scholarly journals Commutativity of the Haagerup tensor product and base change for operator modules

2021 ◽  
pp. 1-11
Author(s):  
Tyrone Crisp
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Base Change ◽  
Product Formula ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Mappings ◽  
Continuous Fields ◽  
Bounded Norm

By computing the completely bounded norm of the flip map on the Haagerup tensor product [Formula: see text] associated to a pair of continuous mappings of locally compact Hausdorff spaces [Formula: see text], we establish a simple characterization of the Beck-Chevalley condition for base change of operator modules over commutative [Formula: see text]-algebras, and a descent theorem for continuous fields of Hilbert spaces.

scholarly journals Completely bounded bimodule maps and spectral synthesis

2017 ◽  
Vol 28 (10) ◽  
pp. 1750067 ◽  
Author(s):  
M. Alaghmandan ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Closed Subset ◽  
Spectral Synthesis ◽  
Locally Compact Group ◽  
Product Formula ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

A Compactification due to Fell

1972 ◽  
Vol 15 (1) ◽  
pp. 145-146 ◽  
Author(s):  
Aubrey Wulfsohn
Keyword(s):  
Fundamental System ◽  
Point Of View ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Alternative Construction ◽  
Continuous Fields ◽  
One Point Compactification

We give an alternative construction of a Hausdorff compactification due to Fell [2]. We say that a space is compact if it has the Heine-Borel property, locally compact if each point has a fundamental system of compact neighbourhoods. The interesting spaces from the point of view of this paper, are the non-Hausdorjf ones since for locally compact Hausdorff spaces Fell's compactification is the usual one-point compactification. The motivation for the compactification comes from the theory of continuous fields of C*-algebras: the primitive spectrum of a C*- algebra A is a locally compact T0 space X and Fell [3] realizes A as an algebra of fields of operators over the compactification of X. This note is based on a discussion of the author with Professor Fell.

scholarly journals A COMPLETELY ENTANGLED SUBSPACE OF MAXIMAL DIMENSION

2006 ◽  
Vol 04 (02) ◽  
pp. 325-330 ◽  
Author(s):  
B. V. RAJARAMA BHAT
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Product Formula ◽  
Maximal Dimension ◽  
Maximum Dimension ◽  
Dimension Formula ◽  
Finite Dimensional ◽  
Minimum Dimension ◽  
Orthogonal Product ◽  
Product Vector

Consider a tensor product [Formula: see text] of finite-dimensional Hilbert spaces with dimension [Formula: see text], 1 ≤ i ≤ k. Then the maximum dimension possible for a subspace of [Formula: see text] with no non-zero product vector is known to be d1 d2…dk - (d1 + d2 + … + dk + k - 1. We obtain an explicit example of a subspace of this kind. We determine the set of product vectors in its orthogonal complement and show that it has the minimum dimension possible for an unextendible product basis of not necessarily orthogonal product vectors.

scholarly journals On the Tensor Convolution and the Quantum Separability Problem

2010 ◽  
Vol 17 (04) ◽  
pp. 331-346
Author(s):  
Gabriel Pietrzkowski
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Hermitian Operator ◽  
Product Formula ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Equivalent Problem ◽  
Finite Dimensional ◽  
Separability Problem ◽  
Convolution Formula

We consider the problem of separability: decide whether a Hermitian operator on a finite dimensional Hilbert tensor product [Formula: see text] is separable or entangled. We show that the tensor convolution [Formula: see text] defined for mappings [Formula: see text] on an almost arbitrary locally compact abelian group G , gives rise to formulation of an equivalent problem to the separability one.

A FEW HOMOLOGICAL CHARACTERIZATIONS OF COMPACT SEMISIMPLE RINGS

2005 ◽  
Vol 04 (05) ◽  
pp. 539-549
Author(s):  
ALINA ALB ◽  
MIHAIL URSUL
Keyword(s):  
Abelian Group ◽  
Tensor Product ◽  
Compact Abelian Group ◽  
Product Formula ◽  
Flat Module ◽  
Compact Ring ◽  
Module Theory ◽  
Flat Modules ◽  
Semisimple Module

Fix any compact ring R with identity. We associate to R the following categories of topological R-modules: (i) R𝔇 (𝔇R) the category of all discrete topological left (right) R-modules; (ii) Rℭ (ℭR) the category of all compact left (right) R-modules. We have introduced the following notions (analogous with classical notions of module theory): (i) the tensor product [Formula: see text] of A ∈ ℭR and B ∈Rℭ ([Formula: see text] has a structure of a compact Abelian group); (ii) a topologically semisimple module; (iii) a compact topologically flat module. We give a characterization of compact semisimple rings by using of flat modules.

scholarly journals CLOSED IDEALS AND LIE IDEALS OF MINIMAL TENSOR PRODUCT OF CERTAIN C*-ALGEBRAS

2020 ◽  
pp. 1-12
Author(s):  
BHARAT TALWAR ◽  
RANJANA JAIN
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hausdorff Space ◽  
Separable Hilbert Space ◽  
Locally Compact ◽  
C Algebras ◽  
Finite Sum ◽  
Locally Compact Hausdorff Space ◽  
Quotient Property

Abstract For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗minA is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.

LOCALLY COMPACT GROUPS ADMITTING FAITHFUL STRONGLY CHAOTIC ACTIONS ON HAUSDORFF SPACES

2013 ◽  
Vol 23 (09) ◽  
pp. 1350158 ◽  
Author(s):  
FRIEDRICH MARTIN SCHNEIDER ◽  
SEBASTIAN KERKHOFF ◽  
MIKE BEHRISCH ◽  
STEFAN SIEGMUND
Keyword(s):  
Hausdorff Space ◽  
Geometric Characterization ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Action

In this paper we provide a geometric characterization of those locally compact Hausdorff topological groups which admit a faithful strongly chaotic continuous action on some Hausdorff space.

The Duality of Distributive Continuous Lattices

1980 ◽  
Vol 32 (2) ◽  
pp. 385-394 ◽  
Author(s):  
B. Banaschewski
Keyword(s):  
Dual Space ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
Continuous Lattice ◽  
Prime Spectrum ◽  
Hausdorff Spaces ◽  
Locally Compact ◽  
Continuous Lattices ◽  
Prime Elements

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

scholarly journals The core of a weakly group-theoretical braided fusion category

2018 ◽  
Vol 29 (02) ◽  
pp. 1850012 ◽  
Author(s):  
Sonia Natale
Keyword(s):  
Tensor Product ◽  
Product Formula ◽  
Fusion Category ◽  
Braided Category ◽  
The Core ◽  
Modular Category ◽  
Braided Fusion Category

We show that the core of a weakly group-theoretical braided fusion category [Formula: see text] is equivalent as a braided fusion category to a tensor product [Formula: see text], where [Formula: see text] is a pointed weakly anisotropic braided fusion category, and [Formula: see text] or [Formula: see text] is an Ising braided category. In particular, if [Formula: see text] is integral, then its core is a pointed weakly anisotropic braided fusion category. As an application we give a characterization of the solvability of a weakly group-theoretical braided fusion category. We also prove that an integral modular category all of whose simple objects have Frobenius–Perron dimension at most 2 is necessarily group-theoretical.

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