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Zbl 0536.46046
Bratteli, O.; Jørgensen, Palle E.T.; Kishimoto, A.; Robinson, D.W.
A $C\sp*$-algebraic Schoenberg theorem. (English)
[J] Ann. Inst. Fourier 34, No.3, 155-187 (1984). ISSN 0373-0956; ISSN 1777-5310/e

Let ${\frak A}$ be a $C\sp*$-algebra, G a compact abelian group, $\tau$ an action of G by *-automorphisms of ${\frak A},{\frak A}\sp{\tau}$ the fixed point algebra of $\tau$ and ${\frak A}\sb F$ the dense sub-algebra of G-finite elements in ${\frak A}$. Further let H be a linear operator from ${\frak A}\sb F$ into ${\frak A}$ which commutes with $\tau$ and vanishes on ${\frak A}\sp{\tau}$. We prove that H is a complete dissipation if and only if H is closable and its closure generates a $C\sb 0$-semigroup of completely positive contractions. These complete dissipations are classified in terms of certain twisted negative definite maps from the dual group $\hat G$ into dissipative operators affiliated with the center of the multiplier algebra of ${\frak A}\sp{\tau}$. We also argue that the complete dissipation property is strictly stronger than the usual dissipation property, except in special circumstances such as when ${\frak A}$ is abelian.

MSC 2000:: *46L55 Noncommutative dynamical systems
47D03 (Semi)groups of linear operators
47B44 Accretive operators, etc. (linear)

Keywords: $C\sp*$-algebraic Schoenberg theorem; fixed point algebra; complete dissipation; $C\sb 0$-semigroup of completely positive contractions; twisted negative definite maps; dual group; center of the multiplier algebra

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