scholarly journals An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center

2001 ◽  
pp. 1-10 ◽  
Author(s):  
Hellmut Baumgärtel ◽  
Fernando Lledó
Keyword(s):  
Duality Theory ◽  
Compact Groups ◽  
C Algebras ◽  
Nontrivial Center

scholarly journals DUALITY OF COMPACT GROUPS AND HILBERT C*-SYSTEMS FOR C*-ALGEBRAS WITH A NONTRIVIAL CENTER

2004 ◽  
Vol 15 (08) ◽  
pp. 759-812 ◽  
Author(s):  
HELLMUT BAUMGÄRTEL ◽  
FERNANDO LLEDÓ
Keyword(s):  
Technical Condition ◽  
Minimality Condition ◽  
Compact Groups ◽  
Character Group ◽  
C Algebra ◽  
C Algebras ◽  
Dual Object ◽  
Nontrivial Center ◽  
Chain Group ◽  
Fixed Point Algebra

In this paper we present duality theory for compact groups in the case when the C*-algebra [Formula: see text], the fixed point algebra of the corresponding Hilbert C*-system [Formula: see text], has a nontrivial center [Formula: see text] and the relative commutant satisfies the minimality condition [Formula: see text] as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*-system is expressed by means of an inclusion of C*-categories [Formula: see text], where [Formula: see text] is a suitable DR-category and [Formula: see text] a full subcategory of the category of endomorphisms of [Formula: see text]. Both categories have the same objects and the arrows of [Formula: see text] can be generated from the arrows of [Formula: see text] and the center [Formula: see text]. A crucial new element that appears in the present analysis is an abelian group [Formula: see text], which we call the chain group of [Formula: see text], and that can be constructed from certain equivalence relation defined on [Formula: see text], the dual object of [Formula: see text]. The chain group, which is isomorphic to the character group of the center of [Formula: see text], determines the action of irreducible endomorphisms of [Formula: see text] when restricted to [Formula: see text]. Moreover, [Formula: see text] encodes the possibility of defining a symmetry ∊ also for the larger category [Formula: see text] of the previous inclusion.

scholarly journals On $C*$-algebras associated with locally compact groups

1996 ◽  
Vol 124 (10) ◽  
pp. 3151-3158 ◽  
Author(s):  
M. B. Bekka ◽  
E. Kaniuth ◽  
A. T. Lau ◽  
G. Schlichting
Keyword(s):  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
C Algebras

scholarly journals AN ALGEBRAIC DUALITY THEORY FOR MULTIPLICATIVE UNITARIES

2001 ◽  
Vol 12 (04) ◽  
pp. 415-459 ◽  
Author(s):  
SERGIO DOPLICHER ◽  
CLAUDIA PINZARI ◽  
JOHN E. ROBERTS
Keyword(s):  
Hilbert Spaces ◽  
Duality Theory ◽  
Duality Theorem ◽  
Separable Hilbert Space ◽  
Regular Representation ◽  
Cuntz Algebra ◽  
Relative Depth ◽  
Minimal Representation ◽  
C Algebras ◽  
Algebraic Duality

Multiplicative unitaries are described in terms of a pair of commuting shifts of relative depth two. They can be generated from ambidextrous Hilbert spaces in a tensor C*-category. The algebraic analogue of the Takesaki–Tatsuuma Duality Theorem characterizes abstractly C*-algebras acted on by unital endomorphisms that are intrinsically related to the regular representation of a multiplicative unitary. The relevant C*-algebras turn out to be simple and indeed separable if the corresponding multiplicative unitaries act on a separable Hilbert space. A categorical analogue provides internal characterizations of minimal representation categories of a multiplicative unitary. Endomorphisms of the Cuntz algebra related algebraically to the grading are discussed as is the notion of braided symmetry in a tensor C*-category.

scholarly journals Superselection Structures for C*-Algebras with Nontrivial Center

1997 ◽  
Vol 09 (07) ◽  
pp. 785-819 ◽  
Author(s):  
Hellmut Baumgärtel ◽  
Fernando Lledó
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
C Algebras ◽  
Nontrivial Center ◽  
Relative Commutant ◽  
Fixed Point Algebra

We present and prove some results within the framework of Hilbert C*-systems [Formula: see text] with a compact group [Formula: see text]. We assume that the fixed point algebra [Formula: see text] of [Formula: see text] has a nontrivial center [Formula: see text] and its relative commutant w.r.t. ℱ coincides with [Formula: see text], i.e. we have [Formula: see text]. In this context we propose a generalization of the notion of an irreducible endomorphism and study the behaviour of such irreducibles w.r.t. [Formula: see text]. Finally, we give several characterizations of the stabilizer of [Formula: see text].

scholarly journals Equivariant KK-Theory and the Continuous Rokhlin Property

2020 ◽  
Author(s):  
Eusebio Gardella
Keyword(s):  
Fixed Point ◽  
Crossed Product ◽  
Compact Groups ◽  
Crossed Products ◽  
Circle Actions ◽  
Technical Result ◽  
C Algebras ◽  
Equivariant Version ◽  
Universal Coefficient Theorem

Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. As an application of the case of ${{\mathbb{Z}}}_3$-actions, we answer a question of Phillips–Viola about algebras not isomorphic to their opposites. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb{T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb{T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb{T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb{T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

scholarly journals Gabor duality theory for Morita equivalent C∗-algebras

2020 ◽  
Vol 31 (10) ◽  
pp. 2050073 ◽  
Author(s):  
Are Austad ◽  
Mads S. Jakobsen ◽  
Franz Luef
Keyword(s):  
Duality Theory ◽  
Morita Equivalence ◽  
Gabor Frames ◽  
Duality Principle ◽  
Gabor Analysis ◽  
Matrix Algebras ◽  
C Algebras ◽  
Morita Equivalent ◽  
Twisted Group ◽  
Density Theorems

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

scholarly journals APPROXIMATION PROPERTIES FOR CROSSED PRODUCTS BY ACTIONS AND COACTIONS

2001 ◽  
Vol 12 (05) ◽  
pp. 595-608 ◽  
Author(s):  
MAY M. NILSEN ◽  
ROGER R. SMITH
Keyword(s):  
Approximation Property ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Crossed Products ◽  
Approximation Properties ◽  
Locally Compact ◽  
Amenable Groups ◽  
C Algebras ◽  
Approximation Constant

We investigate approximation properties for C*-algebras and their crossed products by actions and coactions by locally compact groups. We show that Haagerup's approximation constant is preserved for crossed products by arbitrary amenable groups, and we show why this is not always true in the non-amenable case. We also examine similar questions for other forms of the approximation property.

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