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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A generalization of a theorem by Bochner


Author: Dorte Olesen
Journal: Proc. Amer. Math. Soc. 57 (1976), 115-118
MSC: Primary 46L99
DOI: https://doi.org/10.1090/S0002-9939-1976-0512385-2
MathSciNet review: 0512385
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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".


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DOI: https://doi.org/10.1090/S0002-9939-1976-0512385-2
Keywords: $ {C^ \ast }$-algebras, automorphism groups, spectral subspaces, quasi-invariant functionals
Article copyright: © Copyright 1976 American Mathematical Society