Let X be a Leibniz algebra with unit e, i.e. an algebra with a right invertible linear operator D satisfying the Leibniz condition: D(xy) = xDy + (Dx)y for x,y belonging to the domain of D. If logarithmic mappings exist in X, then cosine and sine elements C(x) and S(x) defined by means of antilogarithmic mappings satisfy the Trigonometric Identity, i.e. whenever x belongs to the domain of these mappings. The following question arises: Do there exist non-Leibniz algebras with logarithms such that the Trigonometric Identity is satisfied? We shall show that in non-Leibniz algebras with logarithms the Trigonometric Identity does not exist. This means that the above question has a negative answer, i.e. the Leibniz condition in algebras with logarithms is a necessary and sufficient condition for the Trigonometric Identity to hold.
@article{bwmeta1.element.bwnjournal-article-bcpv53z1p177bwm,
author = {Przeworska-Rolewicz, D.},
title = {Non-Leibniz algebras with logarithms do not have the trigonometric identity},
journal = {Banach Center Publications},
volume = {51},
year = {2000},
pages = {177-189},
zbl = {0973.13011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv53z1p177bwm}
}
Przeworska-Rolewicz, D. Non-Leibniz algebras with logarithms do not have the trigonometric identity. Banach Center Publications, Tome 51 (2000) pp. 177-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv53z1p177bwm/
[000] DB[1] A. di Bucchianico, Banach algebras, logarithms and polynomials of convolution type, J. Math. Anal. Appl. 156 (1991), 253-273. | Zbl 0765.46027
[001] PR[1] D. Przeworska-Rolewicz, Algebraic Analysis, PWN-Polish Scientific Publishers and D. Reidel, Warszawa-Dordrecht, 1988.
[002] PR[2] D. Przeworska-Rolewicz, Consequences of the Leibniz condition, in: Different Aspects of Differentiability. Proc. Conf. Warsaw, September 1993. Ed. D. Przeworska-Rolewicz. Dissertationes Math. 340 (1995), 289-300.
[003] PR[3] D. Przeworska-Rolewicz, Logarithms and Antilogarithms. An Algebraic Analysis Approach, With Appendix by Z. Binderman. Kluwer Academic Publishers, Dordrecht, 1998.
[004] PR[4] D. Przeworska-Rolewicz, Linear combinations of right invertible operators in commutative algebras with logarithms, Demonstratio Math. 31 (1998), 887-898.
[005] PR[5] D. Przeworska-Rolewicz, Postmodern logarithmo-technia, International Journal of Computers and Mathematics with Applications (to appear). | Zbl 0987.05014
[006] PR[6] D. Przeworska-Rolewicz, Some open questions in Algebraic Analysis, in: Unsolved Problems on Mathematics for the 21th Century - A Tribute to Kiyoshi Iséki's 80th Birthday, IOS Press, Amsterdam (to appear).