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?

What I have in mind is the following: a (sequence of) functor(s) $K_\bullet$ 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) $K_\bullet (A)=K_\bullet ^{\text{op}}(A)$ when $A$ is a $C^*$-algebra and (ii) $K_\bullet (A)=K_\bullet ^{\text{alg}}(A)$ 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). (I suppose one possible issue with extending this to the higher $K$-groups in the most obvious way is determining the 'right' topology to put on the tensor product of topological $^*$-algebras in order to define the suspension.)

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

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?

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?