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- W2615559987 abstract "Let A be an ordered Banach algebra with a unit $$mathbf{e}$$ and a cone $$A^+$$ . An element p of A is said to be an order idempotent if $$p^2 = p$$ and $$0 le ple mathbf{e}$$ . An element $$ain A^+$$ is said to be irreducible if the relation $$(mathbf{e}-p)ap = 0$$ , where p is an order idempotent, implies $$p = 0$$ or $$p = mathbf{e}$$ . For an arbitrary element a of A the peripheral spectrum $$sigma _mathrm{per}(a)$$ of a is the set $$sigma _mathrm{per}(a) = {lambda in sigma (a):|lambda | = r(a)}$$ , where $$sigma (a)$$ is the spectrum of a and r(a) is the spectral radius of a. We investigate properties of the peripheral spectrum of an irreducible element a. Conditions under which $$sigma _mathrm{per}(a)$$ contains or coincides with $$r(a)H_m$$ , where $$H_m$$ is the group of all $$m^mathrm{th}$$ roots of unity, and the spectrum $$sigma (a)$$ is invariant under rotation by the angle $$frac{2pi }{m}$$ for some $$min {mathbb N}$$ , are given. The correlation between these results and the existence of a cyclic form of a is considered. The conditions under which a is primitive, i.e., $$sigma _mathrm{per}(a) = {r(a)}$$ , are studied. The necessary assumptions on the algebra A which imply the validity of these results, are discussed. In particular, the Lotz–Schaefer axiom is introduced and finite-rank elements of A are defined. Other approaches to the notions of irreducibility and primitivity are discussed. Conditions under which the inequalities $$0 le b < a$$ imply $$r(b) < r(a)$$ are studied. The closedness of the center $$A_mathbf{e}$$ , i.e., of the order ideal generated by $$mathbf{e}$$ in A, is proved." @default.
- W2615559987 created "2017-05-26" @default.
- W2615559987 creator A5001955525 @default.
- W2615559987 date "2018-03-09" @default.
- W2615559987 modified "2023-09-26" @default.
- W2615559987 title "On the peripheral spectrum of positive elements" @default.
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- W2615559987 doi "https://doi.org/10.1007/s11117-018-0562-9" @default.
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