An extension of $K$-theory to topological $^*$-algebras

Question

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?

Answer 1

You already mentioned one possible extension, but there is one even simpler:

Just forget both the topology and the $*$ and take the algebraic K-theory. Indeed for a $C^*$-algebra the algebraic $K_0$-group is the same as the 'topological' $K_0$-group, topological and algebraic $K$-theory only become different after $K_1$ (and there the topological $K$⁻groups are considerably simpler than their algebraic counterpart...).

The difficulty is not to find generalization of $K$-theory, the problem is to find one where there is interesting things to says : $K_1$ and higher $K$-groups, long or cyclic exact sequence, excision etc... At this level of generality I don't think that one can find something relevant other than just taking the algebraic $K$-groups...

Answer 2

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 .