RIGHT CANCELLATION IN THE ${\cal L}{\cal U}{\cal C}$ -COMPACTIFICATION OF A LOCALLY COMPACT GROUP

Finite-dimensional left ideals in the duals of introverted spaces

“…The theorem shows that in general for a Banach algebra A, there may be no non-trivial, finite-dimensional right ideals in A * * . In Filali-Pym [16] and Filali-Salmi [17] the authors showed a similar phenomenon for L 1 (G) * * and LU C(G) * for any non-compact locally compact group G. In this paper we will show that the situation is different with regard to finite-dimensional left ideals in A * * , or more generally in X * , if X is a faithful introverted subspace of A * . In fact, using representations of A we obtain complete characterization of finite-dimensional left ideals in X * (Theorems 2.7 and 2.8).…”

“…and A p (G) * * , Baker-Filali [3,4], Filali [13], Filali-Pym [16], and Filali-Salmi [17] for (among other things) X * , L 1 (G) * * , and LU C(G) * , where X is an introverted subspace of C(G), Derighetti et al [10] for duals of introverted subspaces of ppseudo measures P M p (G), and the most recent work by the present authors and Neufang [14] on A(G) * * and UC 2 (G) * . Let G be a locally compact group, 1 < p < ∞, and L (L p (G)) be the space of continuous linear operators on L p (G).…”

Involutions on the second duals of group algebras versus subamenable groups

“…In that case it also follows from Filali and J. Pym ( [FP,Theorem 5]) simply because for discrete G,…”

On the spectra of C⁎-algebras of slowly oscillating functions on locally compact groups

“…The most important applications are maybe those related to the cardinality of the set of left invariant means when G is amenable and to Arens irregularity. Indeed, a careful reader will quickly notice that most of the arguments giving the number of left invariant means or leading to the topological center of LUC(G) * being the measure algebra M (G) and that of the second Banach dual of the group algebra L 1 (G) being L 1 (G) are based on sets (or nets) of points taken from the LUC-compactification G LU C of G. See, for example, [8,11,14,20], for the number of left invariant means and other related results, and [7,9,15,16,18,19,22], for the results on topological centers.…”