On C*-algebras associated with C*-correspondences

Katsura, Takeshi

doi:10.1016/s0022-1236(04)00093-x

Cited by 110 publications

(279 citation statements)

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“…, are called covariant representations [19]. We define the Cuntz-Pimsner-Toeplitz algebra T X as the universal C * -algebra for all Toeplitz representations of (X, A, ϕ X ).…”

Section: Preliminariesmentioning

confidence: 99%

“…It is easy to verify that (π ∞ , t ∞ ) is a representation of X which is called the Fock representation of X. Fowler and Raeburn [11] (resp. Katsura [19]) have shown that the C * -algebra C…”

Section: Preliminariesmentioning

confidence: 99%

“…Hence, for ξ, η ∈ X, the element [ξ, η] ∈ A is identified with the unique element a ∈ J X such that ϕ X (a) = Θ X ξ,η . This is the big advantage (and a characterizing property [19]) of Hilbert bimodules; i.e., J X is * -isomorphic to K(X).…”

Section: Lemma 62 ([18]mentioning

confidence: 99%

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Contributions to the theory of $\mathrm{C}^{*}$-correspondences with applications to multivariable dynamics

Kakariadis

Katsoulis

2012

Trans. Amer. Math. Soc.

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Motivated by the theory of tensor algebras and multivariable C *dynamics, we revisit two fundamental techniques in the theory of C *-correspondences, the "addition of a tail" to a non-injective C *-correspondence and the dilation of an injective C *-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for "adding a tail" to a non-injective C *-correspondence; our scheme includes the "tail" of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable C *-dynamics. We also exhibit a transparent picture for the dilation of an injective C *-correspondence to an essential Hilbert bimodule. As an application of our constructs, we prove two results in the theory of multivariable dynamics that extend earlier results. We also discuss the impact of our results on the description of the C *envelope of a tensor algebra as the Cuntz-Pimsner algebra of the associated C *-correspondence.

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“…, are called covariant representations [19]. We define the Cuntz-Pimsner-Toeplitz algebra T X as the universal C * -algebra for all Toeplitz representations of (X, A, ϕ X ).…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Contributions to the theory of $\mathrm{C}^{*}$-correspondences with applications to multivariable dynamics

Kakariadis

Katsoulis

2012

Trans. Amer. Math. Soc.

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“…There are two main classes of C*-algebras one can associate to a C*-correspondence X over A: the Toeplitz-Pimsner algebra T (X ) generated by the Fock representation, and the Cuntz-Pimsner algebra O(X ) which is minimal in containing an isometric copy of the transformation. In his influential work, Katsura [16] gave the correct * -relations that define O(X ) and answered a range of questions that are now in the core of the theory.…”

Section: Introductionmentioning

confidence: 99%

“…However, the converse is not true even at the level of Z + . Katsura [16] presents an example of Ozawa for a C*-correspondence whose O(X ) is nuclear but A is not. He also gave the correct equivalent, namely that the embedding A → O(X ) is nuclear.…”

Section: Introductionmentioning

confidence: 99%

Finite-dimensional approximations for Nica–Pimsner algebras

Kakariadis¹

2019

Ergod. Th. Dynam. Sys.

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We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.

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