# Prof. dr. Erlend Fornæss Wold: Algebras generated by holomorphic and pluriharmonic functions

Source: Mathematics colloquium

**Prvotno najavljeno kolokvijsko predavanje prof. Lamela je zaradi bolezni predavatelja zamenjano s predavanjem prof. Wolda.**

Erlend Fornæss Wold

Univerza v Oslu

In 1953 John Wermer proved the following, now classical result of complex analysis: let A be a norm closed algebra on the boundary *b***D** of the unit disk in **C**, containing the algebra *P* generated by all holomorphic polynomials. Then either A = *P* or A = C(*b***D**). Hence *P* is a maximal subalgebra of the algebra of all continuous functions on the circle. There is a very natural way of thinking about this result in several complex variables: consider the graph *X* = G_{f}(*b***D**) in **C**^{2} of a continuous function *f* on *b***D**; then either *X* is convex with respect to the polynomials in two complex variables, or *f* is the boundary value of a holomorphic function on the disc **D**. So the failure of the graph *X* being polynomially convex is explained by the presence of analytic structure. A lot of work over several decades went into proving that one could always find analytic structure in polynomially convex hulls in several complex variables, but eventually it turned out that this was not the case. So, when trying to generalize Wermer’s theorem, one has to be more restrictive. In this lecture I will begin by discussing the classical result of Wermer and then give similar statements involving graphs of pluriharmonic functions in several complex variables.

**Pozor: predavanje bo ob 17:15!**

http://wiki.fmf.uni-lj.si/wiki/MatematicniKolokviji