We found a match
Your institution may have rights to this item. Sign in to continue.
- Title
<img src="/fulltext-image.asp?format=htmlnonpaginated&src=2JL57R127J0G10L2_html\209_2008_455_Article_IEq1.gif" border="0" alt="$${\fancyscript{R}}$$" />-diagonal dilation semigroups.
- Authors
Todd Kemp
- Abstract
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.
- Subjects
SEMIGROUPS of operators; OPERATOR algebras; FREE probability theory; SELFADJOINT operators; MATHEMATICAL mappings; VON Neumann algebras
- Publication
Mathematische Zeitschrift, 2010, Vol 264, Issue 1, p111
- ISSN
0025-5874
- Publication type
Article