Vol. 44, No. 1, 1973

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Idempotents in the boundary of a Lie group

Frank Leroy Knowles

Vol. 44 (1973), No. 1, 191–200

Let S be a locally compact semigroup consisting of a dense connected Lie group G and its boundary L; and let e be an idempotent in L. This paper is concerned with the proof of three principal results: (1) If L = Ge is simply connected, then S is homeomorphic to G∕Gr(e) × Gr(e)∕Gl(e) × Gl(e), where Gl(e) is a connected locally compact group with zero. (2) For any connected Lie group G and closed normal subgroup H such that G∕H is simply connected and H is the direct product of the multiplicative group of positive real numbers and a connected compact group, there is a locally compact semigroup S which contains a dense subgroup isomorphic to G whose boundary is a group isomorphic to G∕H. (3) If G = V Gl(e), for some subspace V G, then Ge is locally compact if and only if there is a local cross-section to the global orbits of G at e.

Mathematical Subject Classification 2000
Primary: 22A15
Secondary: 22E99
Received: 3 March 1971
Revised: 28 July 1972
Published: 1 January 1973
Frank Leroy Knowles