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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 \(\text{\textbf{1}}\), 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 \(|x^2|= |x|^2\).

MSC:

46K15 Hilbert algebras
46C15 Characterizations of Hilbert spaces
46H70 Nonassociative topological algebras

Citations:

Zbl 0118.32005
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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
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