A Generalization of A Theorem of Gleason

Abstract

Since D. Hilbert proposed the famous Hilbert's fifth problem in 1900 which conjectured that every locally euclidean group is a Lie group, many mathematicians have dealt with this problem, but few satisfactory results were obtained before 1925. In 1928 Peter and Weyl [18] proved that any representation of a compact group is completely reducible and later in 1933 this problem was solved by von Neumann [17] for compact groups by using the above theorem. In the next year L. Pontrjagin [19] solved this problem for abelian groups. As a step to the final goal C. Chevalley [2] conjectured that every locally compact group with small is a Lie group. In 1948 Kuranishi [10] obtained a very important result; that is, if a factor group over a normal abelian Lie group is a Lie group, then the whole group is a Lie group with some reasonable conditions which do not spoil the importance of this theorem. This theorem completed Iwasawa's theorem [8] which asserts that if a factor group over a normal Lie group is a Lie group, then the whole group is a Lie group. A simplified proof was given by A. M. Gleason [6] which depends only upon the preceding Kuranishi's theorem. Gleason [4] also introduced a very important idea which is called a Gleason semi-group of compact sets in this paper. By making use of this he proved many important theorems. Also Iwasawa [8] and Gleason [6] had built up the theory of generalized Lie groups (L group in the sense of Iwasawa), and from this point of view it became natural to generalize this Hilbert conjecture to the following conjecture made by Iwasawa [8] and Gleason [6]: every locally compact group is a generalized Lie group. Compared with these group-theoretical points of view, Montgomery and Zippin [13] introduced the homology theory in the theory of finite dimensional groups and obtained many important results, such as the invariance of a domain in locally connected finite dimensional groups, the determination of three or four dimensional groups, the elimination of finite groups, etc. [12, 14, 15]. Recently Gleason proved that every locally compact finite dimensional group with small is a Lie group and using this theorem Montgomery and Zippin [16] proved that every locally compact finite dimensional group is a generalized Lie group; this gave an affirmative answer to the Hilbert's fifth problem. In a preceding paper [22] the author showed that the condition no small normal subgroup is equivalent to the condition no small subgroup and in this paper he will prove without the assumption of finite dimension that every locally compact group with small is a Lie group. Combining these results with the Iwasawa-Gleason generalized Lie group theory we prove that every locally compact group is a generalized Lie group. This gives an 351