Analysis and Probability seminar

Abstract: Given an action $G\curvearrowright X$ of a discrete group $G$ on a compact Hausdorff space $X$, it is natural to consider the convolution algebra $\ell^1(G\curvearrowright X)$, whose underlying Banach space is $\ell^1(G,C(X))$, and where multiplication is (twisted) convolution. Recent works of de  Jeu-Tomiyama and of Roland show that this Banach algebra  encodes more information than the more widely studied  C*-algebraic crossed product $C(X)\rtimes G$, which is the completion of $\ell^1(G\curvearrowright X)$ with respect to covariant representations on Hilbert spaces. In this talk, which is based on joint work with Tabare Roland, I will discuss the  isomorphism problem for such \ell^1-algebras, trying to give an answer to the following question: what can be said about two dynamical systems $G\curvearrowright X$ and $H\curvearrowright Y$ if there is an isometric isomorphism $\ell^1(G\curvearrowright X)\cong \ell^1(H\curvearrowright Y)$?