Title: Geometric Measure of Arens Irregularity
Speaker: Roberto Hernandez Palomares (OSU)
Abstract: We will study Banach Algebras and its second duals. The question we ask is, if we start with a Banach Algebra $B$ can we define a canonical product on $B^{**}$ so that it becomes a Banach Algebra itself? The answer is positive and (often) not unique. There are two ways to extend the multiplication, known as the Right and Left Arens Products. Examples are known for which these operations are not equal and we wish to measure how different they are. To do so, we introduce a recently defined geometric invariant which is a number in the $[0,2]$ interval.
In the following we specialize to $\ell^1(G),$ for infinite and discrete groups $G$ and describe an embedding $\beta (G) \hookrightarrow {\ell^1}(G)^{**}$. Here, we regard the second dual of $\ell^1(G)$ with an Arens product and the ultrafilters with their addition. This technique will allow us to compute the invariant. Next, we will focus and characterize the very particular subsets of $G$ which reveal the difference between the Arens products. Finally, we prove such sets must exist for any such $G$ and exhibit some examples.
