Metrics on the space of $C^{*}$ algebras 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"
 A: 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.
A: Probably the most standard metric is Banach-Mazur distance, and there is indeed a theorem due to Amir which says that if the Banach-Mazur distance between $C(K)$ and $C(L)$ is less than $2$ then $K$ and $L$ are homeomorphic. There's also something called the Kadets distance which is basically a linearized Gromov-Hausdorff distance. I don't know what the Kadets distance says about commutative C*-algebras.

Adding an edit to call attention to Caleb Eckhardt's comment. Caleb point out that CB Banach-Mazur distance is a better candidate (I agree) and cites this paper which shows that if two C*-algebras have sufficiently small CB Banach-Mazur distance, and one of them is nuclear (in particular, commutative) then they are isomorphic.
