Rochdi, Abdellatif; Rodriguez Palacios, Ángel Absolute valued algebras with involution. (English) Zbl 1177.46039 Commun. Algebra 37, No. 4, 1151-1159 (2009). Absolute valued algebras with involution (avawi, for short), as defined in K.Urbanik’s paper [Fundam.Math.49, 247–258 (1961; Zbl 0267.17005)], are dealt with in this paper. The authors prove that any avawi satisfying the identity \((x, x^2, x)=0\), with \((\cdot, \cdot, \cdot)\) denoting the associator, is finite-dimensional, and that in dimension different from two, isomorphisms between avawi’s are in fact \(*\)-isomorphisms. They also classify finite-dimensional avawi’s and prove that there exists an eight-dimensional avawi which contains a nonzero central idempotent and where the group of its automorphisms is trivial. Reviewer: Antonio Fernández López (Malaga) Cited in 1 ReviewCited in 8 Documents MSC: 46K70 Nonassociative topological algebras with an involution 46H70 Nonassociative topological algebras 46E15 Banach spaces of continuous, differentiable or analytic functions 46B04 Isometric theory of Banach spaces Keywords:absolute valued algebra; involution Citations:Zbl 0267.17005 PDFBibTeX XMLCite \textit{A. Rochdi} and \textit{Á. Rodriguez Palacios}, Commun. Algebra 37, No. 4, 1151--1159 (2009; Zbl 1177.46039) Full Text: DOI References: [1] Albert A. A., Ann. Math. 48 pp 495– (1947) · Zbl 0029.01001 · doi:10.2307/1969182 [2] Becerra J., J. Algebra 293 pp 448– (2005) · Zbl 1133.46028 · doi:10.1016/j.jalgebra.2005.06.020 [3] El-Mallah M. L., Arch. Math. 51 pp 39– (1988) · Zbl 0649.17007 · doi:10.1007/BF01194152 [4] El-Mallah M. L., J. Algebra 128 pp 180– (1990) · Zbl 0688.17001 · doi:10.1016/0021-8693(90)90048-S [5] El-Mallah M. L., Linear Algebra Appl. 414 pp 295– (2006) · Zbl 1134.46033 · doi:10.1016/j.laa.2005.10.005 [6] Ramírez M. I., Proceedings of the International Conference on Jordan Structures (Málaga, June 1997) pp 169– (1999) [7] Rochdi A., Int. J. Math. Math. Sci. 70 pp 4447– (2003) · Zbl 1068.17003 · doi:10.1155/S016117120320538X [8] Rodríguez A., Advanced Courses of Mathematical Analysis I, Proceedings of the First International School Cádiz pp 22– (2004) [9] Urbanik K., Fundamenta Math. 49 pp 247– (1961) [10] Urbanik K., Proc. Amer. Math. Soc. 11 pp 861– (1960) · doi:10.1090/S0002-9939-1960-0120264-6 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.