Module maps over locally compact quantum groups

Abstract

We study locally compact quantum groups G and their module maps through a general Banach algebra approach. As applications, we obtain various characterizations of compactness and discreteness, which in particular generalize a result by Lau (1978) and recover another one by Runde (2008). Properties of module maps on L∞(G) are used to characterize strong Arens irregularity of L 1 (G) and are linked to commutation relations over G with several double commutant theorems established. We prove the quantum group version of the theorems by Young (1973), Lau (1981), and Forrest (1991) regarding Arens regularity of the group algebra L 1 (G) and the Fourier algebra A(G). We also extend the classical Eberlein theorem on the inclusion B(G) ⊆ W AP (G) to all locally compact quantum groups.

Completely bounded multipliers over locally compact quantum groups

“…However, the left (T (L 2 (G)), )-module structure is significantly complicated than the right (T (L 2 (G)), )-module structure. We will study these module structures on B(L 2 (G)) in [22] and a subsequent work.…”

“…In the case where G = L ∞ (G), the above equality becomes LUC(G) * = M (G) (since it is known from Lau [30] that Z t (LUC(G) * ) = M (G)), which holds if and only if G is compact. We showed in [22] that LUC(G) * being a dual Banach algebra is equivalent to the compactness of G when either L 1 (G) is quotient strongly Arens irregular or G is amenable with L 1 (G) separable. It was also shown in [22] that the compactness of G is equivalent to LUC(G) * = M (G).…”

“…We showed in [22] that LUC(G) * being a dual Banach algebra is equivalent to the compactness of G when either L 1 (G) is quotient strongly Arens irregular or G is amenable with L 1 (G) separable. It was also shown in [22] that the compactness of G is equivalent to LUC(G) * = M (G). We note that this equivalence was given in [44,Proposition 4.2] under the condition that G is co-amenable.…”

Module maps on duals of Banach algebras and topological centre problems

“…This in particular shows that for all amenable locally compact groups G, the strong Arens irregularity of the Fourier-Stieltjes algebra B(G) implies that of the Fourier algebra A(G). In the subsequent work [22], we will investigate module maps over locally compact quantum groups through a general Banach algebra approach as used in the present paper.…”

“…Then there exists a central projection p in C 0 (G) * * such that L 1 (G) ⊥ = (1 − p)C 0 (G) * * , and thus we haveC 0 (G) * * = pC 0 (G) * * ⊕ ∞ L 1 (G) ⊥ ∼ = L ∞ (G) ⊕ ∞ L 1 (G) ⊥ via x ⊕ y −→ x| L 1 (G) ⊕ y.Let κ : L ∞ (G) −→ C 0 (G) * * be the induced normal and injective * -homomorphism. By[22, Proposition 3.6],L 1 (G) = M(G) if and only if κ(C 0 (G)) = C 0 (G) (respectively, κ(M(C 0 (G))) = M(C 0 (G))). It follows that the following statements are equivalent:(i) L 1 (G) = M(G); (ii) κ(LUC(G)) = LUC(G); (iii) κ(L ∞ (G)) = C 0 (G) * * .Therefore, we do not have κ(LUC(G)) = M(G) * L 1 (G) in general.…”