A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

$p$-Completely bounded weighted homomorphisms on the $p$-analog of the Fourier-Stieltjes algebra

“…In Section 3, some crucial and previously obtained properties of the algebra B p (G) will be listed, and as a consequence in Section 4 we will investigate p-completely boundedness of some well-known operators on the p-analog of the Fourier-Stieltjes algebras (Theorem 5). The obtained results will be applied in the final section, Section 5, to generalize Ilie's results on homomorphisms of the Figà-Talamanca-Herz algebras in [20].…”

“…It should be noticed that, converse of Lemma 2 is not necessarily true, unless X be a closed subspace of B(E), for some L p -space E. 2. In comparison to [20], because of above explanations, a major difference in our work is that we need to study predual of some crucial p-completely bounded maps (See Theorem 5), instead of their duals.…”

“…In Section 5, we will study the homomorphisms on the p-analog of the Fourier-Stieltjes algebras induced by the continuous map α : Y ⊆ H → G, on locally compact groups G and H, in the cases that α is homomorphism, affine and piecewise affine map, and Y in the coset ring of H. So, we give some preliminaries here. For a locally compact topological group H, let Ω 0 (H) denote the ring of subsets which generated by open cosets of H. By [20] we have…”

“…Daws in [9] introduced the p-operator space structure, with an extensive application to A p (G), that generalizes the operator space structure of A(G). Oztop and Spronk in [24], and Ilie in [20] studied the p-completely bounded homomorphisms on the Figà-Talamanca-Herz algebras, using the p-operator space structure. In [20] it is shown that the map Φ α : A p (G) → A p (H), defined via…”

“…Oztop and Spronk in [24], and Ilie in [20] studied the p-completely bounded homomorphisms on the Figà-Talamanca-Herz algebras, using the p-operator space structure. In [20] it is shown that the map Φ α : A p (G) → A p (H), defined via…”