scholarly journals Tensor Products and Transferability of Semilattices

1999 ◽  
Vol 51 (4) ◽  
pp. 792-815 ◽  
Author(s):  
G. Grätzer ◽  
F. Wehrung
Keyword(s):  
Tensor Product ◽  
Finite Lattice ◽  
Tensor Products ◽  
Locally Finite ◽  
General Lattice ◽  
Effective Condition ◽  
Transferable Lattice

AbstractIn general, the tensor product, A ⊗ B, of the lattices A and B with zero is not a lattice (it is only a join-semilattice with zero). If A ⊗ B is a capped tensor product, then A ⊗ B is a lattice (the converse is not known). In this paper, we investigate lattices A with zero enjoying the property that A ⊗ B is a capped tensor product, for every lattice B with zero; we shall call such lattices amenable.The first author introduced in 1966 the concept of a sharply transferable lattice. In 1972, H. Gaskill defined, similarly, sharply transferable semilattices, and characterized them by a very effective condition (T).We prove that a finite lattice A is amenable iff it is sharply transferable as a join-semilattice.For a general lattice A with zero, we obtain the result: A is amenable iff A is locally finite and every finite sublattice of A is transferable as a join-semilattice.This yields, for example, that a finite lattice A is amenable iff A ⊗ F(3) is a lattice iff A satisfies (T), with respect to join. In particular, M3 ⊗ F(3) is not a lattice. This solves a problem raised by R. W. Quackenbush in 1985 whether the tensor product of lattices with zero is always a lattice.

Finite Lattices of Projections in Factors and Approximately Finite C*-Algebras

1992 ◽  
Vol 35 (1) ◽  
pp. 116-125
Author(s):  
S. C. Power
Keyword(s):  
Tensor Product ◽  
General Result ◽  
Finite Lattice ◽  
Tensor Products ◽  
C Algebras ◽  
Finite Lattices

AbstractA unique factorisation theorem is obtained for tensor products of finite lattices of commuting projections in a factor. This leads to unique tensor product factorisations for reflexive subalgebras of the hyperfinite II1 factor which have irreducible finite commutative invariant projection lattices. It is shown that the finite refinement property fails for simple approximately finite C*-algebras, and this implies that there is no analogous general result for finite lattice subalgebras in this context.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

Extension of normal functionals on W*-tensor products

1975 ◽  
Vol 78 (2) ◽  
pp. 301-307 ◽  
Author(s):  
Simon Wassermann
Keyword(s):  
Representation Theory ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
General Result ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Hilbert Algebras ◽  
Special Cases

A deep result in the theory of W*-tensor products, the Commutation theorem, states that if M and N are W*-algebras faithfully represented as von Neumann algebras on the Hilbert spaces H and K, respectively, then the commutant in L(H ⊗ K) of the W*-tensor product of M and N coincides with the W*-tensor product of M′ and N′. Although special cases of this theorem were established successively by Misonou (2) and Sakai (3), the validity of the general result remained conjectural until the advent of the Tomita-Takesaki theory of Modular Hilbert algebras (6). As formulated, the Commutation theorem is a spatial result; that is, the W*-algebras in its statement are taken to act on specific Hilbert spaces. Not surprisingly, therefore, known proofs rely heavily on techniques of representation theory.

Noetherian Tensor Products

1972 ◽  
Vol 15 (2) ◽  
pp. 235-238
Author(s):  
E. A. Magarian ◽  
J. L. Motto
Keyword(s):  
Tensor Product ◽  
Jacobson Radical ◽  
Tensor Products ◽  
Minimum Condition ◽  
Ideal Structure ◽  
The Ideal

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

scholarly journals Tensor Products and Bimorphisms

1976 ◽  
Vol 19 (4) ◽  
pp. 385-402 ◽  
Author(s):  
Bernhard Banaschewski ◽  
Evelyn Nelson
Keyword(s):  
Tensor Product ◽  
Commutative Ring ◽  
Tensor Products ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bilinear Maps ◽  
Special Situation ◽  
Dual Spaces ◽  
Finite Dimensional

The binary tensor product, for modules over a commutative ring, has two different aspects: its connection with universal bilinear maps and its adjointness to the internal hom-functor. Furthermore, in the special situation of finite-dimensional vector spaces, the tensor product can also be described in terms of dual spaces and the internal hom-functor. The aim of this paper is to investigate these relationships in the setting of arbitrary concrete categories.

scholarly journals On the Structure of Monodromy Algebras and Drinfeld Doubles

1997 ◽  
Vol 09 (03) ◽  
pp. 371-395
Author(s):  
Florian Nill
Keyword(s):  
Hopf Algebra ◽  
Tensor Product ◽  
Tensor Products ◽  
Spin Chains ◽  
Coadjoint Action ◽  
Braided Group ◽  
Gauge Transformations ◽  
Loop Algebras ◽  
Algebraic Tensor Product ◽  
Quasitriangular Hopf Algebra

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double [Formula: see text]. Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

scholarly journals NuclearJC-algebras and tensor products of types

1993 ◽  
Vol 16 (4) ◽  
pp. 717-723
Author(s):  
Fatmah B. Jamjoom
Keyword(s):  
Tensor Product ◽  
Tensor Products

This article is a continuation of [1], to which the reader is referred for the definition and properties of theJC-tensor product of twoJC-algebras. Our standard references for nuclear and postliminalC*-algebras are[2,3,4,5,6,7]. We extend the notion of nuclearity toJC-algebras and prove that postliminalJC-algebras are nuclear. In contrast with the situation which occurs forC*-algebras, theJC-tensor product of two postliminalJC-algebras turns out, in general, to be non-postliminal and can even be anitliminal.

An approximation property for operator systems

2019 ◽  
Vol 22 (01) ◽  
pp. 1950005
Author(s):  
Wei Wu
Keyword(s):  
Tensor Product ◽  
Convex Sets ◽  
Approximation Property ◽  
Tensor Products ◽  
Order Unit ◽  
Operator System ◽  
Operator Systems ◽  
Natural Sense

Motivated by an observation of Namioka and Phelps on an approximation property of order unit spaces, we introduce the [Formula: see text]-tensor product and the [Formula: see text]-tensor product of two compact matrix convex sets. We define a new approximation property for operator systems, and give a characterization using the [Formula: see text]- and [Formula: see text]-tensor products in the spirit of Grothendieck. Thus, an operator system has the operator system approximation property if and only if it is [Formula: see text]-nuclear in a natural sense.

scholarly journals Counterexamples in intersections for C*-tensor products

1984 ◽  
Vol 27 (3) ◽  
pp. 301-302 ◽  
Author(s):  
Seung-Hyeok Kye
Keyword(s):  
Tensor Product ◽  
Short Note ◽  
Tensor Products

Let A and B be C*-algebras and A ⊗ B denote the minimal C*-tensor product of A and B. T. Huruya [1] gave examples of C*-tensor products A ⊗ B with C*-subalgebras A1⊗B2 and A2⊗B2 such that strictly contains , answering a question of S. Wassermann [3, Remark 23]. In this short note, we show that the same situation can occur even if A1 = A2

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