Quadratic and quasi-quadratic functionals

Šemrl, Peter

doi:10.1090/S0002-9939-1993-1158008-3

Quadratic and quasi-quadratic functionals
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by Peter Šemrl

Proc. Amer. Math. Soc. 119 (1993), 1105-1113

DOI: https://doi.org/10.1090/S0002-9939-1993-1158008-3

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Abstract:

In this note we show how Jordan ${\ast }$-derivations arise as a "measure" of the representability of quasi-quadratic functionals by sesquilinear ones. Our main result can be considered as an extension of the Jordan-von Neumann characterization of pre-Hilbert space.

References

T. M. K. Davison, Jordan derivations and quasibilinear forms, Comm. Algebra 12 (1984), no. 1-2, 23–32. MR 732183, DOI 10.1080/00927878408822987
Andrew M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966), 1049–1056. MR 207728, DOI 10.2307/2314635
P. Jordan and J. Von Neumann, On inner products in linear, metric spaces, Ann. of Math. (2) 36 (1935), no. 3, 719–723. MR 1503247, DOI 10.2307/1968653
Svetozar Kurepa, The Cauchy functional equation and scalar product in vector spaces, Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 19 (1964), 23–36 (English, with Serbo-Croatian summary). MR 171100
Svetozar Kurepa, Quadratic and sesquilinear functionals, Glasnik Mat.-Fiz. Astronom. Društvo Mat. Fiz. Hrvatske Ser. II 20 (1965), 79–92 (English, with Serbo-Croatian summary). MR 193390
Peter emrl, On quadratic and sesquilinear functionals, Aequationes Math. 31 (1986), no. 2-3, 184–190. MR 867516, DOI 10.1007/BF02188187
Peter emrl, On quadratic functionals, Bull. Austral. Math. Soc. 37 (1988), no. 1, 27–28. MR 926973, DOI 10.1017/S0004972700004111
Peter emrl, Quadratic functionals and Jordan $*$-derivations, Studia Math. 97 (1991), no. 3, 157–165. MR 1100685, DOI 10.4064/sm-97-3-157-165
J. Vukman, A result concerning additive functions in Hermitian Banach $^{\ast }$-algebras and an application, Proc. Amer. Math. Soc. 91 (1984), no. 3, 367–372. MR 744631, DOI 10.1090/S0002-9939-1984-0744631-9
J. Vukman, Some results concerning the Cauchy functional equation in certain Banach algebras, Bull. Austral. Math. Soc. 31 (1985), no. 1, 137–144. MR 772638, DOI 10.1017/S0004972700002343
J. Vukman, Some functional equations in Banach algebras and an application, Proc. Amer. Math. Soc. 100 (1987), no. 1, 133–136. MR 883415, DOI 10.1090/S0002-9939-1987-0883415-0

Commutative algebra