, Volume 264, Issue 1, pp 111-136
Date: 02 Dec 2008

\({\fancyscript{R}}\) -diagonal dilation semigroups

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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 *-free \({\fancyscript{R}}\) -diagonal operators in a II1 factor, then \({D_t(a_{i_1}\cdots a_{i_n}) = e^{-nt} a_{i_1}\cdots a_{i_n}}\) 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.

This work was partially supported by NSF Grant DMS-0701162.