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• Title: Innerness of Higher Derivations (Report)
• Author : M. Naranjani, K. Niknam, A. Mirzavaziri
• Release Date : January 01, 2010
• Genre: Mathematics,Books,Science & Nature,
• Pages : * pages
• Size : 63 KB

Description

1. INTRODUCTION Let A be an algebra. A linear mapping [delta] : A [right arrow] A is called a derivation if it satisfies the Leibniz rule [delta](ab) = [delta](a)b + a[delta](b) for all a,b [member of] A. A typical example of a derivation is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] : A [right arrow] A given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (a) = [a.sub.0]a - [aa.sub.0], where [a.sub.0] [member of] A. A derivation of this form is called inner. One of the important questions in the theory of derivations is that "When are all bounded derivations on a Banach algebra inner?" Forty years ago, R. V. Kadison [3] and S. Sakai [10] independently proved that every derivation on a von Neumann algebra M is inner; see also [8]. Let [sigma] : A [right arrow] A be a homomorphism. As a generalization of the notion of a derivation, a linear mapping D : A [right arrow] A is called a ([sigma], [sigma])-derivation if it satisfies the generalized Leibniz rule D(ab) = D (a) [sigma] (b) + a(a)D(b) for all a, b [member of] A (see [7]).

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