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On Hilbert spaces with unital multiplication. (English) Zbl 0843.46040
A celebrated theorem of {\it 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 $\text{{\bf 1}}$, if $|\cdot|$ is a norm on the vector space of $A$ satisfying $|\text{{\bf 1}}|= 1$ and $|xy|\le |x||y|$ for all $x$, $y$ in $A$, and if the norm $|\cdot|$ derives from an inner product, then $A$ is isomorphic to $\bbfR$, $\bbfC$, or $\bbfH$ (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 $|xy|\le |x||y|$ is relaxed to $|x^2|\le |x|^2$ or the assumption of the existence of a unit is dropped and the inequality $|xy|\le |x||y|$ is replaced by the equality $|x^2|= |x|^2$.

##### MSC:
 46K15 Hilbert algebras 46C15 Characterizations of Hilbert spaces 46H70 Nonassociative topological algebras
##### Keywords:
inner product; quaternions; Ingelstam’s theorem
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