R-diagonal dilation semigroups

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