论文标题
在takai二元性上，$ l^{p} $运营商交叉产品
On the Takai duality for $L^{p}$ operator crossed products
论文作者
论文摘要
The aim of this paper is to study a problem raised by N. C. Phillips concerning the existence of Takai duality for $L^p$ operator crossed products $F^{p}(G,A,α)$, where $G$ is a locally compact Abelian group, $A$ is an $L^{p}$ operator algebra and $α$ is an isometric action of $G$ on $A$.受$ C^*$ - 代数的交叉产品的Takai二元定理的证明的启发，我们从$ f^{p}（\ hat {g}，f^p（g，a，a，a，α），\hatα）$ to to to $ f^{p}（\ hat {g}）构建同构$φ$ $ \ MATHCAL {K}（l^{p}（g））\ otimes_ {p} a $，这是D. Williams的地图的天然$ l^p $ -Analog。对于具有独特的$ l^p $运算符矩阵规范的可数离散的Abelian $ g $和可分开的Unital $ l^p $运算符代数$ A $，我们表明$φ$是同构的，并且仅当$ g $是有限的或$ g $有限或$ $ p = 2 $时特别是，在$ p = 2 $的情况下，$φ$是同构同构。此外，证明$φ$对于$ f^p（\ hat {g}，f^p（g，a，a，α），\hatα）的双重双动作$ \ hat {\hatα} $ of $ g $ of $ g $是等效的$ \ Mathcal {k}（l^p（g））\ otimes_p a $。
The aim of this paper is to study a problem raised by N. C. Phillips concerning the existence of Takai duality for $L^p$ operator crossed products $F^{p}(G,A,α)$, where $G$ is a locally compact Abelian group, $A$ is an $L^{p}$ operator algebra and $α$ is an isometric action of $G$ on $A$. Inspired by D. Williams' proof for the Takai duality theorem for crossed products of $C^*$-algebras, we construct a homomorphism $Φ$ from $F^{p}(\hat{G},F^p(G,A,α),\hatα)$ to $\mathcal{K}(l^{p}(G))\otimes_{p}A$ which is a natural $L^p$-analog of D. Williams' map. For countable discrete Abelian groups $G$ and separable unital $L^p$ operator algebras $A$ which have unique $L^p$ operator matrix norms, we show that $Φ$ is an isomorphism if and only if either $G$ is finite or $p=2$; in particular, $Φ$ is an isometric isomorphism in the case that $p=2$. Moreover, it is proved that $Φ$ is equivariant for the double dual action $\hat{\hatα}$ of $G$ on $F^p(\hat{G},F^p(G,A,α),\hatα)$ and the action $\mathrm{Ad}ρ\otimesα$ of $G$ on $\mathcal{K}(l^p(G))\otimes_p A$.
