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.

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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