In this monograph we examine the connection between the structure of locally compact semi-algebras and spectral theory, in particular, the spectral theory of positive linear operators. A semi-algebra is a subset of a Banach algebra which is closed under addition, multiplication and multiplication by non-negative real numbers. Local compactness of Banach algebras implies finite dimensionality and it follows that the spectrum of any element therein consists of a finite set of poles. This from our point of view is the trivial case. However, locally compact semi-algebras may contain elements whose spectrum is not trivial and this fact gives them an important role in the spectral theory of compact positive operators.
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