Vol. 95, No. 2, 1981

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Some remarks about Cāˆž vectors in representations of connected locally compact groups

Louis Magnin

Vol. 95 (1981), No. 2, 391ā€“400

Given a continuous representation U of a connected locally compact group G in a quasi-complete locally convex topological vector space E, one may introduce the space E of C-vectors which contains the dense space F of regular vectors. Natural questions are then: (1) does F = E hold? (2) is the differential U of U a representation of the Lie algebra of G on E? We here prove that answer to (1) is “yes” when G is a quotient of a direct product of compact connected Lie groups and E has a continuous norm, and that answer to (2) is always “yes”. Of special interest are locally compact groups which are almost Lie in the sense that any subgroup algebraically generated by two continuous one-parameter subgroups is a Lie group in a finer connected topology. We prove that a connected locally compact group is almost Lie if and only if its universal covering in the sense of Lashof is H × A with H simply connected Lie group and A direct product of copies of R.

Mathematical Subject Classification 2000
Primary: 22D10
Received: 11 July 1980
Published: 1 August 1981
Louis Magnin