scholarly journals The $K$-Theory of Some Reduced Inverse Semigroup $C^*$-Algebras

2015 ◽  
Vol 117 (2) ◽  
pp. 186 ◽  
Author(s):  
Magnus Dahler Norling
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
One Dimensional ◽  
C Algebras ◽  
Morita Equivalent ◽  
K Theory

We use a recent result by Cuntz, Echterhoff and Li about the $K$-theory of certain reduced $C^*$-crossed products to describe the $K$-theory of $C^*_r(S)$ when $S$ is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that $C^*_r(S)$ is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

scholarly journals TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

2021 ◽  
pp. 1-36
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Inverse Semigroup ◽  
Cohomology Group ◽  
Group Structure ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
Semigroup Rings ◽  
Abelian Extensions ◽  
Algebraic Setting ◽  
Baer Sum

Abstract Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

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

2021 ◽  
Author(s):  
Xin Li
Keyword(s):  
Inverse Semigroups ◽  
Crossed Products ◽  
Partial Isometries ◽  
C Algebras ◽  
Regular Representations ◽  
K Theory ◽  
New Applications ◽  
Theory Formula

AbstractUsing 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.

scholarly journals Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

2021 ◽  
Vol 127 (2) ◽  
pp. 317-336
Author(s):  
Kazunori Kodaka
Keyword(s):  
Discrete Group ◽  
Morita Equivalence ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Countable Discrete Group ◽  
Relative Commutant ◽  
Strong Morita Equivalence

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

Morita equivalences between fixed point algebras and crossed products

1999 ◽  
Vol 125 (1) ◽  
pp. 43-52 ◽  
Author(s):  
CHI-KEUNG NG
Keyword(s):  
Fixed Point ◽  
Quantum Group ◽  
Type Theorem ◽  
Compact Quantum Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Morita Equivalent ◽  
Fixed Point Algebra

In this paper, we will prove that if A is a C*-algebra with an effective coaction ε by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras.

scholarly journals Extension problems and non-Abelian duality for C*-algebras

2007 ◽  
Vol 75 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Astrid an Huef ◽  
S. Kaliszewski ◽  
Iain Raeburn
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Test Question ◽  
Locally Compact ◽  
Dual Representation ◽  
C Algebras

Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) ⋊ (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

scholarly journals Induced Coactions of Discrete Groups on C*-Algebras

1999 ◽  
Vol 51 (4) ◽  
pp. 745-770 ◽  
Author(s):  
Siegfried Echterhoff ◽  
John Quigg
Keyword(s):  
Quotient Group ◽  
Discrete Group ◽  
Discrete Groups ◽  
Crossed Products ◽  
C Algebras ◽  
Morita Equivalent ◽  
Close Relationship

AbstractUsing the close relationship between coactions of discrete groups and Fell bundles, we introduce a procedure for inducing a C*-coaction δ: D → D ⊗C*(G/N) of a quotient group G/N of a discrete group G to a C*-coaction Ind δ: Ind D → D ⊗C*(G) of G. We show that induced coactions behave in many respects similarly to induced actions. In particular, as an analogue of the well known imprimitivity theorem for induced actions we prove that the crossed products Ind D ×IndδG and D ×δG/N are always Morita equivalent. We also obtain nonabelian analogues of a theorem of Olesen and Pedersen which show that there is a duality between induced coactions and twisted actions in the sense of Green. We further investigate amenability of Fell bundles corresponding to induced coactions.

scholarly journals The C*-algebras of some inverse semigroups

1990 ◽  
Vol 42 (2) ◽  
pp. 335-348 ◽  
Author(s):  
Rachel Hancock ◽  
Iain Raeburn
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Inverse Semigroups ◽  
Bicyclic Semigroup ◽  
C Algebras

We discuss the structure of some inverse semigroups and the associated C* algebras. In particular, we study the bicyclic semigroup and the free monogenic inverse semigroup, following earlier work of Conway, Duncan and Paterson. We then associate to each zero-one matrix A an inverse semigroup CA, and show that the C*-algebra OA of Cuntz and Krieger is closely related to the semigroup algebra C*(CA).

Non-Commutative Deformations of C(T2) and K-Theory

1997 ◽  
Vol 08 (05) ◽  
pp. 555-571
Author(s):  
Cristina Cerri
Keyword(s):  
Positive Element ◽  
Crossed Product ◽  
C Algebra ◽  
Basic Properties ◽  
C Algebras ◽  
The Family ◽  
K Theory

For each α ≥ 0, let Bα be the universal C*-algebra generated by unitary elements uα, vα and a self-adjoint hα such that ||hα|| ≤ α and [Formula: see text]. In this work we prove that the family {Bα}α ∈ [0,∞[ extend the family of soft torus with the same basic properties, i.e., the field of C*-algebras {Bα}α ∈ [0,α0] is continuous and each Bα is a crossed product of a C*-algebra homotopically equivalent to C(S1) by Z. We then show that the K-groups of Bα are isomorphic to Z ⊕ Z. Applying results from the theory of rotation algebras we prove that every positive element (n,m) in K0(Bα) satisfies |m|α ≤ 2πn. It follows that these C*-algebras are not all homotopically equivalent to each other, although they have the same K-groups.

A CONSTRUCTION FOR WEIGHTS ON C*-ALGEBRAS: DUAL WEIGHTS FOR C*-CROSSED PRODUCTS

1999 ◽  
Vol 10 (01) ◽  
pp. 129-157 ◽  
Author(s):  
J. QUAEGEBEUR ◽  
J. VERDING
Keyword(s):  
Dynamical System ◽  
Haar Measure ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebras ◽  
System A ◽  
Lower Semi ◽  
Continuous Weight ◽  
Densely Defined

A method for constructing densely defined lower semi-continuous weights on C*-algebras is presented. The method can be used to construct a "dual weight" on the C*-crossed product A×αG of a C*-dynamical system (A,G,α), starting from a relatively α-invariant densely defined lower semi-continuous weight on A. As an application we show that the Haar measure on the quantum E(2) group is a C*-dual weight.

scholarly journals Crossed products of Hilbert pro-C-bimodules and associated pro-C-algebras

2016 ◽  
Vol 32 (2) ◽  
pp. 195-201
Author(s):  
MARIA JOITA ◽  
◽  
RADU-B. MUNTEANU ◽  
Keyword(s):  
Compact Group ◽  
Inverse Limit ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Locally Compact ◽  
C Algebras

An action (γ, α) of a locally compact group G on a Hilbert pro-C∗-bimodule (X, A) induces an action γ × α of G on A ×X Z the crossed product of A by X. We show that if (γ, α) is an inverse limit action, then the crossed product of A ×α G by X ×γ G respectively of A ×α,r G by X ×γ,r G is isomorphic to the full crossed product of A ×X Z by γ × α respectively the reduced crossed product of A ×X Z by γ × α.

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