On stability of distance under some tensor products and some calculations

TL;DR

This paper demonstrates the stability of certain operator algebra distances under specific tensor products and provides explicit calculations of these distances for subalgebras of crossed-product algebras.

Contribution

It establishes the stability of Kadison-Kastler and Christensen distances under tensor products with unital commutative and general $C^*$-algebras, and performs explicit distance calculations.

Findings

Kadison-Kastler and Christensen distances are stable under specified tensor products.

Explicit calculations of distances between subalgebras of crossed-product algebras.

Results extend understanding of operator algebra stability under tensor operations.

Abstract

We prove that the Kadison-Kastler and Christensen distances are stable under the Banach space injective tensor product (resp., the Banach space projective tensor product) of a Banach space with any unital commutative $C^*$-algebra (resp., of a $C^*$-algebra with any unital $C^*$-algebra). Apart from these stability results, we make some explicit calculations of the Kadison-Kastler, Christensen and Mashood-Taylor distances between certain subalgebras of some crossed-product operator algebras.