Involutions on the second duals of group algebras versus subamenable groups

Involutions and trivolutions in algebras related to second duals of group algebras

“…Using this fact we can prove the following analogue of Singh's result [37,Theorem 2.2] for the non-existence of involutions on LUC(G) * . Our result extends Farhadi-Ghahramani [15,Theorem 3.2(b)], from amenable to subamenable groups.…”

“…In Singh [37], the third author introduced the concept of α-amenability for a locally compact group G. Given a cardinal α, a group G is called α-amenable if there exists a subset F ⊂ L 1 (G) * * containing a mean M (not necessarily left invariant) such that |F | ≤ α and the linear span of F is a left ideal of L 1 (G) * * . The group G is called subamenable if G is α-amenable for some cardinal 1 ≤ α < 2 2 κ(G) , where κ(G) denotes the compact covering number of G, that is, the least cardinality of a compact covering of G. (We remark that at the time of the proofreading of the paper Singh [37], the author did not know that the term subamenable had already been used in different contexts, viz., subamenable semigroups by Lau and Takahashi in [31,32], and initially subamenable groups by Gromov in [22]. She thanks A. T.-M. Lau for interesting exchange of mathematical points in this regard.)…”

“…If G is a non-compact locally compact group, then every non-trivial right ideal in L 1 (G) * * or in LUC(G) * has (vector space) dimension at least 2 2 κ(G) (Filali-Pym [18,Theorem 5], and, Filali-Salmi [19,Theorem 6]). It follows from this lower bound that if G is a subamenable, noncompact, locally compact group, then L 1 (G) * * has no involution (Singh [37,Theorem 2.2]). Singh [37, Theorem 2.9(i)] also showed that every discrete group G is a subgroup of a subamenable discrete group G σ with |G σ | ≤ 2 |G| .…”