On the size of orbits in the duals of 𝐶*-algebras and convolution algebras

Abstract

We solve an open problem raised in [Trans. Amer. Math. Soc. 366 (2014), pp.4151–4171] concerning (infinite-dimensional) commutative semisimple Banach algebras A \mathcal {A} with a bounded approximate identity, namely, whether there always exists a functional f ∈ A ∗ f \in \mathcal {A}^* such that the orbit subspace A ∗ ∗ ◻ f \mathcal {A}^{**} \Box f of A ∗ \mathcal {A}^* is w ∗ w^* -closed and infinite-dimensional. Indeed, on the one hand, we show that no commutative C ∗ C^* -algebra shares this property. On the other hand, we prove that the answer is positive, in a strong sense, in the case of convolution algebras A = L 1 ( G ) \mathcal {A} = L_1(\mathcal {G}) , for large classes of locally compact groups G \mathcal {G} (commutativity is not needed): there exists f ∈ A ∗ f \in \mathcal {A}^* with maximal orbit, in fact, f f satisfies B a l l ( A ∗ ) = B a l l ( A ∗ ∗ ) ◻ f \mathrm {Ball} (\mathcal {A}^*) = \mathrm {Ball}(\mathcal {A}^{**}) \Box f . Moreover, as we shall see, the latter property links the size of orbits to Arens irregularity.