R-diagonal Dilation Semigroups
نویسنده
چکیده
This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If a1, . . . , ak are ∗-free R-diagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −ntai1 · · · ain 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: ‖Dt : L 2 → L‖ ∼ t for small t > 0.
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