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On Hilbert spaces with unital multiplication. (English) Zbl 0843.46040
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 1, if |·| is a norm on the vector space of A satisfying |1|=1 and |xy||x||y| for all x, y in A, and if the norm |·| derives from an inner product, then A is isomorphic to , , or (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||x||y| is relaxed to |x 2 ||x| 2 or the assumption of the existence of a unit is dropped and the inequality |xy||x||y| is replaced by the equality |x 2 |=|x| 2 .
MSC:
46K15Hilbert algebras
46C15Characterizations of Hilbert spaces
46H70Nonassociative topological algebras