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For a complex Banach space $X$ with open unit ball $B_X,$ consider the Banachalgebras $\mathcal H^\infty(B_X)$ of bounded scalar-valued holomorphicfunctions and the subalgebra $\mathcal A_u(B_X)$ of uniformly continuousfunctions on $B_X.$ Denoting either algebra by $\mathcal A,$ we study theGleason parts of the set of scalar-valued homomorphisms $\mathcal M(\mathcalA)$ on $\mathcal A.$ Following remarks on the general situation, we focus onthe case $X = c_0.$

Gleason parts for algebras of holomorphic functions on the ball of $\mathbf{c_0}$