The main purposes of this paper are (i) to construct-and-study weighted-semicircular elements from mutually orthogonal $$\left| \mathbb {Z} \right| $$
 -many projections, and the Banach $$*$$
 -probability space $$\mathbb {L}_{Q}$$
 generated by these operators, (ii) to establish $$*$$
 -isomorphisms on $$ \mathbb {L}_{Q}$$
 induced by shifting processes on the set $$\mathbb {Z}$$
 of integers, (iii) to consider how the $$*$$
 -isomorphisms of (ii) generates Banach-space adjointable operators acting on the Banach $$*$$
 -algebra $$\mathbb {L}_{Q}$$
 , (iv) to investigate operator-theoretic properties of the operators of (iii), and (v) to study how the Banach-space operators of (iii) distorts the original free-distributional data on $$\mathbb {L}_{Q}$$
 . As application, one can check how the semicircular law is distorted by our Banach-space operators on $$\mathbb {L}_{Q}$$
 .

Banach-Space Operators Acting on Semicircular Elements Induced by Orthogonal Projections

Comparing Banach Spaces for Systems of Free Random Variables Followed by the Semicircular Law

Free Random Variables Followed by the Semicircular Law

In this paper, we consider (i) how to establish semicircular elements $\{U_{k}\}_{k=1}^{N}$ induced by N-many mutually orthogonal projections $\{q_{k}\}_{k=1}^{N}$, for N $\in $ $(\mathbb {N}\setminus \{1\})\cup \{\infty \},$ and the corresponding free product Banach $*$-probability space $\mathbb {L}_{Q}^{(N)}$ generated by $\{U_{k}\}_{k=1}^{N}$, (ii) the free-distributional data on $\mathbb {L}_{Q}^{(N)}$, (iii) certain $*$-homomorphisms on $\mathbb {L}_{Q}^{(N)},$ and (iv) how the $*$-homomophisms of (iii) deform the original free-distributional data of (ii).

Deformed Joint Free Distributions of Semicircular Elements Induced by Multi Orthogonal Projections

In this paper, we construct-and-consider free (weighted-)Poisson elements in a topological noncommutative free $*$-probability space generated by countable-infinitely many mutually free, (weighted-)semicircular elements which are induced by mutually orthogonal countable-infinitely many projections in a fixed $C^{*}$-probability space.

Free Poisson Elements Induced by Orthogonal Projections

Let $\nu$ be a finite measure on $\mathbb R$ whose Laplace transform is analytic in a neighborhood of zero. An anyon L\'evy process on $(\mathbb R^d,dx)$ is a certain family of noncommuting operators $\langle\varphi,\omega\rangle$ in the anyon Fock space over $L^2(\mathbb R^d\times\mathbb R,dx\otimes\nu)$. Here $\varphi=\varphi(x)$ runs over a space of test functions on $\mathbb R^d$, while $\omega=\omega(x)$ is interpreted as an operator-valued distribution on $\mathbb R^d$. Let $L^2(\tau)$ be the noncommutative $L^2$-space generated by the algebra of polynomials in variables $\langle \varphi,\omega\rangle$, where $\tau$ is the expectation at the vacuum state. We construct noncommutative orthogonal polynomials in $L^2(\tau)$ of the form $\langle f^{(n)},P_n(\omega)\rangle$, where $f^{(n)}$ is a test function on $(\mathbb R^d)^n$. Using these orthogonal polynomials, we derive a unitary isomorphism $U$ between $L^2(\tau)$ and an extended anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathbf F(L^2(\mathbb R^d,dx))$. The usual anyon Fock space over $L^2(\mathbb R^d,dx)$, denoted by $\mathcal F(L^2(\mathbb R^d,dx))$, is a subspace of $\mathbf F(L^2(\mathbb R^d,dx))$. Using the unitary isomorphism $U$, we realize the operators $\langle \varphi,\omega\rangle$ as a Jacobi (i.e., three-diagonal) field in $\mathbf F(L^2(\mathbb R^d,dx))$. We derive a Meixner-type class of anyon L\'evy processes for which the respective Jacobi field in $\mathbf F(L^2(\mathbb R^d,dx))$ has a relatively simple structure. Each anyon L\'evy process of the Meixner type is characterized by two parameters: $\lambda\in\mathbb R$ and $\eta\ge0$. Furthermore, we get the representation $\omega(x)=\partial_x^\dag+\lambda \partial_x^\dag\partial_x +\eta\partial_x^\dag\partial_x\partial_x+\partial_x$. Here $\partial_x$ and $\partial_x^\dag$ are annihilation and creation operators at point $x$.

/pdf/rasshirennoe-enionnoe-fokovskoe-prostranstvo-i-2qdnfyy4t5.pdf

Расширенное энионное фоковское пространство и некоммутативные ортогональные многочлены типа Мейкснера в бесконечномерном случае@@@An extended anyon Fock space and non-commutative Meixner-type orthogonal polynomials in infinite dimensions

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