An amenable group with a nonsymmetric group algebra

Joe W. Jenkins
1969 Bulletin of the American Mathematical Society  
Let G be a discrete group, h(G) the group algebra of G. Symmetry of h(G) has been considered in [l], [3] . Groups containing a free subgroup on two or more generators are the only groups found to have nonsymmetric group algebras, and in each case the groups found to have symmetric algebras are in the family of amenable groups. In this note we present an example of an amenable group with a nonsymmetric group algebra. LEMMA 1. Let G be a group generated by a and b such that 5, the semigroup
more » ... the semigroup generated by a and b t is free and such that cd" 1 = dc~~1 for {c, d} = {a, b}. Then l\(G) is nonsymmetric. PROOF. We will show that h(G) is not symmetric by showing that the involution is not hermitian. In particular, we will show that -*£sp(#) where x = a + ib + ar l -ib~l (we do not distinguish between G and its canonical image in h(G)). This is accomplished by defining a 6 in m(G), the bounded complex valued functions on G, such that ||0|| = 1 =0(e) and such that H(* + ie)g] = 0 for each gEG, where 0-•fl" is the mapping of m(G) onto h(G)*. Let 5 '^SKJS-1^ {e}, and define 0(g) = 0 for gGG~S'. We divide the elements of G into the five disjoint sets; 5, S-\ Sx = a~lbSU {a-^}, S 2 = ab~lS~l VJ {ab~1} and S z = G^(5U5 L -1 U5iU5 2 ). Let A = {a,b,e, ar\ b~1}. Direct computations yield AgC\S'9*0 if and only if g(£Sz or g=e. Note now that if g G Sz and g?^e then [support (x + ie)g] C\S' = Agr\S' = 0, 1 The results announced here are contained in the author's doctoral dissertation written at the University of Illinois under the guidance of Professor M. M. Day.
doi:10.1090/s0002-9904-1969-12170-1 fatcat:fr6ldw74rfct5ne6uwtsvgrn74