scholarly journals Hecke C*-algebras and semi-direct products

2009 ◽  
Vol 52 (1) ◽  
pp. 127-153 ◽  
Author(s):  
S. Kaliszewski ◽  
Magnus B. Landstad ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Direct Product ◽  
Hecke Algebra ◽  
Quadratic Extension ◽  
Crossed Product ◽  
Direct Products ◽  
C Algebras ◽  
Fixed Point Algebra

AbstractWe analyse Hecke pairs (G,H) and the associated Hecke algebra $\mathcal{H}$ when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of $\mathcal{H}$ in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = ℚ or K = ℤ[p−1]. In particular we look at the ax + b group of a quadratic extension of K.

scholarly journals Discrete coactions on C*-algebras

1996 ◽  
Vol 60 (1) ◽  
pp. 118-127 ◽  
Author(s):  
Chi-Keung Ng
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Amenable Group ◽  
Group Actions ◽  
Crossed Product ◽  
Discrete Groups ◽  
C Algebras ◽  
Fixed Point Algebra

AbstractWe will consider coactions of discrete groups on C*-algebras and imitate some of the results about compact group actions on C*-algebras. In particular, the crossed product of a reduced coaction ∈ of a discrete amenable group G on A is liminal (respectively, postliminal) if and only if the fixed point algebra of ∈ is. Moreover, we will also consider ergodic coactions on C*-algebras.

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 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 A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in E-theory

2015 ◽  
Vol 58 (2) ◽  
pp. 374-380 ◽  
Author(s):  
Gábor Szabó
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Short Note ◽  
Circle Actions ◽  
Continuous Action ◽  
C Algebra ◽  
C Algebras ◽  
Group A ◽  
Fixed Point Algebra ◽  
Universal Coefficient Theorem

AbstractLet G be a metrizable compact group, A a separable C*-algebra, and α:G → Aut(A) a strongly continuous action. Provided that α satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in E-theory passes from Ato the crossed product C*-algebra A⋊α G and the ûxed point algebra Aα. This extends a similar result by Gardella for KK-theory in the case of unital C*-algebras but with a shorter and less technical proof. For circle actions on separable unital C*-algebras with the continuous Rokhlin property, we establish a connection between the Etheory equivalence class of A and that of its fixed point algebra Aα.

scholarly journals Sectional algebras of semigroupoid bundles

2020 ◽  
Vol 30 (06) ◽  
pp. 1257-1304
Author(s):  
Luiz Gustavo Cordeiro
Keyword(s):  
Direct Product ◽  
Tensor Products ◽  
Crossed Product ◽  
Direct Products ◽  
Crossed Products ◽  
Graded Algebras ◽  
Semidirect Products ◽  
Skew Products ◽  
Definition Of ◽  
Product Bundles

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ([Formula: see text]-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever [Formula: see text] is a ∧-preaction of a discrete inverse semigroupoid [Formula: see text] on an ample (possibly non-Hausdorff) groupoid [Formula: see text], the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of [Formula: see text] by [Formula: see text]. This is a far-reaching generalization of analogous results which had been proven in particular cases.

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 Hecke C*-Algebras, Schlichting Completions and Morita Equivalence

2008 ◽  
Vol 51 (3) ◽  
pp. 657-695 ◽  
Author(s):  
S. Kaliszewski ◽  
Magnus B. Landstad ◽  
John Quigg
Keyword(s):  
Normal Subgroup ◽  
Representation Theory ◽  
Direct Product ◽  
Open Subgroup ◽  
Hecke Algebra ◽  
Morita Equivalence ◽  
Compact Open Subgroup ◽  
C Algebras ◽  
Extended Analysis

AbstractThe Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

Crossed Products by Semigroups of Endomorphisms and Groups of Partial Automorphisms

2003 ◽  
Vol 46 (1) ◽  
pp. 98-112 ◽  
Author(s):  
Nadia S. Larsen
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Crossed Product ◽  
Crossed Products ◽  
Partial Action ◽  
Multiplier Algebra ◽  
Fixed Point Algebra

AbstractWe consider a class (A; S; α) of dynamical systems, where S is an Ore semigroup and α is an action such that each αs is injective and extendible (i.e. it extends to a non-unital endomorphism of the multiplier algebra), and has range an ideal of A. We show that there is a partial action on the fixed-point algebra under the canonical coaction of the enveloping group G of S constructed in [15, Proposition 6.1]. It turns out that the full crossed product by this coaction is isomorphic to A ⋊αS. If the coaction is moreover normal, then the isomorphism can be extended to include the reduced crossed product. We look then at invariant ideals and finally, at examples of systems where our results apply.

scholarly journals The noncommutative geometry of k-graph C*-algebras

2007 ◽  
Vol 1 (2) ◽  
pp. 259-304 ◽  
Author(s):  
David Pask ◽  
Adam Rennie ◽  
Aidan Sims
Keyword(s):  
Fixed Point ◽  
Isomorphism Class ◽  
Lower Semicontinuous ◽  
Index Theorem ◽  
Graph Algebras ◽  
Gauge Invariant ◽  
Local Index ◽  
C Algebras ◽  
Local Index Theorem ◽  
Fixed Point Algebra

AbstractThis paper is comprised of two related parts. First we discuss which k-graph algebras have faithful traces. We characterise the existence of a faithful semifinite lower-semicontinuous gauge-invariant trace on C* (Λ) in terms of the existence of a faithful graph trace on Λ.Second, for k-graphs with faithful gauge invariant trace, we construct a smooth (k,∞)-summable semifinite spectral triple. We use the semifinite local index theorem to compute the pairing with K-theory. This numerical pairing can be obtained by applying the trace to a KK-pairing with values in the K-theory of the fixed point algebra of the Tk action. As with graph algebras, the index pairing is an invariant for a finer structure than the isomorphism class of the algebra.

Quartic Algebras

1992 ◽  
Vol 44 (6) ◽  
pp. 1167-1191 ◽  
Author(s):  
Carla Farsi ◽  
Neil Watling
Keyword(s):  
Fixed Point ◽  
Crossed Product ◽  
General Characterization ◽  
Crossed Product Algebra ◽  
Fixed Point Algebra ◽  
Product Algebra ◽  
Positive Integers ◽  
Coprime Positive Integers

AbstractIn this paper we study the fixed point algebra of the automorphism of the rotation algebra , θ = p/q with p, q coprime positive integers, given by u → v-1, v → u. We give a general characterization of the fixed point algebra, determine its K-theory and consider the related crossed-product algebra ⋊Ƭ Z4.

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