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Group representations with empty residual spectrum. (English) Zbl 1220.47005
Integral Equations Oper. Theory 67, No. 1, 95-107 (2010); erratum ibid. 69, No. 1, 149-150 (2011).
Let \(\Gamma\) be a discrete group acting on a Banach space \(X\). Elements of the group algebra \(\mathbb{C}\Gamma\) are therefore represented as bounded linear operators in \(X\). The main result in this paper proves that, if \(X\) is either the reduced group C*-algebra \(C^*_r(\Gamma)\) or a noncommutative \(L^p(M)\) (where \(M\) is the group von Neumann algebra of \(\Gamma\), with respect to the natural trace in \(M\)), with \(\Gamma\) acting on \(X\) by left translation, then the spectrum of every operator \(T\) in \(\mathbb{C}\Gamma\) consists of approximate eigenvalues.

MSC:
47A10 Spectrum, resolvent
47C10 Linear operators in \({}^*\)-algebras
47C15 Linear operators in \(C^*\)- or von Neumann algebras
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