R-diagonal dilation semigroups

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 $\mathcal{R}$-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: $\Vert D_t:L^2\to L^{\infty }\Vert \sim t^{-1}$ for small $t>0$.

22 pages, 6 figures

References in corpus (2)

• Strong hypercontractivity in non-commutative holomorphic spaces
• Random matrices, free probability and the invariant subspace problem relative to a von Neumann Algebra