The Toeplitz Algebra of a Hilbert Bimodule


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.


Citations per Year

56 Citations

Semantic Scholar estimates that this publication has 56 citations based on the available data.

See our FAQ for additional information.

Cite this paper

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