### A Characterization of C(X) Among Algebras on Planar Sets by the Existence of a Finite Universal Korovkin System.

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This note contains an approximation theorem that implies that every compact subset of ${\u2102}^{n}$ is a good compact set in the sense of Martineau. The property in question is fundamental for the extension of analytic functionals. The approximation theorem depends on a finiteness result about certain polynomially convex hulls.

The paper gives sufficient conditions for projections of certain pseudoconcave sets to be open. More specifically, it is shown that the range of an analytic set-valued function whose values are simply connected planar continua is open, provided there does not exist a point which belongs to boundaries of all the fibers. The main tool is a theorem on existence of analytic discs in certain polynomially convex hulls, obtained earlier by the author.

Let $M$ be a two dimensional totally real submanifold of class ${C}^{2}$ in ${\mathbf{C}}^{2}$. A continuous map $F:\stackrel{\u203e}{\Delta}\to {\mathbf{C}}^{2}$ of the closed unit disk $\stackrel{\u203e}{\Delta}\subset \mathbf{C}$ into ${\mathbf{C}}^{2}$ that is holomorphic on the open disk $\Delta $ and maps its boundary $b\Delta $ into $M$ is called an analytic disk with boundary in $M$. Given an initial immersed analytic disk ${F}^{0}$ with boundary in $M$, we describe the existence and behavior of analytic disks near ${F}^{0}$ with boundaries in small perturbations of $M$ in terms of the homology class of the closed curve ${F}^{0}\left(b\Delta \right)$ in $M$. We also prove a regularity theorem...

Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that ${\Gamma}_{g}\left(F\right):=(z,g(z\left)\right):z\in F$ is complete pluripolar in ${\u2102}^{N+1}$. Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that ${\Gamma}_{g}\left(D\right)$ is complete pluripolar in ${\u2102}^{N+1}$. These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86]...

We present here three examples concerning polynomial hulls of some manifolds in C2.1. Some real surfaces with equation w = P (z,z') + G(z) where P is a homogeneous polynomial of degree n and G(z) = o(|z|n) at 0 which are locally polynomially convex at 0.2. Some real surfaces MF with equation w = zn+kz'n + F(z,z') such that the hull of Mf ∩ B'(0,1) contains a neighbourhood of 0.3. A contable union of totally real planes (Pj) such that B'(0,1) ∩ (∪j∈N Pj) is polynomially convex.