scholarly journals Sectional algebras of semigroupoid bundles

2020 ◽  
Vol 30 (06) ◽  
pp. 1257-1304
Author(s):  
Luiz Gustavo Cordeiro
Keyword(s):  
Direct Product ◽  
Tensor Products ◽  
Crossed Product ◽  
Direct Products ◽  
Crossed Products ◽  
Graded Algebras ◽  
Semidirect Products ◽  
Skew Products ◽  
Definition Of ◽  
Product Bundles

In this paper, we use semigroupoids to describe a notion of algebraic bundles, mostly motivated by Fell ([Formula: see text]-algebraic) bundles, and the sectional algebras associated to them. As the main motivational example, Steinberg algebras may be regarded as the sectional algebras of trivial (direct product) bundles. Several theorems which relate geometric and algebraic constructions — via the construction of a sectional algebra — are widely generalized: Direct products bundles by semigroupoids correspond to tensor products of algebras; semidirect products of bundles correspond to “naïve” crossed products of algebras; skew products of graded bundles correspond to smash products of graded algebras; Quotient bundles correspond to quotient algebras. Moreover, most of the results hold in the non-Hausdorff setting. In the course of this work, we generalize the definition of smash products to groupoid graded algebras. As an application, we prove that whenever [Formula: see text] is a ∧-preaction of a discrete inverse semigroupoid [Formula: see text] on an ample (possibly non-Hausdorff) groupoid [Formula: see text], the Steinberg algebra of the associated groupoid of germs is naturally isomorphic to a crossed product of the Steinberg algebra of [Formula: see text] by [Formula: see text]. This is a far-reaching generalization of analogous results which had been proven in particular cases.

scholarly journals Crossed products of Hilbert C*-bimodules by bundles

1998 ◽  
Vol 64 (1) ◽  
pp. 119-135 ◽  
Author(s):  
Tsuyoshi Kajiwara ◽  
Yasuo Watatani
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Tensor Products ◽  
Crossed Product ◽  
General Construction ◽  
Crossed Products ◽  
Finite Group Actions ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Definition Of

AbstractWe present the definition of crossed products of Hilbert C*-bimodules by Hilbert bundles with commuting finite group actions and finite dimensional fibers. This is a general construction containing the bundle construction and crossed products of Hilbert C*-bimodule by finite groups. We also study the structure of endomorphism algebras of the tensor products of bimodules. We also define the multiple crossed products using three bimodules containing more than 2 bundles and show the associativity law. Moreover, we present some examples of crossed product bimodules easily computed by our method.

Semigroup Compactifications of Direct and Semidirect Products

1983 ◽  
Vol 26 (2) ◽  
pp. 233-240 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Classical Result ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Products ◽  
Almost Periodic ◽  
Semidirect Products ◽  
Semigroup Compactifications ◽  
Necessary And Sufficient

AbstractA classical result of I. Glicksberg and K. de Leeuw asserts that the almost periodic compactification of a direct product S × T of abelian semigroups with identity is (canonically isomorphic to) the direct product of the almost periodic compactiflcations of S and T. Some efforts have been made to generalize this result and recently H. D. Junghenn and B. T. Lerner have proved a theorem giving necessary and sufficient conditions for an F-compactification of a semidirect product S⊗σT to be a semidirect product of compactiflcations of S and T. A different such theorem is presented here along with a number of corollaries and examples which illustrate its scope and limitations. Some behaviour that can occur for semidirect products, but not for direct products, is exposed

scholarly journals Crossed Products and Ramification

1966 ◽  
Vol 28 ◽  
pp. 85-111 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Crossed Products ◽  
Quotient Field ◽  
Rank One ◽  
Definition Of

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

scholarly journals COVERINGS OF SKEW-PRODUCTS AND CROSSED PRODUCTS BY COACTIONS

2009 ◽  
Vol 86 (3) ◽  
pp. 379-398 ◽  
Author(s):  
DAVID PASK ◽  
JOHN QUIGG ◽  
AIDAN SIMS
Keyword(s):  
Finite Groups ◽  
Direct Limit ◽  
Projective Limit ◽  
Crossed Product ◽  
Crossed Products ◽  
Skew Products ◽  
Higher Rank

