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"