TITLE: Metrics on the space of $C^{*}$ algebras
QUESTION [6 upvotes]: I think that there is  a metric on the huge space of all $C^{*}$ algebras. What is the explicit
definition of this metric?may you introduce me  a reference?
Moreover  is the restriction of this metric to commutative $C^{*}$ algebras gives us  a discrete metric? by discrete I mean "every commutative $C^{*}$ algebra has a neighborhood, with respect to this metric,  which contains no other commutative $C^{*}$ algebra"

REPLY [3 votes]: Another possibility is the Kadison-Kastler metric on C*-subalgebras of $B(H)$, which is just the Hausdorff distance between their unit balls. This paper gives references to a number of  results about stability under perturbation, which include the case where one of the algebras is separable and abelian; this paper generalizes to the case where one of the algebras is separable and nuclear.