scholarly journals TENSOR-PRODUCT COACTION FUNCTORS

2020 ◽  
pp. 1-16
Author(s):  
S. KALISZEWSKI ◽  
MAGNUS B. LANDSTAD ◽  
JOHN QUIGG
Keyword(s):  
Tensor Product ◽  
Recent Work ◽  
Analogous Result ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Fixed Action ◽  
Product Action ◽  
K Theory

Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$ , then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$ , we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$ -ization functor we defined earlier, where $E$ is a large ideal of $B(G)$ .

scholarly journals Characteristic invatiant of tensor product actions and actions on crossed products

2002 ◽  
Vol 73 (3) ◽  
pp. 357-376
Author(s):  
Yukako Miwa ◽  
Yoshikazu Katayama
Keyword(s):  
Tensor Product ◽  
Discrete Group ◽  
Product Formula ◽  
Crossed Product ◽  
Crossed Products ◽  
Modular Invariant ◽  
Characteristic Invariant ◽  
Extended Action ◽  
Product Action

AbstractThe first purpose of this paper is to give a tensor product formula of the characteristic invariant and modular invariant for a tensor product action of a discrete group G on AFD factors. The second purpose is to describe a characteristic invariant and modular invariant of the extended action to a crossed product in terms of the original invariants.

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 Fullness of crossed products of factors by discrete groups

2019 ◽  
Vol 150 (5) ◽  
pp. 2368-2378 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Discrete Group ◽  
Amenable Group ◽  
Complete Characterization ◽  
Crossed Product ◽  
Discrete Groups ◽  
Sufficient Condition ◽  
Crossed Products ◽  
Product Factor ◽  
Jones Criterion ◽  
Arbitrary Factor

AbstractLet M be an arbitrary factor and $\sigma : \Gamma \curvearrowright M$ an action of a discrete group. In this paper, we study the fullness of the crossed product $M \rtimes _\sigma \Gamma $. When Γ is amenable, we obtain a complete characterization: the crossed product factor $M \rtimes _\sigma \Gamma $ is full if and only if M is full and the quotient map $\overline {\sigma } : \Gamma \rightarrow {\rm out}(M)$ has finite kernel and discrete image. This answers the question of Jones from [11]. When M is full and Γ is arbitrary, we give a sufficient condition for $M \rtimes _\sigma \Gamma $ to be full which generalizes both Jones' criterion and Choda's criterion. In particular, we show that if M is any full factor (possibly of type III) and Γ is a non-inner amenable group, then the crossed product $M \rtimes _\sigma \Gamma $ is full.

scholarly journals Iterated crossed products

2014 ◽  
Vol 13 (07) ◽  
pp. 1450036 ◽  
Author(s):  
Florin Panaite
Keyword(s):  
Tensor Product ◽  
Crossed Product ◽  
Smash Product ◽  
Algebra Structure ◽  
Crossed Products ◽  
Twisted Tensor Product

We define a "mirror version" of Brzeziński's crossed product and we prove that, under certain circumstances, a Brzeziński crossed product D ⊗R,σ V and a mirror version [Formula: see text] may be iterated, obtaining an algebra structure on W ⊗ D ⊗ V. Particular cases of this construction are the iterated twisted tensor product of algebras and the quasi-Hopf two-sided smash product.

A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups

2019 ◽  
Vol 71 (5) ◽  
pp. 1103-1125 ◽  
Author(s):  
Jan Cameron ◽  
Roger R. Smith
Keyword(s):  
Dynamical System ◽  
Discrete Group ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Outer Automorphisms ◽  
Galois Correspondence ◽  
Simple Algebras ◽  
Unital Simple

AbstractLet a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

scholarly journals C*-crossed products by partial actions and actions of inverse semigroups

1997 ◽  
Vol 63 (1) ◽  
pp. 32-46 ◽  
Author(s):  
Nándor Sieben
Keyword(s):  
Inverse Semigroups ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Related Concept ◽  
Semigroup Action ◽  
Covariant Representations

AbstractThe recently developed theory of partial actions of discrete groups on C*-algebras is extended. A related concept of actions of inverse semigroups on C*-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.

scholarly journals Coverings of Directed Graphs and Crossed Products of C*-Algebras by Coactions of Homogeneous Spaces

2003 ◽  
Vol 14 (07) ◽  
pp. 773-789 ◽  
Author(s):  
Klaus Deicke ◽  
David Pask ◽  
Iain Raeburn
Keyword(s):  
Fundamental Group ◽  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Directed Graphs ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
C Algebras

We show that if p:F→E is a covering of directed graphs, then the Cuntz–Krieger algebra C*(F) of F can be viewed as a crossed product of C*(E) by a coaction of a homogeneous space for the fundamental group π1(E). Combining this result with information about Cuntz–Krieger algebras gives some interesting corollaries which suggest conjectures about crossed products by coactions of homogeneous spaces of discrete groups. We then prove these conjectures.

scholarly journals Residually finite actions and crossed products

2011 ◽  
Vol 32 (5) ◽  
pp. 1585-1614 ◽  
Author(s):  
DAVID KERR ◽  
PIOTR W. NOWAK
Keyword(s):  
Free Group ◽  
Metrizable Space ◽  
Crossed Product ◽  
Discrete Groups ◽  
Crossed Products ◽  
Residual Finiteness ◽  
Hausdorff Spaces ◽  
Residually Finite ◽  
Compact Metrizable Space ◽  
Continuous Actions

AbstractWe study a notion of residual finiteness for continuous actions of discrete groups on compact Hausdorff spaces and how it relates to the existence of norm microstates for the reduced crossed product. Our main result asserts that an action of a free group on a zero-dimensional compact metrizable space is residually finite if and only if its reduced crossed product admits norm microstates, i.e., is an MF algebra.

scholarly journals Exotic Coactions

2016 ◽  
Vol 59 (2) ◽  
pp. 411-434 ◽  
Author(s):  
S. Kaliszewski ◽  
Magnus B. Landstad ◽  
John Quigg
Keyword(s):  
Compact Group ◽  
Recent Work ◽  
Group Algebra ◽  
Locally Compact Group ◽  
Crossed Product ◽  
General Context ◽  
Crossed Products ◽  
Locally Compact

AbstractIf a locally compact group G acts on a C*-algebra B, we have both full and reduced crossed products and each has a coaction of G. We investigate ‘exotic’ coactions in between the two, which are determined by certain ideals E of the Fourier–Stieltjes algebra B(G); an approach that is inspired by recent work of Brown and Guentner on new C*-group algebra completions. We actually carry out the bulk of our investigation in the general context of coactions on a C*-algebra A. Buss and Echterhoff have shown that not every coaction comes from one of these ideals, but nevertheless the ideals do generate a wide array of exotic coactions. Coactions determined by these ideals E satisfy a certain ‘E-crossed product duality’, intermediate between full and reduced duality. We give partial results concerning exotic coactions with the ultimate goal being a classification of which coactions are determined by ideals of B(G).

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