Standard homomorphisms and regulated weights on weighted convolution algebras (Q804904)

Let \(\omega_ 1,\omega_ 2\) be locally essentially bounded positive Borel functions on \([0,\infty)\). A homomorphism \(\phi:L^ 1(\omega)\to L^ 1(\omega_ 2)\) is said to be standard if whenever \(L^ 1(\omega_ 1)*f\), the image under convolution, is dense in \(L^ 1(\omega_ 1)\) then \(L^ 1(\omega_ 2)*\phi (f)\) is dense in \(L^ 1(\omega_ 2)\). If \(\omega_ 2(x)\) is regulated at any \(a\geq 0\) then it is shown that all continuous nonzero homomorphisms are standard. In general there are close relations between standard elements and standard homomorphisms.