TITLE: A generalized K- theory via generalized idempotents
QUESTION [6 upvotes]: Edit After the answer by Neil Strickland, I add the word "a ring" in this new version.
In the literature, there is  a concept of generalized idempotent: an n- idempotent is an element $a$ of a Banach algebra or a ring 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?

REPLY [14 votes]: Let $E_n(A)$ be the set of $n$-idempotents in $A$, and let $u_1,\dotsc,u_n$ be the elements of $E_n(\mathbb{C})$.  Let $E'_n(A)$ be the set of $n$-tuples $e_1,\dotsc,e_n\in E_2(A)$ with $e_ie_j=0$ for $i\neq j$, and $\sum_ie_i=1$.  Define $f\colon E'_n(A)\to E_n(A)$ by $f(e_1,\dotsc,e_n)=\sum_iu_ie_i$.  Then it is not hard to see that $f$ is bijective.  Thus, $E_n(A)$ does not really tell you anything that is not already determined by $E_2(A)$.