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 $\mathcal {A}$ with a bounded approximate identity, namely, whether there always exists a functional $f \in \mathcal {A}^*$ such that the orbit subspace $\mathcal {A}^{**} \Box f$ of $\mathcal {A}^*$ is $w^*$-closed and infinite-dimensional. Indeed, on the one hand, we show that no commutative $C^*$-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 $\mathcal {A} = L_1(\mathcal {G})$, for large classes of locally compact groups $\mathcal {G}$ (commutativity is not needed): there exists $f \in \mathcal {A}^*$ with maximal orbit, in fact, $f$ satisfies $\mathrm {Ball} (\mathcal {A}^*) = \mathrm {Ball}(\mathcal {A}^{**}) \Box f$. Moreover, as we shall see, the latter property links the size of orbits to Arens irregularity.