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

Abstract

Let 0 be a continuous nonzero homomorphism of the convolution algebra L 1 ' OC (IR + ) and also the unique extension of this homomorphism to A/| OC (1R + ). We show that the map <f> is continuous in the weak* and strong operator topologies on M^, considered as the dual space of C C (R + ) and as the multiplier algebra of L\ x . Analogous results are proved for homomorphisms from(w\), <ji restricts to a continuous homomorphism from some L'(a>i) to some L'(a>2), and, for each sufficiently large L 1 (coj), <p restricts to a continuous homomorphism from some L' (&>i) to L x (,a>i). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L\ x . We also prove results on convergent nets, continuous semigroups, and bounded sets in M loc that we need in our study of homomorphisms.2000 Mathematics subject classification: primary 43A22, 46J40, 46J45, 43A20,43A10.

Properties of derivations on some convolution algebras

“…We regard C c as the inductive limit of the spaces C 0 [0, n) and equip it with the corresponding inductive limit topology. It follows as in the proof of [13,Proposition 3.3] (see also [9]) that…”