Homomorphisms between algebras of holomorphic functions on the infinite polydisk

Abstract

We study the vector-valued spectrum ${\mathcal{M}}_{\infty }(B_{c_0},B_{c_0})$, that is, the set of non null algebra homomorphisms from ${\mathcal{H}}^{\infty }(B_{c_0})$ to ${\mathcal{H}}^{\infty }(B_{c_0})$ which is naturally projected onto the closed unit ball of ${\mathcal{H}}^{\infty }(B_{c_0},{\ell }_{\infty })$, likewise the scalar-valued spectrum ${\mathcal{M}}_{\infty }(B_{c_0})$ which is projected over ${\bar{B}}_{{\ell }_{\infty }}$. Our itinerary begins in the scalar-valued spectrum ${\mathcal{M}}_{\infty }(B_{c_0})$: by expanding a result by Cole, Gamelin and Johnson (1992) we prove that on each fiber there are $2^c$ disjoint analytic Gleason isometric copies of $B_{{\ell }_{\infty }}$. For the vector-valued case, building on the previous result we obtain $2^c$ disjoint analytic Gleason isometric copies of $B_{{\mathcal{H}}^{\infty }(B_{c_0},{\ell }_{\infty })}$ on each fiber. We also take a look at the relationship between fibers and Gleason parts for both vector-valued spectra ${\mathcal{M}}_{u,\infty }(B_{c_0},B_{c_0})$ and ${\mathcal{M}}_{\infty }(B_{c_0},B_{c_0})$.