In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and establish relationships between the topological centers of module actions factorization properties with results group algebras. We consider under which sufficient necessary conditions Banach algebra $A\widehat{\otimes}B$ is Arens regular.

We develop a Harder-Narasimhan theory for Kisin modules generalizing a similar theory for finite flat group schemes due to Fargues [Far10]. We prove the tensor product theorem, i.e., that the tensor product of semistable objects is again semi-stable. We then apply the tensor product theorem to the study of Kisin varieties for arbitrary connected reductive groups.

Under some assumptions we prove that the Deligne tensor product of categories of constructible perverse sheaves on pseudomanifolds X and Y is the category of constructible perverse sheaves on X × Y . The Deligne external tensor product functor is identified with the geometrical external tensor product.

We demonstrate that many multivariate wavelets are projectable wavelets; that is, they essentially carry the tensor product (separable) structure though themselves may be non-tensor product (nonseparable) wavelets. We show that a projectable wavelet can be replaced by a tensor product wavelet without loss of desirable properties such as spatial localization, smoothness and vanishing moments.