Linear and holomorphic idempotents and retracts in the open unit ball of a commutative C*-algebra with identity

Let A be a commutative C*-algebra with identity, and let B=x∈A:∥ x∥<1 be its open unit ball. We consider norm 1 linear projections of A and holomorphic idempotents and retracts of B. (A holomorphic idempotent of B is a holomorphic function F:B→B such that F&j0;F=F. A holomorphic retract of B is the image of a holomorphic idempotent of B.).;We will prove that the linear term in the homogeneous series expansion of holomorphic idempotent of B taking 0 to 0 is a norm 1 projection. We are motivated by this relationship to derive a specific integral representation for these projections by thinking of A as C(X), the space of complex-valued continuous functions on a compact Hausdorff space X. If P is such a projection and f∈CX , we will prove that the values of Pf are dependent only upon a compact subset of X which can be partitioned, independently of f, into compact sets for which Pf will have constant modulus. We will then see that the values of Pf in a particular partition member are given by integration with respect to a unique Borel probability measure supported on that member. Several extensions and examples utilizing this representation in specific C*-algebras will be provided.;The integral representation of the linear part of a holomorphic idempotent offers significant insight. Several identities will be proven relating linear projections to holomorphic idempotents. We will use these identities to classify the holomorphic retracts of the ball. Specifically, we will find that a set R⊆B with 0∈R is a holomorphic retract if and only if R is the graph of a special type of holomorphic function between the unit balls of two specific complementary subspaces of A. We will use these ideas to construct an example of an idempotent and retract when A is the algebra of continuous complex-valued functions on the closed unit disk.;We conclude by presenting questions whose answers will strengthen or extend our results. Some theorems are proven to provide starting points to answer these questions.