Symmetries of Gaussian measures and operator colligations

Let $H$ be the space of all Hermitian matrices of infinite order and $U(\infty)$ be the inductive limit of the chain $U(1)\subset U(2)\subset...$ of compact unitary groups. The group $U(\infty)$ operates on the space $H$ by conjugations, and our aim is to classify the ergodic $U(\infty)$-invariant probability measures on $H$ by making use of a general asymptotic approach proposed in Vershik's note \cite{V}. The problem is reduced to studying the limit behavior of orbital inte...

We study asymptotics of traces of (noncommutative) monomials formed by images of certain elements of the universal enveloping algebra of the infinite-dimensional unitary group in its Plancherel representations. We prove that they converge to (commutative) moments of a Gaussian process that can be viewed as a collection of simply yet nontrivially correlated two-dimensional Gaussian Free Fields. The limiting process has previously arisen via the global scaling limit of spectra ...

Lecture notes in Russian. Topics: the Haar measure (abstract theorems and explicit descriptions for different groups), measures on infinite-dimensional spaces with large natural groups of symmetries (Gaussian measures, Poisson measures, virtual permutations, inverse limits of unitary groups), zoo of examples of topological groups, generalities for large (infinite-dimensional) groups, polymorphisms.

Let $O(\infty)$ and $U(\infty)$ be the inductively compact infinite orthogonal group and infinite unitary group respectively. The classifications of ergodic probability measures with respect to the natural group action of $O(\infty)\times O(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{R})$ and that of $U(\infty)\times U(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{C})$ are due to Olshanski. The original proofs for these results are based on the asymptotic representati...

We study infinite matrices $A$ indexed by a discrete group $G$ that are dominated by a convolution operator in the sense that $|(Ac)(x)| \leq (a \ast |c|)(x)$ for $x\in G$ and some $a\in \ell ^1(G)$. This class of "convolution-dominated" matrices forms a Banach-*-algebra contained in the algebra of bounded operators on $\ell ^2(G)$. Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that $G$ is amenable and rigidl...

For infinite-dimensional groups $G\supset K$ the double cosets $K\setminus G/K$ quite often admit a structure of a semigroup; these semigroups act in $K$-fixed vectors of unitary representations of $G$. We show that such semigroups can be obtained as limits of double cosets hypergroups (or Iwahori--Hecke type algebras) on finite-dimensional (or finite) groups.

We look at the semigroup generated by a system of heat equations. Applications to testing normality and option pricing are addressed.

We give a general approach to infinite dimensional non-Gaussian Analysis for measures which need not have a logarithmic derivative. This framework also includes the possibility to handle measures of Poisson type.

In the present paper we continue the study of the structure of a Banach algebra B(A, T_g) generated by a certain Banach algebra $A$ of operators acting in a Banach space $D$ and a group {T_g}_{g \in G} of isometries of D such that T_g A T^{-1}_g = A. We investigate the interrelations between the existence of the expectation of B(A, T_g) onto $A$, metrical freedom of the automorphisms of A induced by T_g and the dual action of the group G on B(A, T_g). The results obtained are...

We show that Haar measures of connected semisimple groups, embedded via a representation into a matrix space, have a homogeneous asymptotic limit when viewed from far away and appropriately rescaled. This is still true if the Haar measure of the semisimple group is replaced by the Haar measure of a irreducible lattice of the group, and the asymptotic measure is the same. In the case of an almost simple group of rank greater than 2, a remainder term in obtained. This extends a...