Symmetry and nonsymmetry for locally compact groups

Leptin H, Poguntke D. Symmetry and nonsymmetry for locally compact groups. Journal of functional analysis. 1979;33(2):119-134.The class [S] of locally compact groups G is considered, for which the algebra L to the power of 1(G) is symmetric (=Hermitian). It is shown that [S] is stable under semidirect compact extensions, i.e., H Epsilon [S] and K compact implies K xs H Epsilon [S]. For connected solvable Lie groups inductive conditions for symmetry are given. A construction for nonsymmetric Banach algebras is given which shows that there exists exactly one connected and simply solvable Lie group of dimension less-than-or-equals 4 which is not in [S]. This example shows that G/Z Epsilon [S], Z the center of G, in general does not imply G Epsilon [S]. It is shown that nevertheless for discrete groups and a (possibly) stronger form of symmetry this implication holds, implying a new and shorter proof of the fact that [S] contains all discrete nilpotent groups