scholarly journals On a Crossed Product of a Division Ring

1969 ◽  
Vol 35 ◽  
pp. 47-51 ◽  
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Automorphism Group ◽  
Division Ring ◽  
Crossed Product ◽  
Regular Elements ◽  
Factor Set

1. Let R and C be a ring and its center, and G an automorphism group of R of order n. By a factor set {cα,τ}, we mean a system of regular elements cα,τ (α,τ ∈ G) in C such that(1)

scholarly journals The general Ikehata theorem forH-separable crossed products

2000 ◽  
Vol 23 (10) ◽  
pp. 657-662
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Separable Extension ◽  
Factor Set ◽  
Commutative Subring

LetBbe a ring with1,   Cthe center ofB,   Gan automorphism group ofBof ordernfor some integern,   CGthe set of elements inCfixed underG,   Δ=Δ(B,G,f)a crossed product overBwherefis a factor set fromG×GtoU(CG). It is shown thatΔis anH-separable extension ofBandVΔ(B)is a commutative subring ofΔif and only ifCis a Galois algebra overCGwith Galois groupG|C≅G.

scholarly journals On separable noncyclic extensions of rings

1983 ◽  
Vol 34 (3) ◽  
pp. 394-398 ◽  
Author(s):  
George Szeto ◽  
Yuen-Fat Wong
Keyword(s):  
Cyclic Extension ◽  
Crossed Product ◽  
Special Factor ◽  
Factor Set

AbstractThe separable cyclic extension of rings is generalized to a separable noncyclic extension of rings: a crossed product with a factor set over a ring (not necessarily commutative). A representation of separable idempotents for a separable crossed product is obtained, and simplifications for some special factor sets are also given.

scholarly journals Coarse quotients by group actions and the maximal Roe algebra

2019 ◽  
Vol 11 (04) ◽  
pp. 875-907
Author(s):  
Logan Higginbotham ◽  
Thomas Weighill
Keyword(s):  
Automorphism Group ◽  
Metric Space ◽  
Large Scale ◽  
Countable Group ◽  
Crossed Product ◽  
Property A ◽  
Bounded Geometry ◽  
Roe Algebra ◽  
Scale Spaces ◽  
The Ideal

For a finitely generated group [Formula: see text] acting on a metric space [Formula: see text], Roe defined the warped space [Formula: see text], which one can view as a kind of large scale quotient of [Formula: see text] by the action of [Formula: see text]. In this paper, we generalize this notion to the setting of actions of arbitrary groups on large scale spaces. We then restrict our attention to what we call coarsely discontinuous actions by coarse equivalences and show that for such actions the group [Formula: see text] can be recovered as an appropriately defined automorphism group [Formula: see text] when [Formula: see text] satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group [Formula: see text] on a discrete bounded geometry metric space [Formula: see text] there is a relation between the maximal Roe algebras of [Formula: see text] and [Formula: see text], namely that there is a ∗-isomorphism [Formula: see text], where [Formula: see text] is the ideal of compact operators. If [Formula: see text] has Property A and [Formula: see text] is amenable, then [Formula: see text] has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

COCYCLE CONJUGACY OF ONE PARAMETER AUTOMORPHISM GROUPS OF AFD FACTORS OF TYPE III

2002 ◽  
Vol 13 (06) ◽  
pp. 579-603 ◽  
Author(s):  
UN KIT HUI
Keyword(s):  
Automorphism Group ◽  
Von Neumann Algebras ◽  
Automorphism Groups ◽  
Crossed Product ◽  
Von Neumann ◽  
Type Iii ◽  
Finite Dimensional ◽  
Cocycle Conjugacy

We classify, up to cocycle conjugacy, one-parameter automorphism groups on an approximately finite dimensional (AFD) factor ℳ of type III with trivial Connes spectrum. Our goal is to find the complete cocycle conjugacy invariants for one-parameter automorphism groups on ℳ. We also study the relations between the flow of weights of ℳ and that of the crossed product ℳ ⋊α ℝ of ℳ by a one-parameter automorphism group α with Γ(α) = {0}. Moreover, we also study model realizations. "Model realizations" means that given certain commutative data, they can be realized as the complete cocycle conjugacy invariants of centrally free and centrally ergodic one-parameter automorphism groups on some properly infinite AFD von Neumann algebras.

The schur index of a division ring of a crossed product

1995 ◽  
Vol 23 (5) ◽  
pp. 1989-2002
Author(s):  
R. John ◽  
H. Minty
Keyword(s):  
Division Ring ◽  
Crossed Product

scholarly journals On Galois Extension of Rings

1966 ◽  
Vol 27 (1) ◽  
pp. 43-49 ◽  
Author(s):  
Teruo Kanzaki
Keyword(s):  
Finite Group ◽  
Galois Extension ◽  
Crossed Product ◽  
Free Basis ◽  
A Finite Group ◽  
Factor Set

Let Λ be a ring and G a finite group of ring automorphisms of Λ. The totality of elements of Λ which are left invariant by G is a subring of Λ. We call it the G-fixed subring of Λ. Let be the crossed product of Λ and G with trivial factor set, i.e. {u0} is a Λ-free basis of Δ and , and let Γ be a subring of the G-fixed subring of Λ which has the same identity as Λ.

scholarly journals On a Class of Highly Symmetric k-Factorizations

10.37236/2149 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Tommaso Traetta
Keyword(s):  
Automorphism Group ◽  
London Math ◽  
Disjoint Union ◽  
Arbitrary Group ◽  
Full Automorphism Group ◽  
The One ◽  
Group D ◽  
Factor Set ◽  
General Method

A $k$-factorization of $K_v$ of type $(r, s)$ consists of $k$-factors each of which is the disjoint union of $r$ copies of $K_{k+1}$ and $s$ copies of $K_{k,k}$. By means of what we call the patterned $k$-factorization $F_k(D)$ over an arbitrary group $D$ of order $2s + 1$, it is shown that a $k$-factorization of type $(1, s)$ exists for any $k\ge2$ and for any $s\ge1$ with $D$ being an automorphism group acting sharply transitively on the factor-set. The general method to construct a $k$-factorization $F$ of type $(1, s)$ over an arbitrary 1-factorization $S$ of $K_{2s+2}$ ($F$ is said to be based on $S$) is used to prove that the number of pairwise non-isomorphic $k$-factorizations of this type goes to infinity with $s$. In this paper, we show that the full automorphism group of $F$ is known as soon as we know the one of $S$. In particular, the full automorphism group of $F_k(D)$ is determined for any $k\ge2$, generalizing a result given by P. J. Cameron for patterned 1-factorizations [J London Math Soc 11 (1975), 189-201]. Finally, it is shown that $F_k(D)$ has exactly $(k!)2s+1(2s+1)|Aut(D)|$ automorphisms whenever $D$ is abelian.

scholarly journals The universality of Hughes-free division rings

2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Andrei Jaikin-Zapirain
Keyword(s):  
Division Ring ◽  
Amenable Group ◽  
Crossed Product ◽  
Division Rings ◽  
Ring Of Fractions ◽  
Open Question ◽  
Indicable Group

AbstractLet $$E*G$$ E ∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$ E ∗ G -isomorphism, there exists at most one Hughes-free division $$E*G$$ E ∗ G -ring. However, the existence of a Hughes-free division $$E*G$$ E ∗ G -ring $${\mathcal {D}}_{E*G}$$ D E ∗ G for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$ D E ∗ G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$ D E ∗ G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists $${\mathcal {D}}_{E[G]}$$ D E [ G ] and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$ D E [ G ] when G is a RFRS group.

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