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 (respectively, the Banach space projective tensor product) of a Banach space with any unital commutative [Formula: see text]-algebra (respectively, of a [Formula: see text]-algebra with any unital [Formula: see text]-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.