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Proceedings of the American Mathematical Society

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Nilpotent elements in Banach algebras

Author: H. Behncke
Journal: Proc. Amer. Math. Soc. 37 (1973), 137-141
MSC: Primary 46K05
MathSciNet review: 0315457
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Abstract: Let $ \mathfrak{A}$ be an $ {A^\ast}$-algebra such that any maximal abelian $ ^\ast$-subalgebra is regular and such that any quasinilpotent element x in $ \mathfrak{A}$ satisfies $ {x^N} = 0$, with $ N < \infty $. Then any irreducible Hilbert space $ ^\ast$-representation is at most N-dimensional. If $ \mathfrak{A}$ is a $ {C^\ast}$-algebra, $ \mathfrak{A}$ possesses transcendental quasinilpotent elements if there exists a $ \pi \in \hat{\mathfrak{A}}$ with $ \dim \pi = \infty $.

References [Enhancements On Off] (What's this?)

  • [1] H. Behncke, Nilpotent elements in group algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 19 (1971), 197–198 (English, with Russian summary). MR 0283582
  • [2] Jacques Dixmier, Les 𝐶*-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
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Keywords: Nilpotent elements, Banach $ ^\ast$-algebras, $ {C^\ast}$-algebras
Article copyright: © Copyright 1973 American Mathematical Society