Total minimality of the unitary groups

Abstract

Dierolf and Schwanengel ([-7]) showed that if X is an infinite diescrete space, then the group F(X) of all bijections f : X ~ X provided with the topology of pointwise convergence is a totally minimal topological group. That was the first example of a (totally) minimal group which is not precompact. Other examples of such groups can be found in [8] (see also [21, w 2]). Infinite groups which does not admit non-discrete Hausdorff group topologies were constructed first by Shelah ([22], assuming CH), and then by Hesse ([14]), and A. Ol'ghanski~ (who noted that a quotient of the Adian's group A(m,n) has the property in question, cf. [1, w Note that all the examples mentioned above are examples of non-Abelian groups. Prodanov ([19]) established the totally minimal Abelian groups are precompact, and recently Prodanov and the author ([20]) proved that all minimal Abelian groups are precompact. The main purpose of the present paper is to show that the unitary group of every real or complex Hilbert space provided with the strong operator topology is a totally minimal topological group. This result gives an affirmative answer to a question posed by I. Prodanov. In Sects. 2 and 3 we study the equivariant (with respect to the action of the unitary group) compactifications of the unit sphere S of an infinite-dimensional Hilbert space. It is shown in Sect. 2 that the unit ball endowed with the weak topology is the greatest equivariant compactification of S. This fact is used in Sect. 3 to describe all equivariant compactification of S. The main theorem is proved in Sect. 4. The proof uses the scheme of the proof of [7, (1)], a generalization of which is discussed in Proposition 4.6.