Pushouts of extensions of groupoids by bundles of abelian groups

We analyse extensions $\Sigma$ of groupoids G by bundles A of abelian groups. We describe a pushout construction for such extensions, and use it to describe the extension group of a given groupoid G by a given bundle A. There is a natural action of Sigma on the dual of A, yielding a corresponding transformation groupoid. The pushout of this transformation groupoid by the natural map from the fibre product of A with its dual to the Cartesian product of the dual with the circle is a twist over the transformation groupoid resulting from the action of G on the dual of A. We prove that the full C*-algebra of this twist is isomorphic to the full C*-algebra of $\Sigma$, and that this isomorphism descends to an isomorphism of reduced algebras. We give a number of examples and applications.

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Invariant Relations and Aschbacher Classes of Finite Linear Groups

The Electronic Journal of Combinatorics ◽

10.37236/712 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 2

Author(s):

Jing Xu ◽

Michael Giudici ◽

Cai Heng Li ◽

Cheryl E. Praeger

Keyword(s):

Positive Integer ◽

Vector Space ◽

Permutation Group ◽

Cartesian Product ◽

Permutation Groups ◽

Linear Groups ◽

Natural Action ◽

Fourth Author ◽

Empty Subset ◽

Finite Linear

For a positive integer $k$, a $k$-relation on a set $\Omega$ is a non-empty subset $\Delta$ of the $k$-fold Cartesian product $\Omega^k$; $\Delta$ is called a $k$-relation for a permutation group $H$ on $\Omega$ if $H$ leaves $\Delta$ invariant setwise. The $k$-closure $H^{(k)}$ of $H$, in the sense of Wielandt, is the largest permutation group $K$ on $\Omega$ such that the set of $k$-relations for $K$ is equal to the set of $k$-relations for $H$. We study $k$-relations for finite semi-linear groups $H\leq{\rm\Gamma L}(d,q)$ in their natural action on the set $\Omega$ of non-zero vectors of the underlying vector space. In particular, for each Aschbacher class ${\mathcal C}$ of geometric subgroups of ${\rm\Gamma L}(d,q)$, we define a subset ${\rm Rel}({\mathcal C})$ of $k$-relations (with $k=1$ or $k=2$) and prove (i) that $H$ lies in ${\mathcal C}$ if and only if $H$ leaves invariant at least one relation in ${\rm Rel}({\mathcal C})$, and (ii) that, if $H$ is maximal among subgroups in ${\mathcal C}$, then an element $g\in{\rm\Gamma L}(d,q)$ lies in the $k$-closure of $H$ if and only if $g$ leaves invariant a single $H$-invariant $k$-relation in ${\rm Rel}({\mathcal C})$ (rather than checking that $g$ leaves invariant all $H$-invariant $k$-relations). Consequently both, or neither, of $H$ and $H^{(k)}\cap{\rm\Gamma L}(d,q)$ lie in ${\mathcal C}$. As an application, we improve a 1992 result of Saxl and the fourth author concerning closures of affine primitive permutation groups.

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Pseudo-cartesian product and hamiltonian decompositions of Cayley graphs on abelian groups

Discrete Mathematics ◽

10.1016/0012-365x(95)00035-u ◽

1996 ◽

Vol 158 (1-3) ◽

pp. 49-62 ◽

Cited By ~ 10

Author(s):

Cong Fan ◽

Don R. Lick ◽

Jiuqiang Liu

Keyword(s):

Abelian Groups ◽

Cayley Graphs ◽

Cartesian Product ◽

Hamiltonian Decompositions

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On a dilation of a some class of completely positive maps

Tambov University Reports Series Natural and Technical Sciences ◽

10.20310/2686-9667-2019-24-127-333-339 ◽

2019 ◽

pp. 333-339

Author(s):

Yakub V. ELSAEV

Keyword(s):

Cartesian Product ◽

Group Symmetry ◽

Quantum Field ◽

Completely Positive Maps ◽

Completely Positive ◽

Sesquilinear Forms ◽

C Algebra ◽

Covariant Quantum ◽

Positive Maps ◽

Minimal Representations

In this article we investigate sesquilinear forms defined on the Cartesian product of Hilbert C^*-module M over C^*-algebra B and taking values in B. The set of all such defined sesquilinear forms is denoted by S_B (M). We consider completely positive maps from locally C^*-algebra A to S_B (M). Moreover we assume that these completely positive maps are covariant with respect to actions of a group symmetry. This allow us to view these maps as generalizations covariant quantum instruments which are very important for the modern quantum mechanic and the quantum field theory. We analyze the dilation problem for these class of maps. In order to solve this problem we construct the minimal Stinespring representation and prove that every two minimal representations are unitarily equivalent.

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Extension of partial endomorphisms of abelian groups

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500034699 ◽

1963 ◽

Vol 6 (1) ◽

pp. 45-48 ◽

Cited By ~ 2

Author(s):

C. G. Chehata

Keyword(s):

Abelian Group ◽

Positive Integer ◽

Abelian Groups ◽

Normal Subgroups ◽

Extension Group ◽

Image Position ◽

Necessary And Sufficient

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroupssuch that L1 ƞA is the kernel of μ andfor ι = 1, 2,…, m–1.The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.

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Fibre product approach to index pairings for the generic Hopf fibration of SUq(2)

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is010002019jkt110 ◽

2010 ◽

Vol 7 (1) ◽

pp. 1-17 ◽

Cited By ~ 2

Author(s):

Elmar Wagner

Keyword(s):

Line Bundles ◽

Hopf Fibration ◽

C Algebra ◽

Product Construction ◽

Fibre Product ◽

Product Approach

AbstractA fibre product construction is used to give a description of quantum line bundles over the generic Podlés spheres obtained by gluing two quantum discs along their boundaries. Representatives of the corresponding K0-classes are given in terms of 1-dimensional projections belonging to the C*-algebra, and in terms of analogues of the classical Bott projections. The K0-classes of quantum line bundles derived from the generic Hopf fibration of quantum SU(2) are determined and the index pairing is computed. It is argued that taking the projections obtained from the fibre product construction yields a significant simplification of earlier index computations.

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CARTESIAN PRODUCT ON FUZZY IDEALS OF A TERNARY Γ-SEMIGROUP

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.3.44 ◽

2020 ◽

Vol 9 (3) ◽

pp. 1189-1195 ◽

Cited By ~ 1

Author(s):

Y. Bhargavi ◽

T. Eswarlal ◽

S. Ragamayi

Keyword(s):

Cartesian Product

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A new method for approximation of closed convex subsets of B(H)

Filomat ◽

10.2298/fil1719005i ◽

2017 ◽

Vol 31 (19) ◽

pp. 6005-6013

Author(s):

Mahdi Iranmanesh ◽

Fatemeh Soleimany

Keyword(s):

Best Approximation ◽

Numerical Range ◽

New Method ◽

C Algebra ◽

Convex Subsets

In this paper we use the concept of numerical range to characterize best approximation points in closed convex subsets of B(H): Finally by using this method we give also a useful characterization of best approximation in closed convex subsets of a C*-algebra A.

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