On C∗-algebras associated with C∗-correspondences

Katsura, Takeshi

doi:10.1016/j.jfa.2004.03.010

Cited by 155 publications

(206 citation statements)

References 19 publications

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“…As an immediate consequence of the arguments of Katsura [20] we obtain the following lemma. are nuclear for all n ∈ Z + , and if there is an ideal I ⊳ A such that…”

Section: 3mentioning

confidence: 72%

“…Obviously item (iii) implies item (i), and by [20,Proposition A.13] we have the equivalence of items (ii) and (iii). Therefore it suffices to show that exactness of A implies exactness of N T (A, α) γ .…”

Section: 1mentioning

confidence: 89%

“…There is an analogy of the first part with the results of Katsura on C*-correspondences [20,Section 7] from which we were strongly influenced. Nevertheless the analysis is perplexed because of the multidimensional lattice structure of Z n + .…”

Section: Introductionmentioning

confidence: 85%

“…However, due to the vast complexity of product systems, the defining CNP-representations in the sense of [7, p. 568] involve checking a series of properties. A more flexible algebraic characterization, analogous to the one offered by Katsura [20] for C*-correspondences, is not clear at this point (and perhaps this is a price that needs to be payed).…”

Section: 3mentioning

confidence: 99%

“…In Theorem 2.3 we show that N O(A, α) is exact if and only if A is exact. Nuclearity of A suffices to imply nuclearity of N O(A, α), however it is not necessary even for injective systems (see [20,Example 7.7] and Remark 2.7). In Theorem 2.6 we give a necessary and sufficient condition for nuclearity of N O(A, α) in terms of the automorphic dilation β : Z n → Aut( B).…”

Section: Introductionmentioning

confidence: 99%

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On Nica-Pimsner Algebras of C*-Dynamical Systems Over $\mathbb{Z}_+^n$

Kakariadis

2016

Int Math Res Notices

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Abstract. We examine Nica-Pimsner algebras associated with semigroup actions of Z n + on a C*-algebra A by * -endomorphisms. We give necessary and sufficient conditions on the dynamics for exactness and nuclearity of the Nica-Pimsner algebras. Furthermore we parameterize the KMS states at finite temperature by the tracial states on A. A parametrization is also shown for KMS states at zero temperature (resp. ground states) by the tracial states on A (resp. states on A). Finally we give a formula for obtaining tracial states on the Nica-Pimsner algebras. IntroductionThis project lies in the intersection of three ongoing programs on analysing C*-algebras associated with C*-dynamics. There has been a continuous interest for a generalised C*-crossed product construction of possibly noninvertible semigroup actions. (to mention only a few). The key idea is to consider the quotient of a Fock algebra by an ideal of redundancies. Our standing point of view follows Arveson's program on the C*-envelope [19], and suggests that this ideal is theŠilov ideal of a nonselfadjoint operator algebra in the sense of Arveson [2]. In contrast to the case of the usual C*-crossed products, these redundancies may be very complicated to allow an in-depth exploration. An alternate route is suggested in the joint work of the author with Davidson and Fuller [9, Introduction]: dilate a possibly non-invertible semigroup action α : P → End(A) to a group action β : G → Aut( B) such that the distinguished quotient related to α : Z n + → End(A) is a full corner of the usual C*-crossed product related to β : G → Aut( B). Consequently the examination of the distinguished quotient can be induced via the more exploit-able and more exploit-ed theory of usual C*-crossed products. This pathway was met with success for several semigroup actions including Ore semigroups and spanning cones in [9], and in earlier works of the author for P = Z + [16], and of the author with Katsoulis for P = F n + [18].2010 Mathematics Subject Classification. 46L05, 46L55, 46L30, 58B34.

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“…As an immediate consequence of the arguments of Katsura [20] we obtain the following lemma. are nuclear for all n ∈ Z + , and if there is an ideal I ⊳ A such that…”

Section: 3mentioning

confidence: 72%

Section: 1mentioning

confidence: 89%

Section: Introductionmentioning

confidence: 85%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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