Gleason parts for algebras of holomorphic functions on the ball of ${\mathbf{c}}_0$

Abstract

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 algebra by $\mathcal{A},$ 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 $X=c_0.$