THE FIVE INDEPENDENCES AS QUASI-UNIVERSAL PRODUCTS

2002 ◽  
Vol 05 (01) ◽  
pp. 113-134 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Tensor Product ◽  
Free Product ◽  
Boolean Product ◽  
Probability Spaces

A notion of "quasi-universal product" for algebraic probability spaces is introduced as a generalization of Speicher's "universal product". It is proved that there exist only five quasi-universal products, namely, tensor product, free product, Boolean product, monotone product and anti-monotone product. This result means that, in a sense, there exist only five independences which have nice properties of "associativity" and "(quasi-)universality".

THE FIVE INDEPENDENCES AS NATURAL PRODUCTS

2003 ◽  
Vol 06 (03) ◽  
pp. 337-371 ◽  
Author(s):  
NAOFUMI MURAKI
Keyword(s):  
Probability Theory ◽  
Natural Products ◽  
Tensor Product ◽  
Natural Product ◽  
Free Product ◽  
Noncommutative Probability ◽  
Stochastic Independence ◽  
Boolean Product ◽  
Probability Spaces ◽  
Noncommutative Probability Theory

Let [Formula: see text] be the class of all algebraic probability spaces. A "natural product" is, by definition, a map [Formula: see text] which is required to satisfy all the canonical axioms of Ben Ghorbal and Schürmann for "universal product" except for the commutativity axiom. We show that there exist only five natural products, namely tensor product, free product, Boolean product, monotone product and anti-monotone product. This means that, in a sense, there exist only five universal notions of stochastic independence in noncommutative probability theory.

scholarly journals Tensor Product Decompositions of II1 Factors Arising from Extensions of Amalgamated Free Product Groups

2018 ◽  
Vol 364 (3) ◽  
pp. 1163-1194 ◽  
Author(s):  
Ionut Chifan ◽  
Rolando de Santiago ◽  
Wanchalerm Sucpikarnon
Keyword(s):  
Tensor Product ◽  
Free Product ◽  
Amalgamated Free Product

scholarly journals Weak amenability for subfactors

2015 ◽  
Vol 26 (07) ◽  
pp. 1550048 ◽  
Author(s):  
Arnaud Brothier
Keyword(s):  
Tensor Product ◽  
Free Product ◽  
Finite Index ◽  
Finite Family ◽  
Type Ii ◽  
Weak Amenability

We define the notions of weak amenability and the Cowling–Haagerup constant for extremal finite index subfactors s of type II1. We prove that the Cowling–Haagerup constant only depends on the standard invariant of the subfactor. Hence, we define the Cowling–Haagerup constant for standard invariants. We explicitly compute the constant for Bisch–Haagerup subfactors and prove that it is equal to the constant of the group involved in the construction. Given a finite family of amenable standard invariants, we prove that their free product in the sense of Bisch–Jones is weakly amenable with constant 1. We show that the Cowling–Haagerup of the tensor product of a finite family of standard invariants is equal to the product of their Cowling–Haagerup constants.

scholarly journals Unification of Independence in Quantum Probability

1998 ◽  
Vol 01 (03) ◽  
pp. 383-405 ◽  
Author(s):  
Romuald Lenczewski
Keyword(s):  
Free Product ◽  
Tensor Products ◽  
Quantum Probability ◽  
Product Formula ◽  
Free Algebras ◽  
Noncommutative Probability ◽  
Free Variables ◽  
Probability Spaces

Let [Formula: see text], be the conditionally free product of unital free *-algebras [Formula: see text], where ϕl, ψl are states on [Formula: see text], l∈I. We construct a sequence of noncommutative probability spaces [Formula: see text], m∈N, where [Formula: see text] and [Formula: see text], m∈N, [Formula: see text], and the states [Formula: see text], ϕl are Boolean extensions of ϕl, ψl, l∈I, respectively. We define unital *-homomorphisms [Formula: see text] such that [Formula: see text] converges pointwise to *l∈I(ϕl,ψl). Thus, the variables j(m)(w), where w is a word in [Formula: see text], converge in law to the conditionally free variables. The sequence of noncommutative probability spaces [Formula: see text], where [Formula: see text] and Φ(m) is the restriction of [Formula: see text] to [Formula: see text], is called a hierarchy of freeness. Since all finite joint correlations for known examples of independence can be obtained from tensor products of appropriate *-algebras, this approach can be viewed as a unification of independence. Finally, we show how to make the m-fold free product [Formula: see text] into a cocommutative *-bialgebra associated with m-freeness.

Reference Signal Based Tensor Product Expansion for EOG-Related Artifact Separation in EEG

2017 ◽  
Vol E100.A (11) ◽  
pp. 2230-2237
Author(s):  
Akitoshi ITAI ◽  
Arao FUNASE ◽  
Andrzej CICHOCKI ◽  
Hiroshi YASUKAWA
Keyword(s):  
Tensor Product ◽  
Reference Signal

On degeneracy problem of NFSRs via semi-tensor product

2020 ◽  
Author(s):  
Xinyu Zhao ◽  
Biao Wang ◽  
Shuqian Zhu ◽  
Jun-e Feng
Keyword(s):  
Tensor Product ◽  
Degeneracy Problem
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