AbstractConsider a projective limitGof finite groupsGn. Fix a compatible familyδnof coactions of theGnon aC*-algebraA. From this data we obtain a coactionδofGonA. We show that the coaction crossed product ofAbyδis isomorphic to a direct limit of the coaction crossed products ofAby theδn. IfA=C*(Λ) for somek-graph Λ, and if the coactionsδncorrespond to skew-products of Λ, then we can say more. We prove that the coaction crossed product ofC*(Λ) byδmay be realized as a full corner of theC*-algebra of a (k+1)-graph. We then explore connections with Yeend’s topological higher-rank graphs and theirC*-algebras.

Products of two-sorted structures

1972 ◽  
Vol 37 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Finite Number ◽  
Direct Product ◽  
Direct Products ◽  
Direct Sums ◽  
First Order ◽  
Order Language ◽  
Universal Equivalence ◽  
Relational Systems ◽  
Multiple Operation ◽  
Definition Of

First order properties of direct products and direct sums (weak direct products) of relational systems have been studied extensively. For example, in Feferman and Vaught [3] an effective procedure is given for reducing such properties of the product to properties of the factors, and thus in particular elementary equivalence is preserved. We consider here two-sorted relational systems, with the direct product and sum operations keeping one of the sorts stationary. (See Feferman [4] for a similar definition of extensions.)These considerations are motivated by the example of direct products and sums of modules [8], [9]. In [9] examples are given to show that the direct product of two modules (even having only a finite number of module elements) does not preserve two-sorted (even universal) equivalence for any finite or infinitary language Lκ, λ. So we restrict attention here to direct powers and multiples (many copies of one structure). Also in [9] it is shown (for modules, but the proofs generalize immediately to two-sorted structures with a finite number of relations) that the direct multiple operation preserves first order ∀E-equivalence and the direct power operation preserves first order ∀-equivalence. We show here that these results for general two-sorted structures in a finite first order language are, in a sense, best-possible. Examples are given to show that does not imply , and that does not imply .

scholarly journals Some prime factorization results for free quantum group factors

2017 ◽  
Vol 2017 (722) ◽  
Author(s):  
Yusuke Isono
Keyword(s):  
Quantum Group ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Crossed Product ◽  
Direct Products ◽  
Unique Factorization ◽  
Prime Factorization ◽  
Von Neumann ◽  
Type Iii ◽  
Original Type

AbstractWe prove some unique factorization results for tensor products of free quantum group factors. They are type III analogues of factorization results for direct products of bi-exact groups established by Ozawa and Popa. In the proof, we first take continuous cores of the tensor products, which satisfy a condition similar to condition (AO), and discuss some factorization properties for the continuous cores. We then deduce factorization properties for the original type III factors. We also prove some unique factorization results for crossed product von Neumann algebras by direct products of bi-exact groups.

The crystallographic point groups as semi-direct products

1963 ◽  
Vol 255 (1054) ◽  
pp. 216-240 ◽  
Keyword(s):  
Direct Product ◽  
Invariant Subgroup ◽  
Projection Operators ◽  
Direct Products ◽  
Finite Rotations ◽  
Space Groups ◽  
Symmetry Adapted Functions ◽  
Point Groups ◽  
Definition Of

This paper aims at providing a systematic treatment of the crystallographic point groups. Some well-known properties of them, in terms of the theory of the poles of finite rotations, are first discussed, so as to provide a simple way for recognizing their invariant subgroups. A definition of the semi-direct product is then given, and it is shown that all crystallographic point groups can be expressed as a semi-direct product of one of their invariant subgroups by a cyclic subgroup. Many useful relations between point groups can be obtained by exploiting the properties of the triple and mixed triple semi-direct products, which are defined. Much of the rest of the paper is devoted to the theory of the representations of semi-direct products. The treatment here parallels that given by Seitz (1936) for the reduction of space groups in terms of the representations of its invariant subgroups (the translation groups). The latter, however, are always Abelian and this is not always the case for point groups. The full treatment of the general case, such as given by McIntosh (1958), is laborious and it is shown that, if the emphasis is placed on the bases of the representations, rather than the representations themselves, it is possible to achieve the reduction of the point groups by a method hardly more involved than that required when the invariant subgroup is Abelian. It is also shown that, just as for space groups, the representations of the invariant subgroups can be denoted and visualized by means of a vector, which allows a very rapid classification of the representations, very much as the k vector as used by Bouckaert, Smoluchowski & Wigner (1936) allows the formalism of the Seitz method for space groups to be carried out in a graphical fashion. One of the major consequences of this work is that it affords a substantial simplification in the use of the symmetrizing and projection operators that are required to obtain symmetry-adapted functions: a very systematic alternative to the method given by Melvin (1956) is therefore provided. In the last section of the paper all the techniques discussed are applied in detail, as an example, to the cubic groups. The projection operators are used to obtain symmetry-adapted spherical harmonics for these groups. The paper might be found useful as an introduction to the methods for the reduction of space groups.

