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 A note on the duality of locally compact groups

1968 ◽  
Vol 9 (2) ◽  
pp. 87-91 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Group Topology ◽  
Locally Compact ◽  
Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

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 On a Certain Class of Semigroups of Operators

2011 ◽  
Vol 18 (02) ◽  
pp. 129-142 ◽  
Author(s):  
Paolo Aniello
Keyword(s):  
Banach Space ◽  
Compact Group ◽  
Banach Spaces ◽  
Locally Compact Group ◽  
Semigroups Of Operators ◽  
Probability Measures ◽  
Locally Compact ◽  
Quantum Dynamical ◽  
Interesting Class ◽  
Natural Way

We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced in the early 1970s by Kossakowski. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.

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.

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 Regularity of induced representations and a theorem of Quigg and Spielberg

2002 ◽  
Vol 133 (2) ◽  
pp. 249-259
Author(s):  
ASTRID AN HUEF ◽  
IAIN RAEBURN
Keyword(s):  
Compact Group ◽  
Closed Subgroup ◽  
Morita Equivalence ◽  
Locally Compact Group ◽  
Unitary Representations ◽  
Crossed Products ◽  
Locally Compact ◽  
Induced Representations ◽  
Morita Equivalent ◽  
Special Cases

Mackey's imprimitivity theorem characterizes the unitary representations of a locally compact group G which have been induced from representations of a closed subgroup K; Rieffel's influential reformulation says that the group C*-algebra C*(K) is Morita equivalent to the crossed product C0(G/K)×G [14]. There have since been many important generalizations of this theorem, especially by Rieffel [15, 16] and by Green [3, 4]. These are all special cases of the symmetric imprimitivity theorem of [11], which gives a Morita equivalence between two crossed products of induced C*-algebras.

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