Let $H^\infty $ be the algebra of bounded holomorphic functions on the open unit disk, and let $\mathfrak M$ be its maximal ideal space. Let $\mathfrak M_a$ be the union of nontrivial Gleason parts (analytic disks) of $\mathfrak M$. In this paper, we study the problem of extensions of bounded Banach-valued holomorphic functions and holomorphic maps with values in Oka manifolds from Gleason parts of $\mathfrak M_a\setminus \mathbb {D}$. The resulting extensions satisfy the uniform boundedness principle in the sense that their norms are bounded by constants that do not depend on the choice of the Gleason part. The results extend fundamental results of D. Suárez on the characterization of the algebra of restrictions of Gelfand transforms of functions in $H^\infty $ to Gleason parts of $ \mathfrak M_a\setminus \mathbb {D}$. The proofs utilize our recent advances on $\bar \partial $-equations on quasi-interpolating sets and Runge-type approximations.

For a compact subset K of the boundary of a compact Hausdorff space X, six properties that K may have in relation to the algebra A(X) are considered. It is shown that in relation to the algebra A(Dn), where Dn denotes the n-dimensional polydisc, the property of being totally null is weaker than the other five properties. A general condition is given on the algebra A(X) which implies the existence of a totally null set that is not null, and several conditions are stated for A(X) , each of which is sufficient for a totally null set to be a null set.

Let R be a finite open Riemann surface with analytic boundary Γ. Set and define is analytic on R}. Conditions are given on a function algebra A on a compact Hausdorff space X which imply that A is isomorphic to a subalgebra of A(R) of finite codimension.

Let R be a prime ring possessing a nontrivial automorphism T such that xTx = ±xxT. If R is not of characteristic 3 or R has nonzero center, then R is a commutative integral domain.