The Toeplitz algebra of a Hilbert bimodule

Abstract

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 the Toeplitz-Cuntz-Krieger algebras of 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 X A = A A , and define the left action of A by a · b := α(a)b. In [23], Pimsner constructed a C * -algebra O X from a Hilbert bimodule X in such a way that the O X 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 corresponding O X are the Cuntz-Krieger algebras. The Cuntz algebra O n is O X when C X C 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 O X 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 T X generated by a Toeplitz representation, and prove some general results relating these representations to the induced representations of Rieffel.

A Class of Limit Algebras Associated with Directed Graphs

“…https://doi.org/10.1017/S1446788700036168 354 David W. Kribs and Baruch Solel [10] This establishes part (i) of the definition. For (ii), suppose (e, w') e E(n) 1 …”

“…In fact, the *-homomorphism nft determined by the left regular representation is a *-isomorphism of T{E) onto use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700036168 the C*-algebra generated by the operators {L e } [10,25]. Thus, for our purposes, we may identify the algebra T ( £ ) with this faithful representation n P , L (T(E)).…”

Operator Algebras and C*-correspondences: A Survey

“…For any a ∈ A, we define π ∞ (a) ∈ L(F X ) to be the diagonal operator with ϕ X (c) ⊗ id n−1 at its X ⊗n -th entry. It is easy to verify that (π ∞ , t ∞ ) is a representation of X which is called the Fock representation of X. Fowler and Raeburn [19] (resp. Katsura [32]) have shown that the C * -algebra…”

Index maps in theK-theory of graph algebras

“…If R = E 0 reg , then a Cuntz-Krieger (E, R)-family is the same as a Cuntz-Krieger E-family and C * (E, R) = C * (E). If R = ∅, then C * (E, R) is the Toeplitz algebra T (E) defined in [8,Theorem 4.1]. We will call a Cuntz-Krieger (E, ∅)-family a Toeplitz-Cuntz-Krieger E-family.…”