On the Takai duality for L^p operator algebra crossed products

Zhen Wang 汪镇  (Jilin University)

10:00-11:00, October 24, 2023   Science Building A505 & Tencent Meeting 981672796


We study an open problem raised by N. C. Phillips concerning the existence of Takai duality for $L^p$ operator crossed products $F^{p}(G,A,\alpha)$. Inspired by D. Williams' proof for the Takai duality theorem for crossed products of $C^*$-algebras, we construct a homomorphism $\Phi$ from $F^{p}(\hat{G},F^p(G,A,\alpha),\hat{\alpha})$ 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 $\Phi$ is an isomorphism if and only if either $G$ is finite or $p=2$; in particular, $\Phi$ is an isometric isomorphism in the case that $p=2$. If, in addition, $A$ is $p$-incompressible and $p>1$, then $F^{p}(\hat{G},F^p(G,A,\alpha),\hat{\alpha})$ is isometrically isomorphic to $\mathcal{K}(l^{p}(G))\otimes_{p}A$ if and only if $p=2$.

About the speaker:

汪镇,吉林大学博士后。2018年博士毕业于华东师范大学数学科学学院,现在吉林大学数学学院从事博士后研究。主持国家自然科学基金青年项目、中国博士后基金面上项目,入选吉林大学“鼎新学者”。主要从事Lp-算子代数的研究,相关成果发表于Israel J. Math.、Math. Z.等杂志。