A. Rieffel ([43],[44]) by the existence of an imprimitivity bimodule X , which is a left Hilbert A-module as well as a right Hilbert B-module with full C∗-algebra valued inner products A〈 , 〉 and 〈 , 〉B such that A〈x, y〉z = x〈y, z〉B. If σ-unital C∗-algebras A and B are strongly Morita equivalent, then K∗(A) and K∗(B) are isomorphic by the imprimitivity bimodule, and they are stably isomorphic by a result of L. Brown, P. Green and M. A. Rieffel ([2]). The purpose of this paper is to study a relation between the index theory invented by V. F. R. Jones ([23]) and K-theory for C∗-algebras (cf. [1]). There exists a Ktheoretical obstruction even for the inclusion of simple C∗-algebras of index two. Jones introduced an index for a subfactor N of a type II1 factor M in terms of the coupling constant dimN L(M) ([23]), which can be identified with the elements of K0(N). The index of a subfactor N ⊂M is analyzed by the bimodule NMM . The important point to note here is that Jones index is also regarded as an element of the Kasparov group “KK(N,N)” ([26],[15]). We studied the inclusion of C∗-algebras, introducing the index for C∗-subalgebras in [47]. In this paper, we study the C∗index theory from the viewpoint of bimodules. We introduce the notion of a Hilbert A-B bimodule AXB by replacing the associativity condition A〈x, y〉z = x〈y, z〉B in Rieffel’s imprimitivity bimodule by Pimsner-Popa type inequalities [41]. We are Received by the editors April 28, 1995 and, in revised form, March 18, 1998. 2000 Mathematics Subject Classification. Primary 46L08, 46L80.

Simple c*-algebras arising from β-expansion of real numbers

An electrolytic cell is provided for the electrochemical separation of selected metals from electrodissociatable compounds thereof in the molten state utilizing as electrode separator a plurality of solid electrolyte tubes which, under the influence of an electrical potential, are permeable to the flow of selected cations, but impermeable to fluids and the flow of anions and other cations. Electrical switching means are provided with each tube for starting and stopping production of molten metal therein, and for providing a removable electrical path between a cathodic element of the cell and the molten metal in each tube.

CROSSED PRODUCTS OF HILBERT C*-BIMODULES BY COUNTABLE DISCRETE GROUPS

We introduce a notion of crossed products of Hilbert C*-bimodules by countable discrete groups and mainly study the case of finite groups following Jones index theory. We give a sufficient condition such that the crossed product bimodule is irreducible. We have a bimodule version of Takesaki-Takai duality. We show the categorical structures when the action is properly outer, and give some example of this construction concerning the orbifold constructions.

KMS states and branched points

We completely classify the KMS states for the gauge action on a

Toeplitz-Composition C*-Algebras for certain finite blaschke products

C^*

C*-algebras associated with Mauldin-Williams graphs

-algebra associated with a rational function

Traces on cores of C∗-algebras associated with self-similar maps

R

Perturbations of intermediate C*-subalgebras for simple C*-algebras

introduced in our previous work. The gauge action has a phase transition at

A relation between certain interpolated Cuntz algebras and interpolated free group factors

\beta = \log \deg R

Crossed products of Hilbert C*-bimodules by bundles

. We can recover the degree of