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Banach-Space Operators On C*-Probability Spaces Generated by Multi Semicircular Elements
Ilwoo Cho
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eBook - ePub
Banach-Space Operators On C*-Probability Spaces Generated by Multi Semicircular Elements
Ilwoo Cho
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Banach-Space Operators On C*-Probability Spaces Generated by Multi Semicircular Elements introduces new areas in operator theory and operator algebra, in connection with free probability theory. In particular, the book considers projections and partial isometries distorting original free-distributional data on the C?-probability spaces.
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- Suitable for graduate students and professional researchers in operator theory and/or analysis.
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- Numerous applications in related scientific fields and areas
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Información
Chapter 1 Introduction
DOI: 10.1201/9781003263487-1
The main purposes of this monograph are (i) to consider the C*-probability space 𝔛 = (𝔛, φ) generated by the free semicircular family X = {xj}j∈ℤ, and its free-probabilistic substructures, 𝔛N = (𝔛N , φN ) generated by free semicircular sub-families , for
(ii) to construct suitable ∗-isomorphisms λ = {βk}k∈ℤ acting on 𝔛, and their restrictions λN = {βk |𝔛N}k∈ℤ, which are ∗-homomorphisms acting on 𝔛N, and the adjointable Banach-space operators generated by λ and λN, respectively, for , (iii) to study certain types of adjointable Banach-space operators of (ii), and to consider how they deform the free probability on 𝔛, and those on 𝔛N, (iv) to characterize the deformations on the C∗-probability spaces from the Banach-space operators of (iii), and (v) to study dynamics induced by our Banach-space operators.
In particular, we are interested in certain types of “adjointable” Banach-space operators, induced by λN acting on 𝔛N, especially, where they are projections or partial isometries. Different from the Banach-space operators induced by λ acting on 𝔛, such operators from λN distort the free probability on 𝔛N, in general. We characterize such distortions on 𝔛N occurred by projections, and partial isometries.
1.1 Motivation and Background
The study of semicircular elements is one of the major topics (e.g., [20,21,29,30]), not only in both commutative function theory and noncommutative free probability theory but also in various applied fields, including quantum statistical physics (e.g., [5,6,7 and 8,10,11,12]). The semicircular law, which is the free distributions of semicircular elements, is well characterized analytically, and combinatorially, in classical function theory and in free probability theory (e.g., [1,17,18,21,28,29 and 30]). In particular, it is playing a key role in free-probabilistic operator algebra theory (and hence, in quantum physics) by the (free) central limit theorem(s) (e.g., see [2,17,19,28,29 and 30]), i.e., it becomes a noncommutative analytic analogue of the classical Gaussian (or the normal) distribution (in commutative function theory).
From combinatorial approaches (e.g., [17,22,23]), the semicircular law is universally characterized by the Catalan numbers , where
for all n ∈ ℕ0 = ℕ ∪ {0}. i.e., the semicircular law is characterized by the free-moment sequence,
(1.1.1)
where
and ck are the Catalan numbers, for all k ∈ ℕ.
From the analysis on p-adic number fields ℚp (e.g., [26,27]), one can construct semicircular elements (e.g., [5,12]). By generalizing the constructions of [5,12], semicircular elements are constructed whenever there are |ℤ|-many orthogonal projections in a C∗-algebra (e.g., [6,7 and 8,10] and [11]), different from earlier works (e.g., [20,25,29,30]). In this new approach, the semicircular elements are understood as Banach-space operators acting on a given C∗-algebra, by regarding the C∗-algebra as a Banach space equipped with its C∗-norm (e.g., [13,14]).
Independently, the joint free distributions of mutually free, multi semicircular elements were re-characterized in [9] (See...