Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

Abstract

We study holomorphic maps between C*-algebrasAandB, whenf:BA(0,ϱ)→Bis a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ballU=BA(0,δ). If we assume thatfis orthogonality preserving and orthogonally additive onAsa∩Uandf(U)contains an invertible element inB, then there exist a sequence(hn)inB**and Jordan*-homomorphismsΘ,Θ~:M(A)→B**such thatf(x)=∑n=1∞hnΘ~(an)=∑n=1∞Θ(an)hnuniformly ina∈U. WhenBis abelian, the hypothesis ofBbeing unital andf(U)∩inv(B)≠∅can be relaxed to get the same statement.