# Bounded and semibounded representations of infinite dimensional Lie groups

• ### Karl-Hermann Neeb

Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

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## Abstract

In this note we describe the recent progress in the classi fication of bounded and semibounded representations of in finite dimensional Lie groups. We start with a discussion of the semiboundedness condition and how the new concept of a smoothing operator can be used to construct $C*$-algebras (so called host algebras) whose representations are in one-to-one correspondence with certain semibounded representations of an infi nite dimensional Lie group $G$. This makes the full power of $C*$-theory available in this context.

Then we discuss the classi cation of bounded representations of several types of unitary groups on Hilbert spaces and of gauge groups. After explaining the method of holomorphic induction as a means to pass from bounded representations to semibounded ones, we describe the classifi cation of semibounded representations for hermitian Lie groups of operators, loop groups (with infi nite dimensional targets), the Virasoro group and certain in finite dimensional oscillator groups.