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N. I. Nessonov

Publications and source records attributed to N. I. Nessonov.

7 recordsLinked to original sources
arXiv2019

${\rm II}_1$-factor representations of the infinite symmetric inverse semigroup

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $\left[ g_{ij}\right]_{i,j\in \mathbb{N}}$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a finite set. The binary operation in $R_\infty$ is the ordinary matrix multiplication. It is clear that infinite symmetric group $\mathfrak{S}_\infty$ is a subgroup of $R_\infty$. The map $\star:\left[ g_{ij}\right]\mapsto\left[ g_{ji}\right]$ is an involution on $R_\infty$. We call a function $f$ on $R_\infty$ positive definite if for all $r_1, r_2, \ldots, r_n\in R_\infty$ the matrix $\left[ f\left( r_ir_j^\star\right)\right]$ is Hermitian and non-negatively definite. A function $f$ said to be indecomposable if the corresponding $\star$-representation $π_f$ is a factor-representation. A class of the $R_\infty$-central functions (characters) is defined by the condition $f(rs)=f(sr)$ for all $r,s\in R_\infty$. In this paper we classify all factor-representations of $R_\infty$ that correspond to the $R_\infty$-central positive definite functions.

math.RT↗
arXiv2009

Invariant states on the wreath product

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $Γ$ be a separable topological group. The wreath product $Γ\wr \mathfrak{S}_\infty$ is the semidirect product $Γ^\infty_e \rtimes \mathfrak{S}_\infty$ for the usual permutation action of $\mathfrak{S}_\infty$ on $Γ^\infty_e=\{[γ_i]_{i=1}^\infty : γ_i\in Γ,\textit{only finitely many}γ_i\neq e\}$. In this paper we obtain the full description of indecomposable states $φ$ on the group $Γ\wr\mathfrak{S}_\infty,$ satisfying the condition: φ(sgs^{-1})= φ(g)\text{for each}g\in Γ\wr \mathfrak{S}_\infty,s\in\mathfrak{S}_\infty.

math.RT↗
arXiv2005

A description of characters on the infinite wreath product

Let $\mathfrak{S}_\infty$ be the infinity permutation group and $Γ$ an arbitrary group. Then $\mathfrak{S}_\infty$ admits a natural action on $Γ^\infty$ by automorphisms, so one can form a semidirect product $Γ^\infty\rtimes \mathfrak{S}_\infty$, known as the {\it wreath} product $Γ\wr\mathfrak{S}_\infty$ of $Γ$ by $\mathfrak{S}_{\infty}$. We obtain a full description of unitary $II_1-$factor-representations of $Γ\wr\mathfrak{S}_\infty$ in terms of finite characters of $Γ$. Our approach is based on extending Okounkov's classification method for admissible representations of $\mathfrak{S}_\infty\times\mathfrak{S}_\infty$. Also, we discuss certain examples of representations of type $II_1$, where the {\it modular operator} of Tomita-Takesaki expresses naturally by the asymptotic operators, which are important in the characters-theory of infinite symmetric group.

math.RT↗