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arXiv · 0708.2562

R-diagonal dilation semigroups

Todd Kemp
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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 $t>0$.

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Todd Kemp. 2008-02-11. R-diagonal dilation semigroups. https://arxiv.org/abs/0708.2562

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