scholarly journals CROSSED PRODUCTS BY ENDOMORPHISMS, VECTOR BUNDLES AND GROUP DUALITY, II

2006 ◽  
Vol 17 (01) ◽  
pp. 65-96 ◽  
Author(s):  
EZIO VASSELLI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Crossed Product ◽  
Usual Sense ◽  
Crossed Products ◽  
Noncompact Group ◽  
C Algebra

We study C*-algebra endomorphims which are special in a weaker sense with respect to the notion introduced by Doplicher and Roberts. We assign to such endomorphisms a geometrical invariant, representing a cohomological obstruction for them to be special in the usual sense. Moreover, we construct the crossed product of a C*-algebra by the action of the dual of a (nonabelian, noncompact) group of vector bundle automorphisms. These crossed products supply a class of examples for such generalized special endomorphisms.

scholarly journals CROSSED PRODUCTS BY ENDOMORPHISMS, VECTOR BUNDLES AND GROUP DUALITY

2005 ◽  
Vol 16 (02) ◽  
pp. 137-171 ◽  
Author(s):  
EZIO VASSELLI
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Product Formula ◽  
Crossed Product ◽  
Crossed Products ◽  
General Group

We construct the crossed product [Formula: see text] of a C(X)-algebra [Formula: see text] by an endomorphism ρ, in such a way that ρ becomes induced by the bimodule [Formula: see text] of continuous sections of a vector bundle ℰ → X. Some motivating examples for such a construction are given. Furthermore, we study the C*-algebra of G-invariant elements of the Cuntz-Pimsner algebra [Formula: see text] associated with [Formula: see text], where G is a (noncompact, in general) group acting on ℰ. In particular, the C*-algebra of invariant elements with respect to the action of the group of special unitaries of ℰ is a crossed product in the above sense. We also study the analogous construction on certain Hilbert bimodules, called "noncommutative pullbacks".

scholarly journals Discrete product systems with twisted units

1995 ◽  
Vol 52 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Marcelo Laca
Keyword(s):  
Dynamical System ◽  
Crossed Product ◽  
Type I ◽  
Crossed Products ◽  
Product System ◽  
C Algebra ◽  
Covariant Representations ◽  
Product Systems ◽  
Recent Theorem ◽  
I Factor

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

CROSSED PRODUCTS OF CERTAIN C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450010 ◽  
Author(s):  
JIAJIE HUA ◽  
YAN WU
Keyword(s):  
Simple Formula ◽  
Continuous Functions ◽  
Cantor Set ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Tracial Rank ◽  
Universal Coefficient Theorem ◽  
Unital Simple

Let X be a Cantor set, and let A be a unital separable simple amenable [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the Universal Coefficient Theorem (UCT). We use C(X, A) to denote the algebra of all continuous functions from X to A. Let α be an automorphism on C(X, A). Suppose that C(X, A) is α-simple, [α|1⊗A] = [ id |1⊗A] in KL(1 ⊗ A, C(X, A)), τ(α(1 ⊗ a)) = τ(1 ⊗ a) for all τ ∈ T(C(X, A)) and all a ∈ A, and [Formula: see text] for all u ∈ U(A) (where α‡ and id‡ are homomorphisms from U(C(X, A))/CU(C(X, A)) → U(C(X, A))/CU(C(X, A)) induced by α and id, respectively, and where CU(C(X, A)) is the closure of the subgroup generated by commutators of the unitary group U(C(X, A)) of C(X, A)), then the corresponding crossed product C(X, A) ⋊α ℤ is a unital simple [Formula: see text]-stable C*-algebra with rationally tracial rank no more than one, which satisfies the UCT. Let X be a Cantor set and 𝕋 be the circle. Let γ : X × 𝕋n → X × 𝕋n be a minimal homeomorphism. It is proved that, as long as the cocycles are rotations, the tracial rank of the corresponding crossed product C*-algebra is always no more than one.

scholarly journals Ideal structure of crossed products by endomorphisms via reversible extensions of C*-dynamical systems

