Gleason parts for algebras of holomorphic functions in infinite dimensions

DIMANT, VERÓNICA; ARON, RICHARD M.; MAESTRE, MANUEL; ARON, RICHARD M.; MAESTRE, MANUEL; LASSALLE, SILVIA; LASSALLE, SILVIA; DIMANT, VERÓNICA

REVISTA MATEMATICA COMPLUTENSE

UNIV COMPLUTENSE MADRID

Año: 2020 vol. 33 p. 415 - 436

1139-1138

For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banach algebras$mathcal H^infty(B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra$mathcal A_u(B_X)$ of uniformly continuous functions on $B_X.$ Denoting either algebraby $mathcal A,$ we study the Gleason parts of the set of scalar-valued homomorphisms$mathcal M(mathcal A)$ on $mathcal A.$ Following remarks on the general situation, we focus on the case $X = c_0,$ giving a complete characterization of the Gleason parts of$mathcal M(mathcal A_u(B_{c_0}))$ and, among other things, showing that every fiber in$mathcal M(mathcal H^infty(B_{c_0}))$over a point in $B_{ell_infty}$ contains $2^c$ discs lying in differentGleason parts.