On tensor products of weak mixing vector sequences and their applications to uniquely $E$-weak mixing $C^*$- dynamical systems

TL;DR

This paper establishes conditions under which weak mixing properties of sequences and dynamical systems are preserved under tensor products, with applications to $C^*$-dynamical systems.

Contribution

It introduces new results linking weak mixing of sequences and tensor products, and applies these to demonstrate preservation of unique $E$-weak mixing in $C^*$-dynamical systems.

Findings

Uniform weak mixing of sequences implies weak mixing of their tensor products.

Ergodicity of tensor product sequences implies their weak mixing.

Tensor products of uniquely $E$-weak mixing $C^*$-systems are also uniquely $E$-weak mixing.

Abstract

We prove that, under certain conditions, uniform weak mixing (to zero) of the bounded sequences in Banach space implies uniform weak mixing of its tensor product. Moreover, we prove that ergodicity of tensor product of the sequences in Banach space implies its weak mixing. Applications of the obtained results, we prove that tensor product of uniquely $E$-weak mixing $C^*$-dynamical systems is also uniquely $E$-weak mixing as well.