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.

Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.

A commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

Let A be a uniform algebra and M(A) the maximal ideal space of A. A sequence {an} in M(A) is called l1-interpolating if for every sequence (αn) in l1 there exists a function f in A such that f (an) = αn for all n. In this paper, an l1-interpolating sequence is studied for an arbitrary uniform algebra. For some special uniform algebras, an l1-interpolating sequence is equivalent to a familiar l-interpolating sequence. However, in general these two interpolating sequences may be different from each other.