Holomorphic Idempotents and Retracts in the Unit Ball of a Commutative CU-Algebra with Identity 1

Let A be a commutative CU-algebra with identity and open unit ball B. We study holomorphic functions F: BªB that are idempotent under composition and establish necessary and sufficient conditions for a set R:B to be the image of B under such an idempotent function. In other words, R is a holomorphic retract of B.. In order to achieve this result, a representation for linear idempotents of the algebra must be attained. The linear part of a holomorphic idempotent taking 0 to 0 is idempotent itself, and thus the linear representation can be used to prove several identities relating the linear and holomorphic idempotents, giving the main result.