Homomorphisms Between Algebras of Holomorphic Functions on the Infinite Polydisk

Abstract

We study the vector-valued spectrum M∞(Bc0,Bc0), that is, the set of nonzero algebra homomorphisms from H∞(Bc0) to H∞(Bc0) which is naturally projected onto the closed unit ball of H∞(Bc0,ℓ∞), likewise the scalar-valued spectrum M∞(Bc0) which is projected onto B¯ℓ∞. Our itinerary begins in the scalar-valued spectrum M∞(Bc0): by expanding a result by Cole et al. (Michigan Math J 39(3):551–569, 1992), we prove that in each fiber, there are 2 c disjoint analytic Gleason isometric copies of Bℓ∞. For the vector-valued case, building on the previous result we obtain 2 c disjoint analytic Gleason isometric copies of BH∞(Bc0,ℓ∞) in each fiber. We also take a look at the relationship between fibers and Gleason parts for both vector-valued spectra Mu,∞(Bc0,Bc0) and M∞(Bc0,Bc0).