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 nonzero 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 onto $\overline{B}_{\ell _\infty }$. Our itinerary begins in the scalar-valued spectrum ${\mathcal {M}}_\infty (B_{c_0})$: 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 $B_{\ell _\infty }$. For the vector-valued case, building on the previous result we obtain $2^c$ disjoint analytic Gleason isometric copies of $B_{{\mathcal {H}}^\infty (B_{c_0},\ell _\infty )}$ in 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 ${\mathcal {M}}_\infty (B_{c_0},B_{c_0})$.