A C * -algebraic Schoenberg theorem

Ola Bratteli; Palle E. T. Jorgensen; Akitaka Kishimoto; Donald W. Robinson

Annales de l'institut Fourier (1984)

  • Volume: 34, Issue: 3, page 155-187
  • ISSN: 0373-0956

Abstract

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

How to cite

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Bratteli, Ola, et al. "A $C^*$-algebraic Schoenberg theorem." Annales de l'institut Fourier 34.3 (1984): 155-187. <http://eudml.org/doc/74641>.

@article{Bratteli1984,
abstract = {Let $\{\frak A\}$ be a $C^*$-algebra, $G$ a compact abelian group, $\tau $ an action of $G$ by $*$-automorphisms of $\{\frak A\},\{\frak A\}^\{\tau \}$ the fixed point algebra of $\tau $ and $\{\frak A\}_F$ the dense sub-algebra of $G$-finite elements in $\{\frak A\}$. Further let $H$ be a linear operator from $\{\frak A\}_ F$ into $\{\frak A\}$ which commutes with $\tau $ and vanishes on $\{\frak A\}^\{\tau \}$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a $C_0$-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 $\{\frak A\}^\{\tau \}$. 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.},
author = {Bratteli, Ola, Jorgensen, Palle E. T., Kishimoto, Akitaka, Robinson, Donald W.},
journal = {Annales de l'institut Fourier},
keywords = {-algebraic Schoenberg theorem; fixed point algebra; complete dissipation; -semigroup of completely positive contractions; twisted negative definite maps; dual group; center of the multiplier algebra},
language = {eng},
number = {3},
pages = {155-187},
publisher = {Association des Annales de l'Institut Fourier},
title = {A $C^*$-algebraic Schoenberg theorem},
url = {http://eudml.org/doc/74641},
volume = {34},
year = {1984},
}

TY - JOUR
AU - Bratteli, Ola
AU - Jorgensen, Palle E. T.
AU - Kishimoto, Akitaka
AU - Robinson, Donald W.
TI - A $C^*$-algebraic Schoenberg theorem
JO - Annales de l'institut Fourier
PY - 1984
PB - Association des Annales de l'Institut Fourier
VL - 34
IS - 3
SP - 155
EP - 187
AB - Let ${\frak A}$ be a $C^*$-algebra, $G$ a compact abelian group, $\tau $ an action of $G$ by $*$-automorphisms of ${\frak A},{\frak A}^{\tau }$ the fixed point algebra of $\tau $ and ${\frak A}_F$ the dense sub-algebra of $G$-finite elements in ${\frak A}$. Further let $H$ be a linear operator from ${\frak A}_ F$ into ${\frak A}$ which commutes with $\tau $ and vanishes on ${\frak A}^{\tau }$. We prove that $H$ is a complete dissipation if and only if $H$ is closable and its closure generates a $C_0$-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 ${\frak A}^{\tau }$. 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.
LA - eng
KW - -algebraic Schoenberg theorem; fixed point algebra; complete dissipation; -semigroup of completely positive contractions; twisted negative definite maps; dual group; center of the multiplier algebra
UR - http://eudml.org/doc/74641
ER -

References

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