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We characterize the amenability of $Z\ell^1(G)$, the center of the group algebra for $G$. Moreover, we study the characters on the commutative algebra $Z\ell^1(G)$, and consequently, the existence of the bounded approximate identity for the maximal ideals of $Z\ell^1(G)$ will be considered. We also study when an algebra character of $Z\ell^1(G)$ belongs to $c_0$ or $\ell^p$.
Time permitting, we will mention some results about the amenability constant of the center of the group algebra for some particular finite groups.
This is a joint project with Yemon Choi and Ebrahim Samei.
This talk is based on joint work with Matthew Daws (Leeds) and Ebrahim Samei (Saskatchewan).
The Banach algebra $V = V_{1}$ (known as the Volterra algebra) has been the subject of much study. In [1], [2], [3] and [4] derivations and automorphisms of this algebra were studied. This talk is about our recent work on derivations and automorphisms of $V_{p}$ for $p > 1$, as well as the automorphisms and derivations of the $p$-version of the weighted convolutions algebras on the half-line. This is joint work with Sandy Grabiner.
\smallskip \noindent{\bf References.}
[1] F. Ghahramani, The group of automorphisms of $L^{1}(0,1)$ is connected. {\it Trans.\ Amer.\ Math.\ Soc.} 314 (1989), no.~2, 851--859.
[2] F. Ghahramani, The connectedness of the group of automorphisms of $L^{1}(0,1)$, {\it Trans.\ Amer.\ Math.\ Soc.} 302 (1987), no.~2, 647--659.
[3] N. P. Jewell, A. M. Sinclair, Epimorphisms and derivations on $L^{1}(0,1)$ are continuous, {\it Bull.\ London Math.\ Soc.} 8 (1976), no.~2, 135--139.
[4] H. Kamowitz, and S. Scheinberg, Derivations and automorphisms of $L^{1}(0,1)$, {\it Trans.\ Amer.\ Math.\ Soc.} 135 (1969) 415--427.
If we consider a locally compact groupoid $G$, we can define a Fourier algebra $A(G)$. In this talk we are going to present a map that extends $q_0$ to the groupoid context. In particular we need to define a trace-class type groupoid product on spaces that are projective tensor products of amplified $L^2$ row and column spaces.
This represents joint work with Y.-H. Cheng and B.E. Forrest.
Using this approximate diagonal for a group algebra as a motivating example, this talk will discuss the relationship between these tensors and approximate identities and approximate translation invariant means. A general approach for approximate diagonals for predual algebras of locally compact quantum groups will be presented.