scholarly journals Equivariance and imprimivity for discrete Hopf C*-coactions

2000 ◽  
Vol 62 (2) ◽  
pp. 253-272
Author(s):  
S. Kaliszewski ◽  
John Quigg
Keyword(s):  
Tensor Products ◽  
Crossed Product ◽  
Crossed Products ◽  
Natural Equivalence

Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of SV and ŜV, their restrictions to SW and ŜU, their dual coactions, and their full and reduced crossed products. If N (A) denotes the imprimitivity bimodule associated to a coaction δ of SV on a C*-algebra A by Ng's imprimitivity theorem, we prove that for a suitably nondegenerate injective right-Hilbert bimodule coaction of SV on AXB, the balanced tensor products and are isomorphic right-Hilbert A×ŜV×rSU − B × ŜW bimodules. This can be interpreted as a natural equivalence between certain crossed-product functors.

scholarly journals Hecke C*-algebras and semi-direct products

2009 ◽  
Vol 52 (1) ◽  
pp. 127-153 ◽  
Author(s):  
S. Kaliszewski ◽  
Magnus B. Landstad ◽  
John Quigg
Keyword(s):  
Fixed Point ◽  
Direct Product ◽  
Hecke Algebra ◽  
Quadratic Extension ◽  
Crossed Product ◽  
Direct Products ◽  
C Algebras ◽  
Fixed Point Algebra

AbstractWe analyse Hecke pairs (G,H) and the associated Hecke algebra $\mathcal{H}$ when G is a semi-direct product N ⋊ Q and H = M ⋊ R for subgroups M ⊂ N and R ⊂ Q with M normal in N. Our main result shows that, when (G,H) coincides with its Schlichting completion and R is normal in Q, the closure of $\mathcal{H}$ in C*(G) is Morita–Rieffel equivalent to a crossed product I⋊βQ/R, where I is a certain ideal in the fixed-point algebra C*(N)R. Several concrete examples are given illustrating and applying our techniques, including some involving subgroups of GL(2,K) acting on K2, where K = ℚ or K = ℤ[p−1]. In particular we look at the ax + b group of a quadratic extension of K.

HYPERDECIDABLE PSEUDOVARIETIES AND THE CALCULATION OF SEMIDIRECT PRODUCTS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 241-261 ◽  
Author(s):  
JORGE ALMEIDA
Keyword(s):  
Membership Problem ◽  
Direct Products ◽  
Type Ii ◽  
Locally Finite ◽  
Semidirect Products ◽  
Free Objects ◽  
Group Case ◽  
Definition Of

This note introduces the notion of a hyperdecidable pseudovariety. This notion appears naturally in trying to prove decidability of the membership problem of semi-direct products of pseudovarieties of semigroups. It turns out to be a generalization of a notion introduced by C. J. Ash in connection with his proof of the "type II" theorem. The main results in this paper include a formulation of the definition of a hyper-decidable pseudovariety in terms of free profinite semigroups, the equivalence with Ash's property in the group case, the behaviour under the operator g of taking the associated global pseudovariety of semigroupoids, and the decidability of V*W in case gV is decidable and has a given finite vertex-rank and W is hyperdecidable. A further application of this notion which is given establishes that the join of a hyperdecidable pseudovariety with a locally finite pseudovariety with computable free objects is again hyperdecidable.

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