Bounded and Semi-bounded Representations of Infinite Dimensional Lie Groups

In this note we describe the recent progress in the classification of bounded and semibounded representations of infinite 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 infinite dimensional Lie group $G$. This makes the full power of $C^*$-theory available in this context. Then we discuss the classification 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 classification of semibounded representations for hermitian Lie groups of operators, loop groups (with infinite dimensional targets), the Virasoro group and certain infinite dimensional oscillator groups.