J. Aust. Math. Soc.
81 (2006), 253278

Homomorphisms of the algebra of locally integrable functions on the half line

Sandy Grabiner
Department of Mathematics
Pomona College
610 North College Avenue
Claremont, CA 91711
USA
sgrabiner@pomona.edu



Abstract

Let
be a continuous nonzero homomorphism of the
convolution algebra and also the unique extension of this
homomorphism to . We show that the map
is continuous in the weak* and strong operator
topologies on , considered as the dual space of
and as the multiplier algebra of
. Analogous results are proved for
homomorphisms from
to
. For each convolution algebra
,
restricts to a continuous homomorphism from
some to some
, and, for each sufficiently large
,
restricts to a continuous homomorphism from some
to
. We also determine which continuous
homomorphisms between weighted convolution
algebras extend to homomorphisms of
. We also prove results on convergent nets,
continuous semigroups, and bounded sets in
that we need in our study of homomorphisms.

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