Nuclearity of semigroup C⁎-algebras

Huef, Astrid an; Nucinkis, Brita E. A.; Sehnem, Camila F.; Yang, Dilian

doi:10.1016/j.jfa.2020.108793

Cited by 11 publications

(15 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nevertheless, for r, s ∈ P , rP = sP precisely when r = su for some unit u ∈ P * . It is easy to see that quasi-lattice ordered semigroups, and more generally the weak quasi-lattice ordered semigroups from [1], are right LCM, but the converse does not hold.…”

Section: Preliminariesmentioning

confidence: 99%

“…is the multiplication operator by α p (1). and, α p (α p −1 (a)) = α p (α −1 p (α p (1)a)) = α p (1)a.…”

Section: Right Lcm Dynamical Systemmentioning

confidence: 99%

“…Proposition 3.7. For a right LCM dynamical system (A, P, α) and for every p, q ∈ P , and a ∈ A we have (1)…”

Section: Right Lcm Dynamical Systemmentioning

confidence: 99%

“…Then condition (3) in Lemma 6.2 also holds. By condition (1) in Lemma 6.2 we can extend φ to a contractive representation of D P , which we also denote by φ, such that (φ, T ) is a contractive covariant pair. Since T satisfies (2), Proposition 6.1 implies that φ is positive, hence completely positive.…”

Section: Covariant Pairs and Dilationmentioning

confidence: 99%

See 3 more Smart Citations

Dilation theory for right LCM semigroup dynamical systems

Laca,

2021

Preprint

View full text Add to dashboard Cite

This paper examines actions of right LCM semigroups by endomorphisms of C*-algebras that encode an additional structure of the right LCM semigroup. We define contractive covariant representations for these semigroup dynamical systems and prove a generalized Stinespring's dilation theorem showing that these representations can be dilated if and only if the map on the C * -algebra is unital and completely positive. This generalizes earlier results about dilations of right LCM semigroups of contractions. In addition, we also give sufficient conditions under which a contractive covariant representation of a right LCM system can be dilated to an isometric representation of the boundary quotient.

show abstract

Section: Preliminariesmentioning

confidence: 99%

“…is the multiplication operator by α p (1). and, α p (α p −1 (a)) = α p (α −1 p (α p (1)a)) = α p (1)a.…”

Section: Right Lcm Dynamical Systemmentioning

confidence: 99%

“…Proposition 3.7. For a right LCM dynamical system (A, P, α) and for every p, q ∈ P , and a ∈ A we have (1)…”

Section: Right Lcm Dynamical Systemmentioning

confidence: 99%

Section: Covariant Pairs and Dilationmentioning

confidence: 99%

See 2 more Smart Citations

Dilation theory for right LCM semigroup dynamical systems

Laca,

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…It is clear that if an element w ∈ P is a right Least Common Multiple for p, q ∈ P then so is wx for every unit x ∈ P * := P ∩ P −1 . Right LCM semigroups that embed in a group and have no nontrivial units, so that least common multiples are unique whenever they exist, have been called weak quasi-lattice ordered semigroups in [31].…”

Section: 4mentioning

confidence: 99%

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

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.

show abstract

K-Theory for Semigroup C*-Algebras and Partial Crossed Products

2021

Commun. Math. Phys.

View full text Add to dashboard Cite

Using the Baum–Connes conjecture with coefficients, we develop a K-theory formula for reduced C*-algebras of strongly 0-E-unitary inverse semigroups, or equivalently, for a class of reduced partial crossed products. This generalizes and gives a new proof of previous K-theory results of Cuntz, Echterhoff and the author. Our K-theory formula applies to a rich class of C*-algebras which are generated by partial isometries. For instance, as new applications which could not be treated using previous results, we discuss semigroup C*-algebras of Artin monoids, Baumslag-Solitar monoids and one-relator monoids, as well as C*-algebras generated by right regular representations of semigroups of number-theoretic origin, and C*-algebras attached to tilings.

show abstract

Nuclearity of semigroup C⁎-algebras

Cited by 11 publications

References 36 publications

Dilation theory for right LCM semigroup dynamical systems

Dilation theory for right LCM semigroup dynamical systems

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

K-Theory for Semigroup C*-Algebras and Partial Crossed Products

Contact Info

Product

Resources

About

Nuclearity of semigroup C⁎-algebras

Cited by 11 publications

References 36 publications

Dilation theory for right LCM semigroup dynamical systems

Dilation theory for right LCM semigroup dynamical systems

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

K-Theory for Semigroup C*-Algebras and Partial Crossed Products

Contact Info

Product

Resources

About

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