SYMMETRY AND NON-SYMMETRY FOR LOCALLY COMPACT GROUPS

The class [S] of locally compact groups G is considered, for which the algebra L1(G) is symmetric (=Hermitian). It is shown that [S] is stable under semidirect compact extensions, i.e., H ε{lunate} [S] and K compact implies K ×s H ε{lunate} [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 ≤4 which is not in [S]. This example shows that G Z ε{lunate} [S]. Z the center of G, in general does not imply G ε{lunate}[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. © 1979.