We reduce the polynomial cluster value problem for the algebra of bounded analytic functions, H ∞ , on the ball of Banach spaces X to the same polynomial cluster value problem for H ∞ but on the ball of those spaces which are `1 -sums of finite dimensional spaces.

Since it is unclear how to define a domain that is strictly pseudoconvex in the infinite-dimensional setting, we develop a general theory with Banach spaces in mind. We first focus on the finite dimensional case and eliminate the need for two degrees of differentiability of the boundary of a domain, since differentiable functions are difficult to find in infinite dimensions. We introduce ℓ -strict pseudoconvexity for ℓ≥ 1 , 1-strict pseudoconvexity at the boundary, ℓ -uniform pseudoconvexity for ℓ≥ 0 , and finally strong pseudoconvexity. Defining ℓ -strict pseudoconvexity and ℓ -uniform pseudoconvexity for ℓ <2 depends on extending a notion of strict plurisubharmonicity to cases lacking C^2 -smoothness, first studying it in the sense of distribution and then considering it in infinite dimensions. Examples of strictly plurisubharmonic functions as well as strongly pseudoconvex domains related to important classical Banach spaces are presented. Finally, some related solutions to the inhomogeneous Cauchy–Riemann equations for ∂ -closed (0, 1)-forms in infinite-dimensional domains are shown.

The polynomial cluster value problem replaces the role of the continuous linear functionals in the original cluster value problem for the continuous polynomials to describe the corresponding cluster sets and fibers. We prove several polynomial cluster value theorems for uniform algebras H(B) between Au(B) and H∞(B), where B is the open unit ball of a complex Banach space X. We also obtain new results about the original cluster value problem, especially for A∞(B). Examples of spaces X considered here are spaces of continuous functions, ℓ1 and locally uniformly convex spaces.

We survey cluster value problems for Banach algebras H(B) of analytic functions on the open unit ball B of a Banach space X that contain X* and 1. For the Banach spaces X we focus on, we report on cluster value theorems for a Banach algebra H(B) and a point x** is an element of (B) over bar**. We also draw attention to a reduction of the cluster value problem for H (By) for any separable Banach space X that is an l(1)-sum of a sequence of finite-dimensional spaces. We conclude this work by describing the related 5 problem and surveying on strong pseudoconvexity as well as uniform pseudoconvexity in the context of Banach spaces.

We study the cluster value problem for certain Banach algebras of holomorphic functions defined on the unit ball of a complex Banach space X. The main results are for spaces of the form X = C(K).

The main result is that the cluster value problem in separable Banach spaces, for the Banach algebras $A_u$ and $H^{\infty}$, can be reduced to the cluster value problem in those spaces which are $\ell_1$ sums of a sequence of finite dimensional spaces.

Excess charge carriers on semiconducting nanotubes immersed in sluggish polar environments can undergo self- localization into polaronic states. Using a simplified model of electrons and holes of equal effective masses and confined to a cylindrical surface in the three-dimensional polar medium, we evaluate the binding energy E-b(pol) of adiabatic Frohlich - Pekar polarons and compare it to the corresponding exciton binding energy E-b(exc). The ratio E-b(pol)/E-b(exc) is found to be a non- monotonic function of the cylinder radius R which, in an idealized model, can reach values of about 0.35, substantially larger than values of about 0.2 for two- dimensional ( 2D) or three- dimensional ( 3D) systems. We argue that these findings represent a more general crossover effect that could manifest itself in other semiconductor nanostructures in 3D polar environments. As a result of the strong polaronic effect, the activation energy of exciton dissociation into polaron pairs is significantly reduced, which may lead to enhanced charge separation.

Electrons and holes on semiconducting nanotubes immersed in sluggish polar media can undergo self-localization into polaronic states. We evaluate the binding energy of adiabatic Fröhlich-Pekar polarons confined to a cylindrical surface and compare it to the corresponding exciton binding energy . The ratio / is found to be a non-monotonic function of the cylinder radius R that can reach values of about 0.35, substantially larger than values of about 0.2 for 2d or 3d systems. We argue that these findings represent a more general crossover effect that could manifest itself in other semiconductor nanostructures in 3d polar environments. As a result of the strong polaronic effect, the activation energy of exciton dissociation into polaron pairs is significantly reduced which may lead to enhanced charge separation.