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Research Report MRR01-013
Weak* Properties of Weighted Convolution Algebras
Sandy Grabiner
Abstract:
Suppose that L1(\omega) is a weighted convolution
algebra on
R+ = [0, 1) with the weight \omega(t)$ normalized
so that
the corresponding space M(\omega) of measures is the dual space of
the space C0(1/\omega) of continuous functions. Suppose
that $\phi
: L1(\omega) -> L1(\omega') is a
continuous nonzero homomorphism,
where L1(\omega') is also a convolution algebra. If
L1(\omega) *
f is norm dense in L1(\omega), we show that
L1(\omega') *
\phi(f) is (relatively) weak* dense in
L1(\omega'),
and
we identify the norm closure of L1(\omega') *
\phi(f) with the
convergence set for a particular semigroup. When \phi is
weak* continuous it is enough for L1(\omega)
* f to be
weak* dense in L1(\omega). We also give
sufficient
conditions and characterizations of weak* continuity of \phi.
In
addition, we show that, for all nonzero f in
L1(\omega), the
sequence fn / || fn || converges
weak* to 0. When
\omega is regulated, fn+1 / ||
fn || converges to 0 in norm.
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