In the literature, there is  a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach algebra with $a^{n}=a$.



Can the  3 equivalent relations, Murray-Von Neumann, similarity and homotopy on 2-idempotents be generalized to n-idempotents,for arbitrary $n>2$? Does this processes gives us  a useful and new type of K theory? 

We know that "Vector bundles" are the topological analogy of 2-idempotents. Now what is a topological analogy for generalized idempotents?