John Sawatzky, Department of Pure Mathematics, University of Waterloo
"Mind the GAP: Amenability Constants and Arens Regularity of Fourier Algebras"
For a Banach algebra $\mathcal{A}$ there are two natural multiplication operations on the double dual $\mathcal{A}^{**}$ introduced by Arens in 1971, and if these operations agree then the algebra $\mathcal{A}$ is said to be Arens regular. I will discuss Arens regularity of the closures of the Fourier algebra $A(G)$ in the multiplier and completely bounded multiplier norms, denoted $A_M(G)$ and $A_{cb}(G)$ respectively.
Amenable Banach algebras were first studied by B.E. Johnson in 1972. For an amenable Banach algebra $\mathcal{A}$ we can consider its amenability constant $AM(\mathcal{A}) \geq 1$. It is particularly interesting to look at collections of amenable Banach algebras for which there exists a constant $\lambda > 1$ such that the values in the interval $(1,\lambda)$ cannot be attained as amenability constants. If $G$ is a compact group, then the central Fourier algebra is defined as $ZA(G) = ZL^1(G) \cap A(G)$ and endowed with the $A(G)$ norm. I will discuss my investigations into the amenability constant theory of $ZA(G)$ when $G$ is a finite group.
MC 2009 and Online (Contact John Sawatzky at jmsawatzky@uwaterloo.ca for details on how to attend online.)