Is the $K_0$ group for a $C^*$ algebras $A$ same as that for the $K_0$ group of ring $A$ from algebraic $K$ theory?
We assume $A$ is unital (I am not sure if this matters), i.e. what is an example where the these to $K$ theory groups are distinct when restricted to the category of $C^*$ algebras?
The $K_0$ group of a C*-algebra is the same as the (algebraic) $K_0$ group of it considered as a ring.