Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette. The definition is based on invertible groups $\mathrm{GL}_n(A)$where $A$ is a Banach algebra.
What would we obtain if we replace these invertible groups with the space (semigroup) of right or left invertible matrices? Is the resulting structures somehow the same as standard $k$ groups defined with full invertible group $\mathrm{GL}_n(A)$?
The motivation comes from the following situation: One can find in the literature the concepts left and right topological stable rank which are based on left or right invertibility of a certain element $\sum_{i=1}^n a_i a_i^*$ associated to an $n$-tuple $(a_1,a_2,\ldots,a_n)$ in a $C^*$-algebra. With this motivation we try to consider the left and right K-theories on category of Banach algebras. Does this trivially leads us to the standard K-theory?
