Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I am Sorry if the later question is elementary)
Edit: What is a generalization of the following theorem by Taylor, to non commutative Banach algebras(In particular finite dimensional non commutative Banach algebras".(I read this theorem in paper By J. Rosenberg "Comparison between algebraic and Topological K theory")
" Theorem 2: Taylor If A is a unital commutative Banach algebra and ifX is its maximal ideal space, then the Gelfand transform A → C(X) induces an isomorphism on topological K-theory."
Note that this theorem implies that for a commutative Banach algebra of finite dimension, the K_{0} group can not be a non trivial finite group.
Let $A$ be a finite-dimensional Banach algebra. The group $K_0(A)$ only depends on the underlying ring structure of $A$. In what follows, $A$ can be any finite-dimensional algebra over an arbitrary field $k$, well $A\neq 0$. The map $\{\text{finitely generated projective $A$-modules}\}\mapsto \mathbb Z$ given by $P\mapsto\dim_kP$ is well defined (being $A$ finite-dimensional so is any f.g. $A$-module) isomorphism invariant, additive, and non-trivial since $A\neq 0$, therefore it extends to a non-trivial homomorphism $K_0(A)\rightarrow \mathbb{Z}$. The image is a non-trivial subgroup of $\mathbb Z$, all of which are infinite cyclic (and obviously projective), hence $K_0(A)$ cannot be finite (it actually contains an infinite cyclic direct summand).