A Very Simple and Elementary Proof of a Theorem of Ingelstam

Kulkarni, S. H.

doi:10.1080/00029890.2004.11920049

Cited by 3 publications

(1 citation statement)

References 6 publications

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“…This result answers a previous conjecture by Kaplansky. Later proofs have been given in [28,35,51,54]. In the non-associative and non-unital cases very little is known on real normed algebras with norm deriving from an inner product, despite the fact that then different and interesting algebras appear, as the real division algebra O of the octonions, or the algebras C, H or O.…”

Section: Introductionmentioning

confidence: 99%

On structure theory of pre-Hilbert algebras

Mira¹

2009

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Let A be a real (non-associative) algebra which is normed as real vector space, with a norm · deriving from an inner product and satisfying ac a c for any a, c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a 2 c)a 2 holds in A, then the existence of an idempotent e such that e = 1 and ea = a = ae , a ∈ A, implies that A is isometrically isomorphic to R, C, H, O, C, H, O or P. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

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Section: Introductionmentioning

confidence: 99%

On structure theory of pre-Hilbert algebras

Mira¹

2009

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Gelfand–Mazur Theorems in normed algebras: A survey

Bhatt

Kulkarni

2018

Expositiones Mathematicae

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Symmetric Quantum Sets and L-Algebras

Rump

2020

International Mathematics Research Notices

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Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size $>3$ is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size $>3$ are shown to be representable by Hilbert spaces over ${\mathbb{R}}$, ${\mathbb{C}}$, or ${\mathbb{H}}$. Symmetric quantum sets are characterized as a class of $L$-algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.

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A Very Simple and Elementary Proof of a Theorem of Ingelstam

Cited by 3 publications

References 6 publications

On structure theory of pre-Hilbert algebras

On structure theory of pre-Hilbert algebras

Gelfand–Mazur Theorems in normed algebras: A survey

Symmetric Quantum Sets and L-Algebras

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