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Noncommutative Geometry Seminar - Roberto Hernandez Palomares

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March 1, 2018
1:50PM - 2:50PM
Cockins Hall 240

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Add to Calendar 2018-03-01 13:50:00 2018-03-01 14:50:00 Noncommutative Geometry Seminar - Roberto Hernandez Palomares 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. Cockins Hall 240 Department of Mathematics math@osu.edu America/New_York public

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.

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