Splitting of tensor products and intermediate factor theorem: Continuous version

Let (Formula presented.) be a discrete group. Given unital (Formula presented.) - (Formula presented.) -algebras (Formula presented.) and (Formula presented.), we give an abstract condition under which every (Formula presented.) -subalgebra (Formula presented.) of the form (Formula presented.) is a tensor product. This generalizes the well-known splitting results in the context of (Formula presented.) -algebras by Zacharias and Zsido. As an application, we prove a topological version of the intermediate factor theorem (IFT). When a product group (Formula presented.) acts (by a product action) on the product of corresponding (Formula presented.) -boundaries (Formula presented.), using the abstract condition, we show that every intermediate subalgebra (Formula presented.) is a tensor product (under some additional assumptions on (Formula presented.)). This can be considered as a topological version of the IFT. We prove that our assumptions are necessary and cannot generally be relaxed. We also introduce the notion of a uniformly rigid action for (Formula presented.) -algebras and use it to give various classes of inclusions (Formula presented.) for which every invariant intermediate algebra is a tensor product.