Mathematische Zeitschrift

, Volume 264, Issue 1, pp 111–136

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



This paper addresses extensions of the complex Ornstein–Uhlenbeck semigroup to operator algebras in free probability theory. If a1, . . . , ak 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 a1, . . . , ak. We show that Dt extends (in two different ways) to a semigroup of completely positive maps on the von Neumann algebra generated by a1, . . . , ak. Moreover, we show that Dt satisfies an optimal ultracontractive property: \({\|D_t\colon L^2\to L^\infty\| \sim t^{-1}}\) for small t > 0.


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© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsMITCambridgeUSA

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