Natural *-representations of inverse semigroups by operators (Q1120807)

For an inverse semigroup S, the \(\lambda\) representation of S on \(\ell^ 2(S)\) is not a *-representation with respect to the standard inner product on \(\ell^ 2(S)\). The auhor finds another natural inner product (obtained by expressing a trace functional combinatorially) with respect to which \(\lambda\) is always a *-representation on \(\ell^ 2(S)\). The connections of this representation with the \(\pi\) representation of Barnes are discussed.