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Some classes of pre-Hilbert algebras with norm-one central idempotent. (English) Zbl 1292.46033

It is well known that every alternative pre-Hilbert algebra with a unit \(1\) such that \(||1||=1\) is isomorphic to \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) or \(\mathbb{O}\). By studying certain extensions of this result, the class of pre-Hilbert algebras satisfying the identity \((x((xy)x))x=(x^2y)x^2\) and with a norm-one idempotent \(e\) such that \(||ex||=||x||=||xe||\) for any \(x\), appears in a natural way.
In the paper under review, the author gives a characterization of the class of pre-Hilbert algebras satisfying the above identity, \((x((xy)x))x=(x^2y)x^2\), and having a norm-one central idempotent \(e\) such that \(||ex||=||x||\). Since the existence of a norm-one central idempotent \(e\) is not guaranteed in every pre-Hilbert algebra \(A\neq 0\), it is also shown that, if \(A \) is power-associative and \(||a^2||=||a||^2\) for all \(a \in A\), then \(A\) has only a nonzero idempotent, which is the unit element in \(A\). Several conditions making a real algebra, which is also a pre-Hilbert space, isomorphic to \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) or \(\mathbb{O}\), are also provided.

MSC:

46K70 Nonassociative topological algebras with an involution
17D05 Alternative rings
46K15 Hilbert algebras
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References:

[1] A. A. Albert, Power associative rings, Transactions of the American Mathematical Society 64 (1948), 552–593. · Zbl 0033.15402 · doi:10.1090/S0002-9947-1948-0027750-7
[2] A. Cedilnik and A. Rodríguez, Continuity of homomorphisms into complete normed algebraic algebras, Journal of Algebra 264 (2003), 6–14. · Zbl 1032.17002 · doi:10.1016/S0021-8693(03)00137-6
[3] A. Cedilnik and B. Zalar, Nonassociative algebras with submultiplicative bilinear form, Acta Mathematica Universitatis Comenianae 63 (1994), 285–301. · Zbl 0823.17003
[4] J. A. Cuenca, One-sided division infinite dimensional normed real algebras, Publicacions Matematiques 36 (1992), 485–488. · Zbl 0783.17001 · doi:10.5565/PUBLMAT_362A92_14
[5] J. A. Cuenca, On composition and absolute valued algebras, Proceedings of the Royal Society of Edinburgh. Section A 136A (2006), 717–731. · Zbl 1153.17002
[6] J. A. Cuenca, On an Ingelstam’s theorem, Communications in Algebra 35 (2007), 4057–4067. · Zbl 1137.17003 · doi:10.1080/00927870701511343
[7] J. A. Cuenca, On structure theory of pre-Hilbert algebras, Proceedings of the Royal Society of Edinburgh. Section A 139 (2009), 303–319. · Zbl 1222.46035 · doi:10.1017/S0308210507000753
[8] J. Froelich, Unital multiplications on a Hilbert space, Proceedings of the American Mathematical Society 117 (1993), 757–759. · Zbl 0795.46038 · doi:10.1090/S0002-9939-1993-1116259-8
[9] L. Ingelstam, Hilbert algebras with identity, American Mathematical Society, Bulletin 69 (1963), 794–795. · Zbl 0118.32005 · doi:10.1090/S0002-9904-1963-11035-6
[10] L. Ingelstam, Non-associative normed algebras and Hurwitz’ problem, Arkiv för Matematik 5 (1964), 231–238. · Zbl 0136.02201 · doi:10.1007/BF02591125
[11] S. H. Kulkarni, A very simple and elementary proof of a theorem of Ingelstam, The American Mathematical Monthly 111 (2004), 54–58. · Zbl 1062.46035 · doi:10.2307/4145018
[12] A. Moutassim and A. Rochdi, Sur les algèbres préhilbertiennes vérifiant |a 2| |a|2, Advances in Applied Clifford Algebras 18 (2008), 269–278. · Zbl 1152.46045 · doi:10.1007/s00006-008-0071-1
[13] J. M. Osborn, Quadratic division algebras, Transactions of the American Mathematical Society 105 (1962), 202–221. · Zbl 0136.30303 · doi:10.1090/S0002-9947-1962-0140550-X
[14] A. Rodríguez, Nonassociative normed algebras spanned by hermitian elements, Proceedings of the London Mathematical Society 47 (1983), 258–274. · Zbl 0521.47036
[15] A. Rodríguez, One-sided division absolute valued algebras, Publicacions Matematiques 36 (1992), 925–954. · Zbl 0797.46040 · doi:10.5565/PUBLMAT_362B92_12
[16] R. D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York/San Francisco/London, 1966. · Zbl 0145.25601
[17] M. F. Smiley, Real Hilbert algebras with identity, Proceedings of the American Mathematical Society 16 (1965), 440–441. · Zbl 0161.10904 · doi:10.1090/S0002-9939-1965-0176316-2
[18] K. Urbanik and F. B. Wright, Absolute valued algebras, Proceedings of the American Mathematical Society 11 (1960), 861–866. · Zbl 0156.03801 · doi:10.1090/S0002-9939-1960-0120264-6
[19] B. Zalar, On Hilbert spaces with unital multiplication, Proceedings of the American Mathematical Society 123 (1995), 1497–1501. · Zbl 0843.46040 · doi:10.1090/S0002-9939-1995-1233986-4
[20] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov and A. I. Shirshov, Rings that are nearly associative, Academic Press, New York, 1982. · Zbl 0487.17001
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