scholarly journals EPSILON-STRONGLY GROUPOID-GRADED RINGS, THE PICARD INVERSE CATEGORY AND COHOMOLOGY

2019 ◽  
Vol 62 (1) ◽  
pp. 233-259 ◽  
Author(s):  
PATRIK NYSTEDT ◽  
JOHAN ÖINERT ◽  
HÉCTOR PINEDO
Keyword(s):  
Cohomology Group ◽  
Equivalence Classes ◽  
Crossed Products ◽  
Graded Rings ◽  
Classical Construction

AbstractWe introduce the class of partially invertible modules and show that it is an inverse category which we call the Picard inverse category. We use this category to generalize the classical construction of crossed products to, what we call, generalized epsilon-crossed products and show that these coincide with the class of epsilon-strongly groupoid-graded rings. We then use generalized epsilon-crossed groupoid products to obtain a generalization, from the group-graded situation to the groupoid-graded case, of the bijection from a certain second cohomology group, defined by the grading and the functor from the groupoid in question to the Picard inverse category, to the collection of equivalence classes of rings epsilon-strongly graded by the groupoid.

Moore Cohomology and Central Twisted Crossed Product C*-Algebras

1996 ◽  
Vol 48 (1) ◽  
pp. 159-174 ◽  
Author(s):  
Judith A. Packer
Keyword(s):  
Cohomology Group ◽  
Hausdorff Space ◽  
Countable Group ◽  
Equivalence Classes ◽  
Locally Compact ◽  
C Algebras ◽  
Integer Coefficients ◽  
Direct Sum ◽  
Cech Cohomology ◽  
The Relationship

AbstractLet G be a locally compact second countable group, let X be a locally compact second countable Hausdorff space, and view C(X, T) as a trivial G-module. For G countable discrete abelian, we construct an isomorphism between the Moore cohomology group Hn(G, C(X, T)) and the direct sum Ext(Hn-1(G), Ȟl(βX, Ζ)) ⊕ C(X, Hn(G, T)); here Ȟ1 (βX, Ζ) denotes the first Čech cohomology group of the Stone-Čech compactification of X, βX, with integer coefficients. For more general locally compact second countable groups G, we discuss the relationship between the Moore group H2(G, C(X, T)), the set of exterior equivalence classes of element-wise inner actions of G on the stable continuous trace C*-algebra C0(X) ⊗ 𝒦, and the equivariant Brauer group BrG(X) of Crocker, Kumjian, Raeburn, and Williams. For countable discrete abelian G acting trivially on X, we construct an isomorphism is the group of equivalence classes of principal Ĝ bundles over X first considered by Raeburn and Williams.

scholarly journals Object-unital groupoid graded rings, crossed products and separability

2020 ◽  
pp. 1-21
Author(s):  
Juan Cala ◽  
Patrik Lundström ◽  
Hector Pinedo
Keyword(s):  
Crossed Products ◽  
Graded Rings

Strong Morita Equivalence for Heisenberg C*-Algebras and the Positive Cones of Their K 0-Groups

1988 ◽  
Vol 40 (04) ◽  
pp. 833-864 ◽  
Author(s):  
Judith A. Packer
Keyword(s):  
Heisenberg Group ◽  
Morita Equivalence ◽  
Equivalence Classes ◽  
Crossed Products ◽  
C Algebras ◽  
Projective Representations ◽  
Discrete Heisenberg Group ◽  
Strong Morita Equivalence

In [14] we began a study of C*-algebras corresponding to projective representations of the discrete Heisenberg group, and classified these C*-algebras up to *-isomorphism. In this sequel to [14] we continue the study of these so-called Heisenberg C*-algebras, first concentrating our study on the strong Morita equivalence classes of these C*-algebras. We recall from [14] that a Heisenberg C*-algebra is said to be of class i, i ∊ {1, 2, 3}, if the range of any normalized trace on its K 0 group has rank i as a subgroup of R; results of Curto, Muhly, and Williams [7] on strong Morita equivalence for crossed products along with the methods of [21] and [14] enable us to construct certain strong Morita equivalence bimodules for Heisenberg C*-algebras.

scholarly journals On the symmetric algebra of quotients of a C*-algebra

1990 ◽  
Vol 32 (3) ◽  
pp. 377-379 ◽  
Author(s):  
Pere Ara
Keyword(s):  
Galois Theory ◽  
Semiprime Ring ◽  
Equivalence Classes ◽  
Crossed Products ◽  
Prime Rings ◽  
C Algebra ◽  
Essential Ideal ◽  
Module Homomorphism ◽  
Semiprime Rings ◽  
Ring Of Quotients

