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Article overview
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$*$-freeness in Finite Tensor Products | Benoit Collins
; Pierre Yves Gaudreau Lamarre
; | Date: |
1 May 2016 | Abstract: | In this paper, we consider the following question and variants thereof: given
$mathbf D:=ig(a_{1;i}otimescdotsotimes a_{K;i}:iin Iig)$, a collection
of elementary tensor non-commutative random variables in the tensor product of
probability spaces $(mathcal A_1otimescdotsotimesmathcal
A_K,phi_1otimescdotsotimesphi_K)$, when is $mathbf D$ $*$-free? (See
Section 1.2 for a precise formulation of this problem.)
Settling whether or not freeness occurs in tensor products is a recurring
problem in operator algebras, and the following two examples provide a natural
motivation for the above question:
(A) If $(a_{1;i}:iin I)$ is a $*$-free family of Haar unitary variables and
$a_{k,i}$ are arbitrary unitary variables for $kgeq2$, then the $*$-freeness
persists at the level of the tensor product $mathbf D$.
(B) A converse of (A) holds true if all variables $a_{k;i}$ are group-like
elements (see Corollary 1.7 of Proposition 1.6).
It is therefore natural to seek to understand the extent to which such simple
characterizations hold true in more general cases. While our results fall short
of a complete characterization, we make notable steps toward identifying
necessary and sufficient conditions for the freeness of $mathbf D$. For
example, we show that under evident assumptions, if more than one family
$(a_{k,i}:iin I)$ contains non-unitary variables, then the tensor family fails
to be $*$-free (see Theorem 1.8 (1)). | Source: | arXiv, 1605.0288 | Services: | Forum | Review | PDF | Favorites |
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