Nilpotent elements in Banach algebras
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- by H. Behncke
- Proc. Amer. Math. Soc. 37 (1973), 137-141
- DOI: https://doi.org/10.1090/S0002-9939-1973-0315457-8
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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
- H. Behncke, Nilpotent elements in group algebras, Bull. Acad. Polon. Sci. SĂ©r. Sci. Math. Astronom. Phys. 19 (1971), 197â198 (English, with Russian summary). MR 283582
- Jacques Dixmier, Les $C^{\ast }$-algĂšbres et leurs reprĂ©sentations, Cahiers Scientifiques [Scientific Reports], Fasc. XXIX, Gauthier-Villars & Cie, Ăditeur-Imprimeur, Paris, 1964 (French). MR 171173
- Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 115101
- David M. Topping, Transcendental quasi-nilpotents in operator algebras, J. Functional Analysis 2 (1968), 342â351. MR 233212, DOI 10.1016/0022-1236(68)90011-6
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 137-141
- MSC: Primary 46K05
- DOI: https://doi.org/10.1090/S0002-9939-1973-0315457-8
- MathSciNet review: 0315457