scholarly journals Two-sided crossed products of groups

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 1005-1012 ◽  
Author(s):  
Esra Cetinalp ◽  
Eylem Karpuz ◽  
Firat Ates ◽  
Sinan Cevik
Keyword(s):  
Normal Forms ◽  
New Product ◽  
Crossed Product ◽  
Crossed Products ◽  
Separate Section ◽  
Cyclic Groups ◽  
Rewriting System ◽  
Products Of Groups ◽  
Group Construction ◽  
Product Of Groups

In this paper, we first define a new version of the crossed product of groups under the name of two-sided crossed product. Then we present a generating and relator sets for this new product over cyclic groups. In a separate section, by using the monoid presentation of the two-sided crossed product of cyclic groups, we obtain the complete rewriting system and normal forms of elements of this new group construction.

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 The palindromic width of a free product of groups

2006 ◽  
Vol 81 (2) ◽  
pp. 199-208 ◽  
Author(s):  
Valery Bardakov ◽  
Vladimir Tolstykh
Keyword(s):  
Free Product ◽  
Free Groups ◽  
Uniform Bound ◽  
Free Products ◽  
Cyclic Groups ◽  
Similar Fact ◽  
Products Of Groups ◽  
Free Product Of Groups ◽  
Verbal Subgroups ◽  
Product Of Groups

AbstractPalindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups G has infinite palindromic width, provided that G is not the free product of two cyclic groups of order two (Theorem 2.4). This means that there is no uniform bound k such that every element of G is a product of at most k palindromes. Earlier, the similar fact was established for non-abelian free groups. The proof of Theorem 2.4 makes use of the ideas by Rhemtulla developed for the study of the widths of verbal subgroups of free products.

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.

scholarly journals Supramenable groups and partial actions

2016 ◽  
Vol 37 (5) ◽  
pp. 1592-1606 ◽  
Author(s):  
EDUARDO P. SCARPARO
Keyword(s):  
Semidirect Product ◽  
Crossed Product ◽  
Probability Measures ◽  
Crossed Products ◽  
Hausdorff Spaces ◽  
Tracial States ◽  
Product Of Groups

We characterize supramenable groups in terms of the existence of invariant probability measures for partial actions on compact Hausdorff spaces and the existence of tracial states on partial crossed products. These characterizations show that, in general, one cannot decompose a partial crossed product of a $\text{C}^{\ast }$-algebra by a semidirect product of groups into two iterated partial crossed products. However, we give conditions which ensure that such decomposition is possible.

scholarly journals The Dixmier-Douady class of groupoid crossed products

2004 ◽  
Vol 76 (2) ◽  
pp. 223-234 ◽  
Author(s):  
Paul S. Muhly ◽  
Dana P. Williams
Keyword(s):  
Crossed Product ◽  
Crossed Products ◽  
Continuous Trace

AbstractWe give a formula for the Dixmier-Douady class of a continuous-trace groupoid crossed product that arises from an action of a locally trivial, proper, principal groupoid on a bundle of elementary C*-algebras that satisfies Fell's condition.

scholarly journals A rigidity result for crossed products of actions of Baumslag–Solitar groups

2015 ◽  
Vol 26 (14) ◽  
pp. 1550117
Author(s):  
Niels Meesschaert
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Crossed Product ◽  
Crossed Products ◽  
Orbit Equivalence ◽  
Rigidity Result ◽  
Profinite Completions ◽  
Abelian Subgroups ◽  
Measure Equivalence ◽  
Measure Preserving Actions

Let [Formula: see text] and [Formula: see text] be two ergodic essentially free probability measure preserving actions of nonamenable Baumslag–Solitar groups whose canonical almost normal abelian subgroups act aperiodically. We prove that an isomorphism between the corresponding crossed product II1 factors forces [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. This improves an orbit equivalence rigidity result obtained by Houdayer and Raum in [Baumslag–Solitar groups, relative profinite completions and measure equivalence rigidity, J. Topol. 8 (2015) 295–313].

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.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

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 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.

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