Anjidani, E. On Gelfand-Mazur theorem on a class of \(F\)-algebras. (English) Zbl 1316.46040 Topol. Algebra Appl. 2, 19-23 (2014). The paper generalizes the Gelfand-Mazur theorem for a special class of \(F\)-algebras: fundamental division \(F\)-algebras with bounded elements and with dual of the algebra separating the points of algebra. It is shown that in this case the algebra under consideration is isomorphic to the field of complex numbers. As an addition, some examples of non-locally bounded \(F\)-algebras and non-locally convex \(F\)-algebras belonging to that class of \(F\)-algebras are given. It is also shown that the spectrum of an arbitrary element of a fundamental \(F\)-algebra whose dual separates its points is a nonempty compact set. Reviewer: Mart Abel (Tartu) Cited in 1 Review MSC: 46H05 General theory of topological algebras 46H20 Structure, classification of topological algebras Keywords:Gelfand-Mazur theorem; \(F\)-algebra; topological algebra with bounded elements; fundamental algebra; dual separating the points; division algebra PDFBibTeX XMLCite \textit{E. Anjidani}, Topol. Algebra Appl. 2, 19--23 (2014; Zbl 1316.46040) Full Text: DOI References: [1] M. Abel, Topological algebras with idempotently pseudoconvex von Neumann bornology, Contemp. Math. 427 (2007) 15- 29.; · Zbl 1125.46037 [2] M. Abel and W. ˙Z elazko, Topologically invertible elements and topological spectrum, Bull. Pol. Acad. Sc. Math. 54 (2006) 257-271.; · Zbl 1114.46036 [3] G. R. Allan, A spectral theory for locally convex algebra, Proc. Landon Math. Soc. 115 (1965) 399-421.; · Zbl 0138.38202 [4] E. Ansari Piri, A class of factorable topological algebra, Proceedings of the Edinburgh Mathematical Sociaty 33 (1990) 53- 59.; · Zbl 0699.46027 [5] R. Arens, Linear topological division algebras, Bull. Amer. Math. Soc. 53 (1947) 623-630.; · Zbl 0031.25103 [6] V. M. Bogdan, On Frobenius, Mazur, and Gelfand-Mazur Theorems on Division Algebras, Quaestiones Mathematicae 29 (2006) 171-209.; · Zbl 1126.46033 [7] F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, Heidelberg and Berlin, 1973.; · Zbl 0271.46039 [8] A. Ya. Helemskii, Banach and locally convex algebras, Oxford university press, 1993.; [9] A. Mallios, Topological algebras, selected topics, Mathematical Studies, North Holland, Amsterdam, 1986.; · Zbl 0597.46046 [10] W. Zelazko, A theorem on B0 division algebras, Bull. Polon. Acad. Sc. 8 (1960) 373-375.; · Zbl 0095.31303 [11] W. Zelazko, F-algebras: Some results and open problems, Functional Analysis and its Applications 197 (2004) 317-326.; · Zbl 1096.46028 [12] W. Zelazko, On the locally bounded and m-convex topological algebras, Studia Math. 19 (1960) 333-356.; · Zbl 0096.08303 [13] W. Zelazko, continuous characters and joint topological spectrum, Control and Cybernetics 36 (2007) 859-864.; · Zbl 1189.46040 [14] W. Zelazko, Selected Topics in Topological Alghebras, Aarhus Univ. Lecture Notes 31 1971.; · Zbl 0221.46041 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.