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.

COCYCLE CONJUGACY OF ONE PARAMETER AUTOMORPHISM GROUPS OF AFD FACTORS OF TYPE III

2002 ◽  
Vol 13 (06) ◽  
pp. 579-603 ◽  
Author(s):  
UN KIT HUI
Keyword(s):  
Automorphism Group ◽  
Von Neumann Algebras ◽  
Automorphism Groups ◽  
Crossed Product ◽  
Von Neumann ◽  
Type Iii ◽  
Finite Dimensional ◽  
Cocycle Conjugacy

We classify, up to cocycle conjugacy, one-parameter automorphism groups on an approximately finite dimensional (AFD) factor ℳ of type III with trivial Connes spectrum. Our goal is to find the complete cocycle conjugacy invariants for one-parameter automorphism groups on ℳ. We also study the relations between the flow of weights of ℳ and that of the crossed product ℳ ⋊α ℝ of ℳ by a one-parameter automorphism group α with Γ(α) = {0}. Moreover, we also study model realizations. "Model realizations" means that given certain commutative data, they can be realized as the complete cocycle conjugacy invariants of centrally free and centrally ergodic one-parameter automorphism groups on some properly infinite AFD von Neumann algebras.

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.

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.

On tensor products of von Neumann algebras

1996 ◽  
Vol 123 (1) ◽  
pp. 453-466 ◽  
Author(s):  
L. Ge ◽  
R. Kadison
Keyword(s):  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann

Ergodic Actions of Compact Groups on Operator Algebras II: Classification of Full Multiplicity Ergodic Actions

1988 ◽  
Vol 40 (6) ◽  
pp. 1482-1527 ◽  
Author(s):  
Antony Wassermann
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Crossed Product ◽  
Ergodic Action ◽  
Compact Groups ◽  
Type I ◽  
List Type ◽  
Von Neumann ◽  
C Algebra ◽  
Set Up

In the first paper of this series [17], we set up some general machinery for studying ergodic actions of compact groups on von Neumann algebras, namely, those actions for which . In particular we obtained a characterisation of the full multiplicity ergodic actions:THEOREM A. If α is an ergodic action of G on , then the following conditions are equivalent:(1) Each spectral subspace has multiplicity dim π for π in .(2) Each π in admits a unitary eigenmatrix in .(3) The W* crossed product is a (Type I) factor.(4) The C* crossed product of the C* algebra of norm continuity is isomorphic to the algebra of compact operators on a Hilbert space.

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