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"



Metrics on the space of $C^{*}$ algebras
Probably the most standard metric is [Banach-Mazur distance](http://en.wikipedia.org/wiki/Banach-Mazur_compactum), and there is indeed [a theorem due to Amir](http://link.springer.com/article/10.1007%2FBF03008398#page-1) 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](http://arxiv.org/pdf/math/9709211v1.pdf) which is basically a linearized Gromov-Hausdorff distance. I don't know what the Kadets distance says about commutative C\*-algebras.




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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](http://arxiv.org/abs/1108.1970) 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.


