Hello all , I am reading a research paper that talks about the (P,Q) outer generalised inverse. In order to find some examples in arbitrary Banach spaces, I am excepting the answer for the above question.
Every dual C* algebra has a lot of idempotents, but a general Banach algebra can be free from non-trivial idempotents. Say, in C[0, 1] the only idempotents are the zero element and the unit element of the algebra.
In addition to what has been observed by Vladimir Kadets, a good Ph.D. thesis that speaks to the question for this thread is
M. Almus, Structure in Operator Algebras, Ph.D. thesis, University of Houston, 2011:
http://www.math.uh.edu/analysis/Theses/2011-Melahat-Almus.pdf
For some interesting idempotents, see Example 4.1.20, p. 58, continuing on p. 59.
The best-known example of a Banach-algebra is formed by the bounded operators of a Hilbert space H. Here the self-adoint idempotents are the projectors and these are in one-to-one correspondence to the closed linear subspaces of H. I'm not aware of a structure theorem on the non-selfadjoint idempotents.
For commutative Banach algebras the Shilov Idempotent Theorem is definitive. Taking the unital case for simplicity, there is a 1-1 correspondence between the clopen subsets of the character space and the idempotents in the commutative unital Banach algebra.
Concerning self-adjoint and non self-adjoint idempotents. In the paper
Julio Becerra Guerrero, Ángel Rodríguez Palacios. Non self-adjoint idempotents in C * - and JB * -algebras, Manuscr. Math. 124, No. 2, 183-193 (2007)
it is proved that every JB * -algebra containing a non self-adjoint idempotent also contains a nonzero selfadjoint idempotent
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