Product systems of C*-correspondences and Takai duality

Katsoulis, Elias G.

doi:10.1007/s11856-020-2063-3

Cited by 6 publications

(3 citation statements)

References 30 publications

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“…As α g defines automorphisms on the compact operators then a proof similar to [20, Proposition 6.3] (see also [36,Proposition 3.2] for a shorter proof) shows that compact alignment of X implies compact alignment of X ⋊ α,λ G. By construction we see that…”

Section: It Follows Thatmentioning

confidence: 76%

C-envelopes for operator algebras with a coaction and co-universal C-algebras for product systems

Dor-On¹,

Kakariadis²,

Katsoulis³

et al. 2020

Preprint

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A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittadello, for the Fock tensor algebra equipped with its natural coaction. This yields the existence of a co-universal C*-algebra, generalizing previous results of Carlsen, Larsen, Sims and Vittadello, and of Dor-On and Katsoulis. We also realize the C*-envelope of the tensor algebra as the reduced cross sectional algebra of a Fell bundle introduced by Sehnem, which, under a mild assumption of normality, we then identify to the quotient of the Fock algebra by the image of Sehnem's strong covariance ideal. In another application, we obtain a reduced Hao-Ng isomorphism theorem for the co-universal algebras.

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Section: It Follows Thatmentioning

confidence: 76%

C-envelopes for operator algebras with a coaction and co-universal C-algebras for product systems

Dor-On¹,

Kakariadis²,

Katsoulis³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…□ Remark 3.6. Katsoulis obtained similar results for the so-called generalized gauge action on a product system over a semigroup P that is the positive cone of an abelian group, see Theorem 3.8 in [10]. Moreover, using a Fourier transform, he proved a Takai-duality result and generalized some results of Schafhauser from [15].…”

mentioning

confidence: 74%

Group Actions on Product Systems

Deaconu,

Huang

2023

NZ J Math

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We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{\ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.

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“…Recall that the Cuntz algebra contains a canonical Hilbert space of dimension n and it can be constructed as the closure of the linear span of using embeddings where and is the identity map. This linear span becomes a -algebra with a multiplication given by composition and an involution (see [8] and Proposition 2.5 in [16]).…”

Section: The Doplicher–roberts Algebramentioning

confidence: 99%

-Algebras From Group Representations

Deaconu

2022

J. Aust. Math. Soc.

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We introduce certain $C^*$ -algebras and k-graphs associated to k finite-dimensional unitary representations $\rho _1,\ldots ,\rho _k$ of a compact group G. We define a higher rank Doplicher-Roberts algebra $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ , constructed from intertwiners of tensor powers of these representations. Under certain conditions, we show that this $C^*$ -algebra is isomorphic to a corner in the $C^*$ -algebra of a row-finite rank k graph $\Lambda $ with no sources. For G finite and $\rho _i$ faithful of dimension at least two, this graph is irreducible, it has vertices $\hat {G}$ and the edges are determined by k commuting matrices obtained from the character table of the group. We illustrate this with some examples when $\mathcal {O}_{\rho _1,\ldots ,\rho _k}$ is simple and purely infinite, and with some K-theory computations.

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Product systems of C*-correspondences and Takai duality

Cited by 6 publications

References 30 publications

C-envelopes for operator algebras with a coaction and co-universal C-algebras for product systems

C-envelopes for operator algebras with a coaction and co-universal C-algebras for product systems

Group Actions on Product Systems

-Algebras From Group Representations

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Product systems of C*-correspondences and Takai duality

Cited by 6 publications

References 30 publications

C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

C*-envelopes for operator algebras with a coaction and co-universal C*-algebras for product systems

Group Actions on Product Systems

-Algebras From Group Representations

Contact Info

Product

Resources

About

C-envelopes for operator algebras with a coaction and co-universal C-algebras for product systems

C-envelopes for operator algebras with a coaction and co-universal C-algebras for product systems