Uniform algebras and approximation on manifolds

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\bar{\Omega})$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $\Omega$ and $F$ we show that the only obstruction to $\mathcal{A} = \mathcal{C}(\bar{\Omega})$ is that there is a holomorphic disk $D \subset \bar{\Omega}$ such that all functions in $F$ are holomorphic on $D$, i.e., the only obstruction is the obvious one. This generalizes work by A. Izzo. We also have a generalization of Wermer's maximality theorem to the (distinguished boundary of the) bidisk

Oka's inequality for currents and applications

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46241/1/208_2005_Article_BF01446636.pd

A Stone-Weierstrass theorem for weak-star approximation by rational functions

AbstractLet K be a compact subset of the complex plane C, and let μ be a finite, positive Borel measure on K. Define R∞(μ) to be the weak-star closure in L∞(μ) of the algebra of rational functions with poles off K. For f ϵ R∞(μ), we consider A∞(f, μ), the weak-star closure in L∞(μ) of the algebra generated by R∞(μ) and the complex conjugate f of f. A problem which arises in connection with subnormal operators is to determine for which f ϵ R∞(μ), A∞(f, μ) = L∞(μ). We obtain a characterization of A∞(f,μ) as precisely the functions in L∞(μ) which belong to R∞(μE) for every restriction of μ to a level set E of f of positive measure. This characterization gives a solution to the problem above