The maximal injective crossed product

2019 ◽  
Vol 40 (11) ◽  
pp. 2995-3014
Author(s):  
ALCIDES BUSS ◽  
SIEGFRIED ECHTERHOFF ◽  
RUFUS WILLETT
Keyword(s):  
Compact Group ◽  
Explicit Construction ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Locally Compact ◽  
Dual Result

A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes _{\text{inj}}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^{\ast }$-algebras; this is a sort of ‘dual’ result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum–Connes conjecture. It turns out that $\rtimes _{\text{inj}}$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.

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 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.

scholarly journals Erratum to “Full and reduced C*-coactions”. Math. Proc. Camb. Phil. Soc. 116 (1994), 435–450

2016 ◽  
Vol 161 (2) ◽  
pp. 379-380
Author(s):  
S. KALISZEWSKI ◽  
JOHN QUIGG
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
The Other ◽  
Crossed Product ◽  
Locally Compact

Proposition 2ċ5 of [5] states that a full coaction of a locally compact group on a C*-algebra is nondegenerate if and only if its normalisation is. Unfortunately, the proof there only addresses the forward implication, and we have not been able to find a proof of the opposite implication. This issue is important because the theory of crossed-product duality for coactions requires implicitly that the coactions involved be nondegenerate. Moreover, each type of coaction — full, reduced, normal, maximal, and (most recently) exotic — has its own distinctive properties with respect to duality, making it crucial to be able to convert from one to the other without losing nondegeneracy.

scholarly journals THE CROSSED PRODUCT BY A POINTWISE UNITARY ACTION ON A C*-ALGEBRA WITH CONTINUOUS TRACE

2001 ◽  
Vol 44 (1) ◽  
pp. 215-218
Author(s):  
Klaus Deicke
Keyword(s):  
Compact Group ◽  
Elementary Proof ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Locally Compact ◽  
Primary 46L55 ◽  
C Algebra ◽  
Group A ◽  
Continuous Trace ◽  
Unitary Action

AbstractLet $G$ be a locally compact group, $A$ a continuous trace $C^*$-algebra, and $\alpha$ a pointwise unitary action of $G$ on $A$. It is a result of Olesen and Raeburn that if $A$ is separable and $G$ is second countable, then the crossed product $A\times_\alpha G$ has continuous trace. We present a new and much more elementary proof of this fact. Moreover, we do not even need the separability assumptions made on $A$ and $G$.AMS 2000 Mathematics subject classification: Primary 46L55

scholarly journals Twisted crossed products by coactions

1994 ◽  
Vol 56 (3) ◽  
pp. 320-344 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Normal Subgroup ◽  
Compact Group ◽  
Duality Theorem ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Locally Compact

AbstractWe consider coactions of a locally compact group G on a C*-algebra A, and the associated crossed product C*-algebra A× G. Given a normal subgroup N of G, we seek to decompose A× G as an iterated crossed product (A× G/ N) × N, and introduce notions of twisted coaction and twisted crossed product which make this possible. We then prove a duality theorem for these twisted crossed products, and discuss how our results might be used, especially when N is abelian.

scholarly journals Categorical constructions in C*-algebra theory

2002 ◽  
Vol 73 (1) ◽  
pp. 97-114
Author(s):  
M. Khoshkam ◽  
J. Tavakoli
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Locally Compact ◽  
C Algebra ◽  
Algebra Theory ◽  
Categorical Constructions

AbstractThe notions of limits and colimits are studied in the category of C*-algebras. It is shown that limits and colimits of diagrams of C*-algebras are stable under tensor product by a fixed C*-algebra, and crossed product by a locally compact group.

scholarly journals Full coactions on Hilbert C*-modules

1995 ◽  
Vol 59 (3) ◽  
pp. 409-420
Author(s):  
Huu Hung Bui
Keyword(s):  
Compact Group ◽  
Morita Equivalence ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Locally Compact ◽  
Natural Notion ◽  
Strong Morita Equivalence ◽  
Natural Way

AbstractWe introduce a natural notion of full coactions of a locally compact group on a Hilbert C*-module, and associate each full coaction in a natural way to an ordinary coaction. We also introduce a natural notion of strong Morita equivalence of full coactions which is sufficient to ensure strong Morita equivalence of the corresponding crossed product C*-algebras.

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).

scholarly journals Interpolation in wavelet spaces and the HRT-conjecture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Eirik Berge
Keyword(s):  
Compact Group ◽  
Wavelet Transforms ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Space Structure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Positive Type ◽  
Time Frequency ◽  
Hrt Conjecture

AbstractWe investigate the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })\subset L^{2}(G)$$ W g ( H π ) ⊂ L 2 ( G ) arising from square integrable representations $$\pi :G \rightarrow \mathcal {U}(\mathcal {H}_{\pi })$$ π : G → U ( H π ) of a locally compact group G. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong restrictions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time–frequency analysis, this problem turns out to be equivalent to the HRT-conjecture. Finally, we consider the problem of whether all the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })$$ W g ( H π ) of a locally compact group G collectively exhaust the ambient space $$L^{2}(G)$$ L 2 ( G ) . We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.

The Superstability of the Generalized D'alembert Functional Equation

2003 ◽  
Vol 10 (3) ◽  
pp. 503-508 ◽  
Author(s):  
Elhoucien Elqorachi ◽  
Mohamed Akkouchi
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Compact Group ◽  
Locally Compact Group ◽  
Complex Numbers ◽  
Locally Compact

Abstract We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

Sign in / Sign up

Export Citation Format

Share Document