On the topology of the group of invertible elements – A Survey –

On the topology of the group of invertible elements – A Survey – The topological structure of the group of invertible elements in a unital Banach algebra (regular group for short) has attracted topologists from the very beginning of homotopy theory. Be it for its own sake simply to show the instrumental power of newly invented methods or because there were important applications, the most notable one being perhaps the Atiyah-Singer index theorem whose topological pillar is Bott's periodicity theorem for the homotopy groups of the stable general linear group. Recently, operator K-theory, which is the homotopy theory of the stable regular group of a C *-algebra, has been used to obtain index theorems in a more general setting. While the properties that are needed in index theory are by now quite well understood, since there one only makes use of the stable regular group, the topological structure of the regular group of a C *-algebra itself is less well studied. Here we want to survey the present state. Although today operator K-theory is merged in KK-theory we do not include these new developments since they would take us to far and beyond our purpose. The prototype of a regular group is the general linear group GL(n, F) of invertible n × n-matrices with entries from F = R, C, or H. Exploring its homotopical structure is intimately linked to the development of algebraic topology, and even nowadays a full understanding is out of reach. Without exposing the highly sophisticated methods which are used to pursue this problem, we will give a short historical account in the first section. In the second section we look at the general linear group GL(E) of invertible continuous linear operators on a Banach space E, and also as an intermediate step from finite to infinite dimensions at the Fredholm group GL c (E), the subgroup of GL(E) that consists of perturbations of the identity by compact operators. General Banach algebras will be studied in section three, in particular, commutative Banach algebras which for nearly forty years have taken an independent development because of the close relation to complex analysis. In the last section we deal exclusively with C *-algebras. We review the important tools and results from operator K-theory and then go on to discuss nonstable K-theory. However, we have to be selective since operator K-theory is still rapidly expanding which makes it impossible …