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.
