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

Abstract

Abstract. 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. IntroductionStrong 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, which can be identified with the elements of K 0 (N ). The index of a subfactor N ⊂ M is analyzed by the bimodule N M M . 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 A X B 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 forced to postulate that Hilbert C * -bimodules have two-sided inner products in order to make the category of them have the notion of conjugation, which plays an important role in subfactor theory. We remark that considering two-sided inner products corresponds to the orientation of tangles [24].A

The Strong Morita Equivalence for Inclusions of -Algebras and Conditional Expectations for Equivalence Bimodules

“…We note that Y is of finite type and that X can be regarded as a full right Hilbert Bmodule of finite type in the sense of Kajiwara and Watatani [4]. Let B D (Y) be the C * -algebra of all right D-linear operators on Y for which has a right adjoint D-linear operator on Y.…”

“…We note that Y is of finite type and that X can be regarded as a full left Hilbert Amodule of finite type in the sense of Kajiwara and Watatani [4]. (ii) A conditional expectation from an equivalence onto its closed subspace in Definition 2.4 is a left and right conditional expectation.…”

Cancellation for inclusions of C*-algebras of finite depth

“…When A and B have trivial centers, Lemma 3.11 of [17] shows that a minimal conditional expectation is pseudominimal.…”

“…This completes the proof. We now interpret B l as a Hilbert module over A using the composition [17].) The basic construction B 2 = C * (B, Proof.…”

Jones Type Basic Construction on Hopf Spin Models

“…Kosaki [8] investigated the index for a conditional expectation of an arbitrary factor onto a subfactor, where he used the spatial theory [10] and the theory on operator-valued weights [11]. Inspired by previous work, Watatani [9] considered the index for a conditional expectation on a C * -algebra, and established a link between transfer in the K-theory of C * -algebras [9,12] and the multiplication map. Based on the quasi-basis, Watatani developed a C * -version of basic construction.…”