Still $c_2(K^3 \times T^2)$ is not an integer. I am joining other people here to make sense of chern classes among other things by some basic readings

Is $T(S)$ the image of the set of some contant polynomials ?

The model of Karoubi of K theory involving gradings (as in the definition of K homology cycles) makes non necessary the introduction of formal inverses. See his introductory book on the subject.

(Nota also that the name diagonal would have been better suited for maps of shape $a \otimes b \mapsto ab$)

What topology do you put on your tensor product $C^* $ algebra ?

Or take any translate of it.

If i give you a $v : h^* \to h$ could you give me a preimage for the matrice with coeeficients $u=0$ and $v$ ?

By bounded you mean for the $L^2$ norm given by the trace ? By 2) do you mean linear independance over $\mathbb C$ ? If so, your question boils down to one for Hilbert spaces, and the answer just depends on the cardinal of $G$

A reference may be Fiber Bundles of Husemoller

And waht is the Opial property for $A$ Hilbert modules ? Do we consider the topology associated to the $A$ linear continuous maps $\phi : M \to A$ for the weak topology?

Do you have a definition for this property ?

In the $II_1$ factor case it reduces to the image of the trace : $\tau(K_0(C_r^*(\Gamma)) \subset \mathbb R$. It is known that in some case the image is $\mathbb Z$. Take for exemple $\Gamma = F_n$ the free group. In full generallity, for any ICC amenable group i don't know.

There is an other proof of Bott periodicity using Banach alegbras that you can find in the book of Blackadar. You use an exact sequence of C* algebras to obain directly the Bott peridicity