Module maps on duals of Banach algebras and topological centre problems

Abstract

We study various spaces of module maps on the dual of a Banach algebra A, and relate them to topological centres. We introduce an auxiliary topological centre Z t ( A * A * ) ♦ for the left quotient Banach algebra A * A * of A * * . Our results indicate that Z t ( A * A * ) ♦ is indispensable for investigating properties of module maps over A * and for understanding some asymmetry phenomena in topological centre problems as well as the interrelationships between different Arens irregularity properties. For the class of Banach algebras of type (M) introduced recently by the authors, we show that strong Arens irregularity can be expressed both in terms of automatic normality of A * * -module maps on A * and through certain commutation relations. This in particular generalizes the earlier work on group algebras by Ghahramani and McClure (1992) [13] and by Ghahramani and Lau (1997) [12]. We link a module map property over A * to the space WAP(A) of weakly almost periodic functionals on A, generalizing a result by Lau and Ülger (1996) [34] for Banach algebras with a bounded approximate identity. We also show that for a locally compact quantum group G, the quotient strong Arens irregularity of L 1 (G) can be obtained from that of M(G) and can be characterized via the canonical C 0 (G)-module structure on LUC(G) * .

Arens irregularity of the trace class convolution algebra

“…As shown in [7], there are another two strong topological centres of A * * , which are intrinsically related to Q-SAI of the algebra A. These two strong topological centres are defined in [7] by…”

“…It is seen that (T (L 2 (G)), ) = (N (L 2 (G)), * ) op when G is the commutative quantum group L ∞ (G). It has been shown in the recent work [5][6][7][8][9][10] that many interesting results in abstract harmonic analysis over locally compact groups can be generalized to the locally compact quantum group setting. In the present paper, we continue to pursue the development of quantum harmonic analysis by studying certain Arens irregularity properties of the convolution algebra (T (L 2 (G)), ).…”

On invariant and natural projections

“…In our proofs we will use the result below which follows from [11, Theorem 3.2 (V)]. Lemma Let $${\mathcal {A}}$$ be a commutative Banach algebra such that $$\overline{\mathrm{lin}} {\mathcal {A}}^2 = {\mathcal {A}}$$ (for example, $${\mathcal {A}}$$ has a BAI).…”

“…Invariant projections play an important role in amenability theory, and in the study of ideals and multipliers in Banach algebras and abstract harmonic analysis; see, for example, [1,7,11,15,18,19,22,25,27]. Here, in the general setting, one considers a Banach algebra , and a projection on the dual  * (that is, a bounded linear map 𝑃 ∶  * →  * such that 𝑃 2 = 𝑃) which is invariant under…”

“…Invariant projections play an important role in amenability theory, and in the study of ideals and multipliers in Banach algebras and abstract harmonic analysis; see, for example, [1, 7, 11, 15, 18, 19, 22, 25, 27]. Here, in the general setting, one considers a Banach algebra $${\mathcal {A}}$$, and a projection on the dual $${\mathcal {A}}^*$$ (that is, a bounded linear map $$P: {\mathcal {A}}^* \rightarrow {\mathcal {A}}^*$$ such that $$P^2 = P$$) which is invariant under the canonical left, respectively right, action of $${\mathcal {A}}$$ on $${\mathcal {A}}^*$$.…”