On stability of distance under some tensor products and some calculations

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.