Fourier algebra and harmonic operators

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We examine the common null spaces of families of Herz- Schur multipliers and apply our results to study jointly harmonic opera- tors and their relation with jointly harmonic functionals. We give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite func- tion. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.