STUDIA MATHEMATICA 206 (1) (2011) Involutions on the second duals of group algebras versus subamenable groups by Ajit Iqbal Singh (New Delhi) Abstract. Let L 1 (G) be the second dual of the group algebra L 1 (G) of a locally compact group G. We study the question of involutions on L 1 (G) . A new class of subamenable groups is introduced which is universal for all groups. There is no involution on L 1 (G) for a subamenable group G. 1. Introduction. Let A be a complex Banach algebra, A its dual and A its second dual. We follow R. Arens ([A1], [A2]) and equip A with the first Arens product  or the second Arens product ♦ defined as follows. For ϕ,ψ ∈ A, f ∈ A , and F,H ∈ A , fϕ(ψ)= f (ϕψ), Ff (ϕ)= F (fϕ),H  F (f )= H (Ff ); whereas ϕf (ψ)= f (ψϕ), fF (ϕ)= F (ϕf ),F ♦ H (f )= H (fF ). The Banach algebra A is said to be Arens regular if  = ♦. For a continuous conjugate linear map T : AA, the conjugate-adjoint T c : A A and the second conjugate-adjoint T cc : A A are defined via T c f = f T, T cc F = FT c for f ∈ A,F ∈ A. Both T c and T cc are conjugate linear mappings and can very well be called the first adjoint and the second adjoint of T respectively. We will denote them by T and T respectively only if no confusion can arise. P. Civin and B. Yood [CY] noted that a continuous (algebra) involution on A, i.e. a continuous conjugate linear anti-homomorphism of period two, can be extended to an involution on A if A is Arens regular. The converse is also true. On the other hand, M. Grosser [G] showed that a necessary 2010 Mathematics Subject Classification : Primary 43A20, 43A22; Secondary 46K99, 54D35. Key words and phrases : Arens product, group algebra, involutions, second dual, Stone– Čech compactification, subamenable group. DOI: 10.4064/sm206-1-4 [51] c  Instytut Matematyczny PAN, 2011