The Toeplitz Algebra of a Hilbert Bimodule

Abstract

Suppose a C-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C-algebra OX which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras On, and the Cuntz-Krieger algebras OB. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for a family of Toeplitz-Cuntz-Krieger algebras for directed graphs, which includes Cuntz’s uniqueness theorem for O∞. A Hilbert bimodule X over a C∗-algebra A is a right Hilbert A-module with a left action of A by adjointable operators. The motivating example comes from an automorphism α of A: take XA = AA, and define the left action of A by a · b := α(a)b. In [23], Pimsner constructed a C∗-algebra OX from a Hilbert bimodule X in such a way that the OX corresponding to an automorphism α is the crossed product A ×α Z. He also produced interesting examples of bimodules which do not arise from automorphisms or endomorphisms, including bimodules over finite-dimensional commutative C∗-algebras for which the correspondingOX are the Cuntz-Krieger algebras. The Cuntz algebra On is OX when CXC is a Hilbert space of dimension n and the left action of C is by multiples of the identity. Here we use methods developed in [18, 9] for analysing semigroup crossed products to study Pimsner’s algebras. These methods seem to apply more directly to Pimsner’s analogue of the Toeplitz-Cuntz algebras rather than his analogue OX of the Cuntz algebras. Nevertheless, our results yield new information about the Cuntz-Krieger algebras of some infinite graphs, giving a whole class of these algebras which behave like O∞. The uniqueness theorems for C∗-algebras generated by algebraic systems of isometries say, roughly speaking, that all examples of a given system in which the isometries are non-unitary generate isomorphic C∗-algebras. We can approach such a theorem by introducing a C∗-algebra which is universal for systems of the given type, and then characterising its faithful representations. Here the systems consist of representations ψ of X and π of A on the same Hilbert space which convert the module actions and the inner product to operator multiplication; we call these Toeplitz representations of X. (The partial isometries and isometries appearing in more conventional systems are obtained by applying ψ to the elements of a basis for X.) In Section 1, we discuss these Toeplitz representations, show that there is a universal C∗-algebra TX generated by a Toeplitz representation, and prove some general results relating these representations to the induced representations of Rieffel. Date: June 18, 1998. 1991 Mathematics Subject Classification. Primary 46L55.

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@inproceedings{FOWLER1998TheTA, title={The Toeplitz Algebra of a Hilbert Bimodule}, author={NEAL J. FOWLER and Iain Raeburn}, year={1998} }