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Almost multiplicative maps and ε-spectrum of an element in Fréchet Q-algebra

Farajzadeh A.P. (Department of Mathematics, Faculty of Science, Razi University, Kermanshah, Iran)
Omidi M.R. (Department of Basic Sciences, Kermanshah University of Technology, Kermanshah, Iran)

Let (A,(pk)) be a Fréchet Q-algebra with unit eA. The ε-spectrum of an element x in A is defined by σε(x) = {λ C:pk0 (λeA-x)pk0(λeA-x)-1 ≥ 1/ε for 0 < ε < 1. We show that there is a close relation between the ε-spectrum and almost multiplicative maps. It is also shown that {φ(x) : φ  Mε alm(A), φ(eA) = 1}  σε(x) for every x  A, where Mε alm(A) is the set of all ε-multiplicative maps from A to C.

Keywords: Q-algebra, Fréchet Q-algebra, almost multiplicative map, ε-spectrum, almost spectrum