Zalar, Borut On Hilbert spaces with unital multiplication. (English) Zbl 0843.46040 Proc. Am. Math. Soc. 123, No. 5, 1497-1501 (1995). A celebrated theorem of L. Ingelstam [Bull. Am. Math. Soc. 69, 794-796 (1963; Zbl 0118.32005)] asserts that, if \(A\) is an associative real algebra with a unit \(\text{\textbf{1}}\), if \(|\cdot|\) is a norm on the vector space of \(A\) satisfying \(|\text{\textbf{1}}|= 1\) and \(|xy|\leq |x||y|\) for all \(x\), \(y\) in \(A\), and if the norm \(|\cdot|\) derives from an inner product, then \(A\) is isomorphic to \(\mathbb{R}\), \(\mathbb{C}\), or \(\mathbb{H}\) (the algebra of Hamilton’s quaternions). This result has been reproved and/or improved many times in the literature. This is the case for the paper we are reviewing. It is shown that Ingelstam’s theorem remains true if either the assumption \(|xy|\leq |x||y|\) is relaxed to \(|x^2|\leq |x|^2\) or the assumption of the existence of a unit is dropped and the inequality \(|xy|\leq |x||y|\) is replaced by the equality \(|x^2|= |x|^2\). Reviewer: A.Rodriguez-Palacios (Granada) Cited in 3 Documents MSC: 46K15 Hilbert algebras 46C15 Characterizations of Hilbert spaces 46H70 Nonassociative topological algebras Keywords:inner product; quaternions; Ingelstam’s theorem Citations:Zbl 0118.32005 PDFBibTeX XMLCite \textit{B. Zalar}, Proc. Am. Math. Soc. 123, No. 5, 1497--1501 (1995; Zbl 0843.46040) Full Text: DOI References: [1] John Froelich, Unital multiplications on a Hilbert space, Proc. Amer. Math. Soc. 117 (1993), no. 3, 757 – 759. · Zbl 0795.46038 [2] Lars Ingelstam, A vertex property for Banach algebras with identity, Math. Scand. 11 (1962), 22 – 32. · Zbl 0122.35003 · doi:10.7146/math.scand.a-10646 [3] M. F. Smiley, Real Hilbert algebras with identity, Proc. Amer. Math. Soc. 16 (1965), 440 – 441. · Zbl 0161.10904 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.