Banach function algebras and approximate identities

When
Start: 03/11/2014 - 4:15pm
End  : 03/11/2014 - 5:15pm

Category
Colloquium

Speaker
H. G. Dales (Lancaster University)

Abstract

Let $ K $ be a locally compact space. Then $ C_o(K) $ is the Banach algebra of all continuous,
complex-valued functions on $ K $ that vanish at infinity, taken with the pointwise algebraic
operations and uniform norm $ |\cdot|_K $. The maximal modular ideals of $ C_o(K) $ have the
form $ M_x =\{ f\in C(K) \ \colon \ f(x) = 0\} $, where $ x\in K $. We see that each of these maximal
ideals has a contractive approximate identity: this is a net $ (f_\alpha) $ in
$ M_x $ with $ |f_\alpha|_K\leqslant 1 $ and $ \lim_\alpha ff_\alpha=f $ for each $ f\in M_x $.

Now suppose that $ A $ is a Banach function algebra on $ K $ such that each maximal modular
ideal has such a contractive approximate identity. Must we have $ A=C_o(K) $? Or can you
think of such an algebra $ A $ with $ A \not= C_o(K) $? How close does the group algebra of
a locally compact group come to having the above properties?

This is joint work with Ali  Ülger of Istanbul

Where
Seeley G. Mudd Science Library 126, Pomona College

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