I search for a noncommutative idempotent-less Banach algebra $A$ which is graded by a finite abelian group $G$ such that a nontrivial homogenous element lies in the same connected component as $1_{A}$ (in the space of invertible elements).
Motivation: This situation is impossible for commutative idempotent-less Banach algebras.
By a nontrivial homogenous element, I mean the following:
Assume $A=\oplus_{g\in G} A_{g}$. A nontrivial homogenous element is an element of $A_{g}$ for some $g\neq e$.
By idempotent-less I mean: $A$ has no nontrivial idempotent.
In particular, can one obtain an example with $A=C^{*}_{red}(F_{2})$ which is graded by each of the two $Z_{2}$-graded structures discussed in the following post?:
