Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version

Abstract

Let $G$ be a discrete group. Given unital $G$-$C^{\ast }$-algebras $\mathcal{A}$ and $\mathcal{B}$, we give an abstract condition under which every $G$-subalgebra $\mathcal{C}$ of the form $\mathcal{A}\subset \mathcal{C}\subset \mathcal{A}{\otimes }_{\text{min}}\mathcal{B}$ is a tensor product. This generalizes the well-known splitting results in the context of $C^{\ast }$-algebras by Zacharias and Zsido. As an application, we prove a topological version of the Intermediate Factor theorem. When a product group $G={\Gamma }_1\times {\Gamma }_2$ acts (by a product action) on the product of corresponding ${\Gamma }_i$-boundaries $\partial {\Gamma }_i$, using the abstract condition, we show that every intermediate subalgebra $C(X)\subset \mathcal{C}\subset C(X){\otimes }_{\text{min}}C(\partial {\Gamma }_1\times \partial {\Gamma }_2)$ is a tensor product (under some additional assumptions on $X$). This can be considered as a topological version of the Intermediate Factor theorem. We prove that our assumptions are necessary and cannot generally be relaxed. We also introduce the notion of a uniformly rigid action for $C^{\ast }$-algebras and use it to give various classes of inclusions $\mathcal{A}\subset \mathcal{A}{\otimes }_{\text{min}}\mathcal{B}$ for which every invariant intermediate algebra is a tensor product.

Final version. It is going to appear in the Journal of the London Mathematical Society

References in corpus (3)

• C*-simplicity and the unique trace property for discrete groups
• Group C*-algebras as decreasing intersection of nuclear C*-algebras
• Relative C*-simplicity and characterizations for normal subgroups