Let R be a semiprime ring (possibly without 1). The symmetric ring of quotients of R is defined as the set of equivalence classes of essentially defined double centralizers (ƒ, g) on R; see [1], [8]. So, by definition, ƒ is a left R-module homomorphism from an essential ideal I of R into R, g is a right R-module homomorphism from an essential ideal J of R into R, and they satisfy the balanced condition ƒ(x)y = xg(y) for x ∈ Iand y ∈ J. This ring was used by Kharchenko in his investigations on the Galois theory of semiprime rings [4] and it is also a useful tool for the study of crossed products of prime rings [7]. We denote the symmetric ring of quotients of a semiprime ring R by Q(R).

scholarly journals TWISTS, CROSSED PRODUCTS AND INVERSE SEMIGROUP COHOMOLOGY

2021 ◽  
pp. 1-36
Author(s):  
BENJAMIN STEINBERG
Keyword(s):  
Inverse Semigroup ◽  
Cohomology Group ◽  
Group Structure ◽  
Inverse Semigroups ◽  
Crossed Product ◽  
Crossed Products ◽  
Semigroup Rings ◽  
Abelian Extensions ◽  
Algebraic Setting ◽  
Baer Sum

Abstract Twisted étale groupoid algebras have recently been studied in the algebraic setting by several authors in connection with an abstract theory of Cartan pairs of rings. In this paper we show that extensions of ample groupoids correspond in a precise manner to extensions of Boolean inverse semigroups. In particular, discrete twists over ample groupoids correspond to certain abelian extensions of Boolean inverse semigroups, and we show that they are classified by Lausch’s second cohomology group of an inverse semigroup. The cohomology group structure corresponds to the Baer sum operation on twists. We also define a novel notion of inverse semigroup crossed product, generalizing skew inverse semigroup rings, and prove that twisted Steinberg algebras of Hausdorff ample groupoids are instances of inverse semigroup crossed products. The cocycle defining the crossed product is the same cocycle that classifies the twist in Lausch cohomology.

scholarly journals Simple semigroup graded rings

2015 ◽  
Vol 14 (07) ◽  
pp. 1550102 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Simple Semigroup ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Crossed Products ◽  
Graded Rings ◽  
Unital Ring ◽  
Nonzero Idempotent ◽  
Hypercentral Group ◽  
Necessary And Sufficient ◽  
Skew Group Ring

We show that if R is a, not necessarily unital, ring graded by a semigroup G equipped with an idempotent e such that G is cancellative at e, the nonzero elements of eGe form a hypercentral group and Re has a nonzero idempotent f, then R is simple if and only if it is graded simple and the center of the corner subring f ReGe f is a field. This is a generalization of a result of Jespers' on the simplicity of a unital ring graded by a hypercentral group. We apply our result to partial skew group rings and obtain necessary and sufficient conditions for the simplicity of a, not necessarily unital, partial skew group ring by a hypercentral group. Thereby, we generalize a very recent result of Gonçalves'. We also point out how Jespers' result immediately implies a generalization of a simplicity result, recently obtained by Baraviera, Cortes and Soares, for crossed products by twisted partial actions.

scholarly journals A lower bound on the homological bidimension of a non-unital C*-algebra

1998 ◽  
Vol 40 (3) ◽  
pp. 435-444
Author(s):  
Olaf Ermert
Keyword(s):  
Cohomology Group ◽  
Jacobson Radical ◽  
Hochschild Cohomology ◽  
Equivalence Classes ◽  
Basic Question ◽  
Homological Dimension ◽  
C Algebra ◽  
Wedderburn Decomposition ◽  
Hochschild Cohomology Group ◽  
Closed Subalgebra

Let A be a C*-algebra. For each Banach A-bimodule X, the second continuous Hochschild cohomology group H2(A, X) of A with coefficients in X is defined (see [6]); there is a natural correspondence between the elements of this group and equivalence classes of singular, admissible extensions of A by X. Specifically this means that H2(A, X) ≠ {0} for some X if and only if there exists a Banach algebra B with Jacobson radical R such that R2 = {0}, R is complemented as a Banach space, and B/R ≅ A, but B has no strong Wedderburn decomposition; i.e., there is no closed subalgebra C of B such that B ≅ C © R. In turn this is equivalent to db A ≥ 2, where db A is the homological bidimension of A; i.e., the homological dimension of A#, the unitization of A, as an,A-bimodule [6, III. 5.15]. This paper is concerned with the following basic question, which was posed in [7].

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