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Register a new userMultilinear function series in conditionally free probability with amalgamation
متوالية الدوال المتعددة الأبعاد في الاحتمالية الحرة بشرط الإدمج
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Authors
Mihai Popa
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كما في حالات الحرية والاستقلال المونوتوني، يكون معنى الحرية المشروطة معنوياً عند استبدال الحالات المقدرة بالقيم المعقدة بتوقعات مشروطة إيجابية. في هذا الإطار، يقدم البحث عدة نتائج إيجابية، نسخة من قانون النهاية المركزي وتحويل مشروط الحر مبني باستخدام سلاسل وظيفية متعددة الأبعاد.
As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.
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In this paper, we develop the notion of free-Boolean independence in an amalgamation setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence with amalgamation. We also provide a characterization of free-Boolean independence by conditions in terms of mixed moments. In addition, we study free-Boolean independence over a $C^*$-algebra and prove a positivity property.
We study Fourier multipliers on free group $mathbb{F}_infty$ associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative $L^p$ spaces $L^p(hat{mathbb{F}}_infty)$ iff their restriction on $L^p(hat{mathbb{F}}_1)=L^p(mathbb{T})$ are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a $L^p$ Fourier multiplier on free groups for all $1<p<infty$.
We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras and probability measures on $mathbb{R}^m$ are replaced by non-commutative laws of $m$-tuples. We prove an analog of the Monge-Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescus non-commutative $L^2$-Wasserstein distance using a new type of convex functions. As a consequence, we show that if $(X,Y)$ is a pair of optimally coupled $m$-tuples of non-commutative random variables in a tracial $mathrm{W}^*$-algebra $mathcal{A}$, then $mathrm{W}^*((1 - t)X + tY) = mathrm{W}^*(X,Y)$ for all $t in (0,1)$. Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of $m$-tuples is not separable with respect to the Wasserstein distance for $m > 1$.
A free semigroupoid algebra is the closure of the algebra generated by a TCK family of a graph in the weak operator topology. We obtain a structure theory for these algebras analogous to that of free semigroup algebra. We clarify the role of absolute continuity and wandering vectors. These results are applied to obtain a Lebesgue-von Neumann-Wold decomposition of TCK families, along with reflexivity, a Kaplansky density theorem and classification for free semigroupoid algebras. Several classes of examples are discussed and developed, including self-adjoint examples and a classification of atomic free semigroupoid algebras up to unitary equivalence.
We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescus fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescus fully matricial function theory. In the end, we study relations between certain convolutions and transforms in $C^*$-operator valued probability.
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