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 (this uses the fact that if self-adjoint elements are conjugate, then they are in fact unitarily equivalent).

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