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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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