On logarithms of measures (Q6068742)

In this paper, measures on a locally compact group having logarithms are identified by means of the following statement. Let \(A\) be a Banach algebra and let \(x \in A\) have the property that its spectrum does not separate \(0\) from infinity. The point is that it is well known that such an \(x\) has a logarithm $y\in A$, meaning $e^y=x$. It is also shown that the converse of the above statement is in general not true. The results obtained are related to infinitely divisible probability measures.