  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 $t>0$.
