EMBEDDINGS OF HILBERT BIMODULES

NIEN-TSU SHEN, Purdue University

Abstract

Let C('*)-algebras A, B and an A-B Hilbert bimodule X be given. Let denote the collection of isomorphism equivalence classes of embeddings f = (f(,A),f(,B),f(,X)) of (A,B,X) into C('*)-algebras C such that (I) f(,A)(A) and f(,B)(B) are hereditary C('*)-subalgebras of C, (II) f(,A) is a ('*)-isomorphism, (III) f(,B) is a ('*)-isomorphism, (IV)(' )f(,A)(A)Cf(,B)(B) = f(,X)(X), (V) f(,X) is an isomorphism of A-B Hilbert bimodules, (VI) f(,A)(A) (UNION) f(,B)(B) generates C as a hereditary C('*)-subalgebra. Let denote the collection of quasi-multipliers T of X such that (VBAR)(VBAR)T(VBAR)(VBAR) (LESSTHEQ) 1. In this paper, we prove that there is a one-to-one correspondence between and . This problem is motivated by a result of L. G. Brown, P. Green and M. A. Rieffel in which case the embedding corresponds to T = 0. We also discuss some related versions of these embeddings. Let denote the collection of embeddings f = (f(,A),f(,B)) of (A,B) into C('*)-algebras C satisfying (I), (II), (III), (VI) and (IV')(' )f(,A)(A)Cf(,B)(B) is isomorphic to X as an A-B Hilbert bimodule. Let denote the collection of {vTu: u,v are central unitary multipliers of X} where T is a quasi-multiplier of X such that (VBAR)(VBAR)T(VBAR)(VBAR) (LESSTHEQ) 1. Then there is a one-to-one correspondence between and . We also study the case where X is not given. The correspondence between various properties of T with various properties of f = (f(,A),f(,B),f(,X)) is discussed, too.

Degree

Ph.D.

Subject Area

Mathematics

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