What I have in mind is the following: a functor $K_0$ on the category of topological $^*$-algebras (with values in the category of commutative groups) that satisfies (among other properties one would wish a $K$-theory functor to possess) (i) $K_0(A)=K_0^{\text{op}}(A)$ when $A$ is a $C^*$-algebra and (ii) $K_0(A)=K_0^{\text{alg}}(A)$ when $A$ has involution the identity and the discrete topology.

As a matter of fact, a quick skim of the construction of $K_0^{\text{op}}(A)$ (as the grothendieck group of unitary equivalence classes of projections in $M_\infty (A)$) seems as if would work verbatim for an arbitrary topological $^*$-algebra, and furthermore, would give the same definition in the case of involution the identity and the discrete topology.

Does there exist such an extension? If not, why not? If so, is there a reference?