2015 ◽  
Vol 26 (03) ◽  
pp. 1550022 ◽  
Author(s):  
Bartosz Kosma Kwaśniewski
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Transfer Operator ◽  
Crossed Product ◽  
Crossed Products ◽  
Ideal Structure ◽  
Gauge Invariant ◽  
C Algebra ◽  
System A ◽  
The Ideal

We consider an extendible endomorphism α of a C*-algebra A. We associate to it a canonical C*-dynamical system (B, β) that extends (A, α) and is "reversible" in the sense that the endomorphism β admits a unique regular transfer operator β⁎. The theory for (B, β) is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for (B, β) in terms of the initial system (A, α). We apply this idea to study the ideal structure of a non-unital version of the crossed product C*(A, α, J) introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal J in (ker α)⊥, and if J = ( ker α)⊥ it is a modification of Stacey's crossed product that works well with non-injective α's. We provide descriptions of the lattices of ideals in C*(A, α, J) consisting of gauge-invariant ideals and ideals generated by their intersection with A. We investigate conditions under which these lattices coincide with the set of all ideals in C*(A, α, J). In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of α or pointwise quasinilpotence of α.

Crossed product of c*-algebras by hypergroups via group coactions

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Massoud Amini
Keyword(s):  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras

AbstractWe define the crossed product of a C*-algebra by a hypergroup via a group coaction. We generalize the results on Hecke C*-algebra crossed products to our setting.

scholarly journals On maximal ideals in certain reduced twisted C*-crossed products

2015 ◽  
Vol 158 (3) ◽  
pp. 399-417 ◽  
Author(s):  
ERIK BÉDOS ◽  
ROBERTO CONTI
Keyword(s):  
Discrete Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Bijective Correspondence ◽  
C Algebra ◽  
Maximal Ideals ◽  
Maximal Invariant

AbstractWe consider a twisted action of a discrete groupGon a unital C*-algebraAand give conditions ensuring that there is a bijective correspondence between the maximal invariant ideals ofAand the maximal ideals in the associated reduced C*-crossed product.

Reduced ${\bi C}^*$-crossed products by free shifts

1998 ◽  
Vol 18 (5) ◽  
pp. 1075-1096 ◽  
Author(s):  
MARIE CHODA ◽  
TOSHIKAZU NATSUME
Keyword(s):  
Free Product ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra

The free shift $\alpha$ of the reduced free product $C^*$-algebra $A$ is studied from both the analytic and non-commutative ergodic theoretic viewpoints. For an automorphism $\beta$ of $B$, we show that the entropy of $\mathop{\rm Ad}\nolimits u(\alpha \otimes \beta)$ is equal to the entropy of $\mathop{\rm Ad}\nolimits u(\beta)$. We also show that if $B$ is unital, nuclear, and simple, and if the crossed product $B \rtimes_\beta {\Bbb Z}$ is simple and purely infinite, then $(O_\infty \otimes B)\rtimes_{\alpha \otimes \beta} {\Bbb Z}$ is isomorphic to $B \rtimes_\beta {\Bbb Z}$.

scholarly journals Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

2021 ◽  
Vol 127 (2) ◽  
pp. 317-336
Author(s):  
Kazunori Kodaka
Keyword(s):  
Discrete Group ◽  
Morita Equivalence ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebra ◽  
C Algebras ◽  
Morita Equivalent ◽  
Countable Discrete Group ◽  
Relative Commutant ◽  
Strong Morita Equivalence

We consider two twisted actions of a countable discrete group on $\sigma$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.

scholarly journals GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS

2009 ◽  
Vol 20 (11) ◽  
pp. 1363-1396 ◽  
Author(s):  
EZIO VASSELLI
Keyword(s):  
Vector Bundle ◽  
Moduli Space ◽  
Chern Class ◽  
Vector Bundles ◽  
Crossed Products ◽  
Tensor Categories ◽  
Dual Object ◽  
Nontrivial Center ◽  
First Chern Class ◽  
The Given

C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.

Euler classes of vector bundles over manifolds

2021 ◽  
Vol 71 (1) ◽  
pp. 199-210
Author(s):  
Aniruddha C. Naolekar
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Euler Class ◽  
Homology Sphere ◽  
Rational Homology ◽  
Rational Homology Sphere ◽  
Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

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