FREE-PRODUCT GROUPS, CUNTZ-KRIEGER ALGEBRAS, AND COVARIANT MAPS

1991 ◽  
Vol 02 (04) ◽  
pp. 457-476 ◽  
Author(s):  
JOHN SPIELBERG
Keyword(s):  
Free Product ◽  
Compact Convex ◽  
Negative Answer ◽  
Crossed Product ◽  
Crossed Products ◽  
Explicit Computation ◽  
Completely Positive ◽  
Cyclic Groups ◽  
Crossed Product Algebra ◽  
Product Algebra

A construction is given relating a finitely generated free-product of cyclic groups with a certain Cuntz-Krieger algebra, generalizing the relation between the Choi algebra and 02. It is shown that a certain boundary action of such a group yields a Cuntz-Krieger algebra by the crossed-product construction. Certain compact convex spaces of completely positive mappings associated to a crossed-product algebra are introduced. These are used to generalize a problem of J. Anderson regarding the representation theory of the Choi algebra. An explicit computation of these spaces for the crossed products under study yields a negative answer to this problem.

Certain properties for crossed products by automorphisms with a certain non-simple tracial Rokhlin property

2012 ◽  
Vol 33 (5) ◽  
pp. 1391-1400 ◽  
Author(s):  
XIAOCHUN FANG ◽  
QINGZHAI FAN
Keyword(s):  
Finite Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Infinite Dimensional ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra ◽  
Tracial Rokhlin Property

AbstractLet $\Omega $ be a class of unital $C^*$-algebras. Then any simple unital $C^*$-algebra $A\in \mathrm {TA}(\mathrm {TA}\Omega )$ is a $\mathrm {TA}\Omega $ algebra. Let $A\in \mathrm {TA}\Omega $ be an infinite-dimensional $\alpha $-simple unital $C^*$-algebra with the property SP. Suppose that $\alpha :G\to \mathrm {Aut}(A)$ is an action of a finite group $G$ on $A$ which has a certain non-simple tracial Rokhlin property. Then the crossed product algebra $C^*(G,A,\alpha )$ belongs to $\mathrm {TA}\Omega $.

TILTED ALGEBRAS AND CROSSED PRODUCTS

2015 ◽  
Vol 58 (3) ◽  
pp. 559-571
Author(s):  
YANAN LIN ◽  
ZHENQIANG ZHOU
Keyword(s):  
Finite Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Artin Algebra ◽  
Tilted Algebra ◽  
Tilted Algebras ◽  
Crossed Product Algebra ◽  
A Finite Group ◽  
Product Algebra

AbstractWe consider an artin algebra A and its crossed product algebra Aα#σG, where G is a finite group with its order invertible in A. Then, we prove that A is a tilted algebra if and only if so is Aα#σG.

CROSSED PRODUCT ORDERS OVER VALUATION RINGS

2001 ◽  
Vol 33 (5) ◽  
pp. 520-526 ◽  
Author(s):  
JOHN S. KAUTA
Keyword(s):  
Crossed Product ◽  
Homological Dimension ◽  
Crossed Products ◽  
Quotient Field ◽  
Group Of Units ◽  
Valuation Rings ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
Product Order ◽  
Natural Way

Let V be a commutative valuation domain of arbitrary Krull-dimension (rank), with quotient field F, and let K be a finite Galois extension of F with group G, and S the integral closure of V in K. If, in the crossed product algebra K [midast ] G, the 2-cocycle takes values in the group of units of S, then one can form, in a natural way, a ‘crossed product order’ S [midast ] G ⊆ K [midast ] G. In the light of recent results by H. Marubayashi and Z. Yi on the homological dimension of crossed products, this paper discusses necessary and/or sufficient valuation-theoretic conditions, on the extension K/F, for the V-order S [midast ] G to be semihereditary, maximal or Azumaya over V.

scholarly journals Semigroups of local homeomorphisms and interaction groups

2007 ◽  
Vol 27 (6) ◽  
pp. 1737-1771 ◽  
Author(s):  
R. EXEL ◽  
J. RENAULT
Keyword(s):  
Compact Space ◽  
Crossed Product ◽  
C Algebra ◽  
Crossed Product Algebra ◽  
Product Algebra

AbstractGiven a semigroup of surjective local homeomorphisms on a compact space X we consider the corresponding semigroup of *-endomorphisms on C(X) and discuss the possibility of extending it to an interaction group, a concept recently introduced by the first named author. We also define a transformation groupoid whose C*-algebra turns out to be isomorphic to the crossed product algebra for the interaction group. Several examples are considered, including one which gives rise to a slightly different construction and should be interpreted as being the C*-algebra of a certain polymorphism.

scholarly journals The block Schur product is a Hadamard product

2020 ◽  
Vol 126 (3) ◽  
pp. 603-616
Author(s):  
Erik Christensen
Keyword(s):  
Fourier Coefficients ◽  
Hadamard Product ◽  
Natural Generalization ◽  
Continuous Functions ◽  
Crossed Product ◽  
Convolution Product ◽  
C Algebra ◽  
Schur Product ◽  
Crossed Product Algebra ◽  
Product Algebra

Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their block Schur product is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the Schur product on scalar matrices is also known as the Hadamard product. We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product. We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

Isomorphism Classes of Solenoidal Algebras I

1993 ◽  
Vol 36 (4) ◽  
pp. 414-418 ◽  
Author(s):  
Berndt Brenken
Keyword(s):  
Dynamical System ◽  
Growth Rate ◽  
Unit Circle ◽  
Compact Space ◽  
Periodic Points ◽  
Crossed Product ◽  
Roots Of Unity ◽  
Isomorphism Classes ◽  
Crossed Product Algebra ◽  
Product Algebra

AbstractEach g ∊ ℤ[x] defines a homeomorphism of a compact space We investigate the isomorphism classes of the C*-crossed product algebra Bg associated with the dynamical system An isomorphism invariant Eg of the algebra Bg is shown to determine the algebra Bg up to * or * anti-isomorphism if degree g ≤ 1 and 1 is not a root of g or if degree g = 2 and g is irreducible. It is also observed that the entropy of the dynamical system is equal to the growth rate of the periodic points if g has no roots of unity as zeros. This slightly extends the previously known equality of these two quantities under the assumption that g has no zeros on the unit circle.

Growth series of crossed and two-sided crossed products of cyclic groups

2018 ◽  
Vol 68 (3) ◽  
pp. 537-548 ◽  
Author(s):  
Eylem Güzel Karpuz ◽  
Esra Kirmizi Çetinalp
Keyword(s):  
Normal Forms ◽  
Crossed Product ◽  
Crossed Products ◽  
Growth Series ◽  
Cyclic Groups ◽  
Rewriting Systems ◽  
Product Construction

Abstract We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

scholarly journals SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

2019 ◽  
Vol 62 (S1) ◽  
pp. S165-S185 ◽  
Author(s):  
CHRISTIAN BROWN ◽  
SUSANNE PUMPLÜN
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Crossed Product ◽  
Finite Chain ◽  
Central Simple ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
Maximal Subfield ◽  
Algebra Over A Field ◽  
Crossed Product Algebras

AbstractFor any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

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