A generalization of a theorem by Bochner

Abstract:

A theorem of Bochner states that if $\mu$ is a complex Borel measure on the $n$-dimensional torus ${{\mathbf {T}}^n}$ with Fourier-coefficients that vanish outside a proper cone in ${{\mathbf {Z}}^n}$, then $\mu$ is absolutely continuous with respect to Haar measure on ${{\mathbf {T}}^n}$. This result is generalized to a ${C^ \ast }$-algebra setting using the concept of spectral subspaces for an $n$-parameter group of automorphisms and its dual group, in the case where the cone is the positive “octant".