Homomorphisms between algebras of holomorphic functions on the infinite polydisk. (English) Zbl 1518.46034
Summary: 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 B. J. Cole et al. [Mich. Math. J. 39, No. 3, 551–569 (1992; Zbl 0792.46016)],
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 \(\mathcal{M}_{\infty} (B_{c_0},B_{c_0})\).
MSC:
| 46J15 | Banach algebras of differentiable or analytic functions, \(H^p\)-spaces |
| 46E50 | Spaces of differentiable or holomorphic functions on infinite-dimensional spaces |
| 32A38 | Algebras of holomorphic functions of several complex variables |
| 30H05 | Spaces of bounded analytic functions of one complex variable |
Citations:
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