In this paper, we present a number of examples of commutative Banach algebras with various Arens irregularity properties. These examples illustrate in particular that strong Arens irregularity and extreme non-Arens regularity, the two natural concepts of "maximal" Arens irregularity for general Banach algebras as introduced by Dales-Lau and Granirer, respectively, are indeed distinct. Thereby, an open question raised by several authors is answered. We also link these two properties to another natural Arens irregularity property.

Arens products, Banach algebras, Fourier algebras, Topological centres, Weakly almost periodic functionals
dx.doi.org/10.1090/S0002-9939-08-09678-0
Proceedings of the American Mathematical Society
School of Mathematics and Statistics

Hu, Z. (Zhiguo), & Neufang, M. (2009). Distinguishing properties of Arens irregularity. Proceedings of the American Mathematical Society, 137(5), 1753–1761. doi:10.1090/S0002-9939-08-09678-0