Homomorphisms Between Algebras of Holomorphic Functions on the Infinite Polydisk
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Homomorphisms Between Algebras of Holomorphic Functions on the Infinite Polydisk Verónica Dimant1,2
· Joaquín Singer3,4
Received: 21 July 2020 / Accepted: 17 September 2020 © Mathematica Josephina, Inc. 2020
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 , ∞ ), likewise the scalar-valued spectrum M∞ (Bc0 ) which is projected onto B ∞ . 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 2c disjoint analytic Gleason isometric copies of B∞ . For the vector-valued case, building on the previous result we obtain 2c disjoint analytic Gleason isometric copies of BH∞ (Bc0 ,∞ ) 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 ). Keywords Spectrum · Algebras of holomorphic functions · Homomorphisms of algebras Mathematics Subject Classification 46J15 · 46E50 · 32A38 · 30H05
Partially supported by Conicet PIP 11220130100483 and ANPCyT PICT 2015-2299.
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Verónica Dimant [email protected] Joaquín Singer [email protected]
1
Departamento de Matemática y Ciencias, Universidad de San Andrés, Vito Dumas 284, Victoria, B1644BID Buenos Aires, Argentina
2
CONICET, Buenos Aires, Argentina
3
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
4
IMAS-CONICET, Buenos Aires, Argentina
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V. Dimant, J. Singer
1 Introduction In 1909, David Hilbert wrote an article [23], which is now considered as one of the pioneering works of the theory of analytic functions in infinitely many variables. There he claimed his intention of transferring the main theorems about analytic functions of several variables to the infinite dimensional setting. Following his path, the open unit ball of c0 appears as a natural domain for analytic functions in infinitely many variables since it is the usual infinite dimensional analogue of the polydisk Dn . While Infinite Dimensional Holomorphy has been developed in many directions besides Hilbert’s initial approach, the open unit ball of c0 has continued to be a favorite domain for the study of bounded holomorphic functions. Our interest here is to describe the set of homomorphisms between algebras of bounded holomorphic functions on the open unit ball of c0 . While pursuing this objective, we give a description of the fibers of the spectrum of the algebra of bounded holomorphic functions on the open unit ball of c0 which improves and completes results of [3,4,10]. In order to explain our goal more precisely, we need to recall some definitions and introduce some notation. For a complex Banach space X , let H∞ (B X ) be the space of bounded holomorphic functions on B X (the open unit ball of X ). The (scalar-valued) spectrum of this uniform a
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