R-diagonal dilation semigroups

TL;DR

This paper extends the complex Ornstein-Uhlenbeck semigroup to free probability operator algebras, demonstrating its extension as a completely positive semigroup and establishing an optimal ultracontractive property.

Contribution

It introduces a dilation semigroup for free $ ext{R}$-diagonal operators and proves its extension as a completely positive map with optimal ultracontractive bounds.

Findings

Dilation semigroup extends to a completely positive map on the von Neumann algebra.

The semigroup satisfies an optimal ultracontractive bound:  D_t  \, L^2 \, o \, L^\u221e \, norm \, \\sim \, t^{-1}.

The extension applies to free iagonal operators in a  factor.

Abstract

This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If $a_1,...,a_k$ are $\ast$-free $\mathscr{R}$-diagonal operators in a $\mathrm{II}_1$ factor, then $D_t(a_{i_1}... a_{i_n}) = e^{-nt} a_{i_1}... a_{i_n}$ defines a dilation semigroup on the non-self-adjoint operator algebra generated by $a_1,...,a_k$. We show that $D_t$ extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by $a_1,...,a_k$. Moreover, we show that $D_t$ satisfies an optimal ultracontractive property: $\|D_t\colon L^2\to L^\infty\| \sim t^{-1}$ for small $t>0$.