Attempts to construct a unifying theory for arbitrary complete metric algebras were undertaken as early as in 70's by Karoubi and Villamayor, You may check this http://webusers.imj-prg.fr/~max.karoubi/Publications/11.pdf for the original paper, and the standard reference is the paper of Cortinas http://arxiv.org/abs/0903.3983 . The major issue one faces when constructing such theories is whether You

a) want them to be homotopy stable, and if yes, what kind of homotopy do You choose;

b) want them to be homology theories.

Since You mention $C^\ast$-algebras - let's even better stick to Banach algebras as it was suggested by Branimir Cacic above - as a theory standing on one side, we may assume the theory in question is supposed to be homotopy-stable. Here, You face two problems on both discrete and Banach sides. (BTW, $K$-theory for Banach algebras was defined by Vincent Lafforgue in http://www.researchgate.net/publication/267481248_Bivariant_K-theory_for_Banach_algebras_and_the_Baum-Connes_conjecture )

From the discrete side, the problem is that $K_\bullet$ is not homotopy-invariant. This may be cured by introducing polynomial homotopy, as did Karoubi and Villamayor. However, although they can construct all positive K-Theory groups, you cannon have negative ones such that they pass into a long exact sequence. This is cured by Weibel, who constructs his homotopy K-Theory $KH$. The problem with it is that in general neither $KH_0$ coincides with $K_0$, nor $KH_1$ with $KV_1$ (i.e. the Karoubi-Villamayor group), and that was one of the reasons Weibel did not publish his theory for about a decade.. But at least for $KH$ you may have definitions via spectra, as $KH_n(R)=\pi_n(BGl_+(R^\Delta))$, where $R^\Delta$ is a simplicial algebra based on the simplices $R^{\Delta^n}:=R[t_0,\dots,t_n]/<1-t_0-\dots-t_n>$. And $KH$ is perfectly a homology theory.

However, if you go up to the the Banach case, another problem occurs: what is the general notion of homotopy. Karoubi and Villamayor propose the homotopy given by convergent formal power series. However, the elementary homotopies provided by convergent power series are not symmetric. You could of course symmetrize them, but still you cannot have a good definition of $A^\Delta$ like Weibel, and with that a definition via spectra. In the case when your ground ring is, say, $\mathbb{C}$, the notion of convergent series homotopy may be proved to coincide with the usual homotopy of Banach algebras, yet it's tricky to generalize this for arbitrary metric rings. It's doable with another notion of homotopy, something like probability measures taking values in the ground ring (shameless self-advertisement here), but it is not known (at least to me) whether this other notion of homotopy may yield any kind of interesting K-Theory (although you could do spectra once again).

By the way, all the problems become much more simple if You're doing ultrametric Banach algebras. A Karoubi-Villamayor-type theory is now being developed by Bunke and Tamme, see e.g. http://arxiv.org/pdf/1111.4109v4.pdf .

properlyat how algebraic and topological K-theory for operator algebras relate to each other is Jonathan Rosenberg. I recommend math.umd.edu/~jmr/algtopK.pdf $\endgroup$ – Yemon Choi Aug 29 '15 at 15:03