On logarithms of measures

Abstract

Let A be a Banach algebra and let xin A have the property that its spectrum does not separate 0 from infinity. It is well known that x has a logarithm, i.e., there exists yin A with x=e^y. We will use this statement to identify measures defined on a locally compact group to have logarithms. Also, we will show that the converse of the above statement is in general not true. Our results will be related to infinitely divisible probability measures.