Let \(w:\mathbb{R}^ +\to\mathbb{R}^ +\) be a positive continuous submultiplicative function, and let \(L^ 1(w)\) be the corresponding Banach algebra of Lebesgue integrable functions on \(\mathbb{R}^ +\) with weight \(w\) and with convolution product. Further, let \(M(w)\) be the Banach algebra of all Radon measures on \(\mathbb{R}^ +\) with weight \(w\) and convolution product. It is well known that \(L^ 1(w)\) and \(M(w)\) are semisimple if and only if \(\lim_{t\to\infty}- (\log w(t))/t\) is finite.
In this paper the following is proved: Suppose \(L^ 1(w_ 1)\) and \(L^ 1(w_ 2)\) are semisimple. If \(\theta\) is a homomorphism from \(M(w_ 1)\) onto \(M(w_ 2)\), then \(\theta\) maps \(L^ 1(w_ 1)\) onto \(L^ 1(w_ 2)\). A necessary and sufficient condition is given for \(M(w_ 1)\) and \(M(w_ 2)\) to be isomorphic. These results are used to describe all the automorphisms of \(M(w)\).