Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems

Carlsen, Toke Meier; Larsen, Nadia S.; Sims, Aidan; Vittadello, Sean T.

doi:10.1112/plms/pdq028

Cited by 53 publications

(165 citation statements)

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“…Cuntz-Nica-Pimsner algebras of such objects have been studied in the influential papers of Fowler [14], Sims and Yeend [33], and Carlsen, Larsen, Sims and Vittadello [7]. From the analysis in [9] it follows that N O(A, α) falls inside this framework.…”

Section: 3mentioning

confidence: 91%

“…These algebras arise naturally in the theory of product systems, e.g. see Fowler [14], Solel [32], Deaconu, Kumjian, Pask and Sims [10], Sims and Yeend [33], Carlsen, Larsen, Sims and Vittadello [7]. The existence of N T (A, α) is readily verified by the existence of a Fock representation.…”

Section: Introductionmentioning

confidence: 97%

“…From the analysis in [9] it follows that N O(A, α) falls inside this framework. Among many others, the key idea in [7] is to compare Cuntz-Nica-Pimsner algebras with co-universal C*-algebras subject to a gauge invariant uniqueness theorem. 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.…”

Section: 3mentioning

confidence: 99%

“…Carlsen, Larsen, Sims, and Vittadello [7] show that Cuntz-Nica-Pimsner algebras over Z n + exist as co-universal objects that satisfy a gauge invariant uniqueness theorem. This remarkable result leaves open the exploration of their algebraic structure.…”

Section: Introductionmentioning

confidence: 99%

“…Dilation theory was used essentially to construct a non-trivial Cuntz-Nica representation in the sense of [9]; then N O(A, α) is shown to be universal with respect to these representations. The meeting point is then achieved by showing that N O(A, α) satisfies a gauge invariant uniqueness theorem [9,Theorem 4.3.17]; thus N O(A, α) coincides with the Cuntz-Nica-Pimsner algebra of [7].…”

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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Section: 3mentioning

confidence: 91%

Section: Introductionmentioning

confidence: 97%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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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Ideal Structure of Nica-Toeplitz Algebras

Bilich

2024

Integr. Equ. Oper. Theory

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We study the gauge-invariant ideal structure of the Nica-Toeplitz algebra $$\mathcal {N}\mathcal {T}(X)$$ N T ( X ) of a product system (A, X) over $$\mathbb {N}^n$$ N n . We obtain a clear description of X-invariant ideals in A, that is, restrictions of gauge-invariant ideals in $$\mathcal {N\hspace{-1.111pt}T}(X)$$ N T ( X ) to A. The main result is a classification of gauge-invariant ideals in $$\mathcal {N\hspace{-1.111pt}T}(X)$$ N T ( X ) for a proper product system in terms of families of ideals in A. We also apply our results to higher-rank graphs.

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KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

Afsar

Larsen

Neshveyev

2020

Commun. Math. Phys.

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Given a quasi-lattice ordered group (G, P ) and a compactly aligned product system X of essential C * -correspondences over the monoid P , we show that there is a bijection between the gauge-invariant KMS β -states on the Nica-Toeplitz algebra N T (X) of X with respect to a gauge-type dynamics, on one side, and the tracial states on the coefficient algebra A satisfying a system (in general infinite) of inequalities, on the other. This strengthens and generalizes a number of results in the literature in several directions: we do not make any extra assumptions on P and X, and our result can, in principle, be used to study KMS-states at any finite inverse temperature β. Under fairly general additional assumptions we show that there is a critical inverse temperature βc such that for β > βc all KMS β -states are of Gibbs type, hence gauge-invariant, in which case we have a complete classification of KMS β -states in terms of tracial states on A, while at β = βc we have a phase transition manifesting itself in the appearance of KMS β -states that are not of Gibbs type. In the case of right-angled Artin monoids we show also that our system of inequalities for traces on A can be reduced to a much smaller system, a finite one when the monoid is finitely generated. Most of our results generalize to arbitrary quasi-free dynamics on N T (X).Acknowledgement. This research was initiated when Z.A. visited the Department of Mathematics at the University of Oslo. She thanks her colleagues for their hospitality.

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Co-universal algebras associated to product systems, and gauge-invariant uniqueness theorems

Cited by 53 publications

References 33 publications

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

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

Ideal Structure of Nica-Toeplitz Algebras

KMS States on Nica-Toeplitz $$\hbox {C}^*$$-algebras

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