Abstract
Let A be a normed algebra, \({\varphi:A\rightarrow \mathbb{C}}\) be a linear functional. Then, the functional \({{(\varphi,n)}^{\vee}}\) is defined as \({{(\varphi,n)}^{\vee}(a_{1},\dots, a_{n})=\varphi(a_{1} \dots a_{n})-\varphi(a_{1})\dots\varphi(a_{n})}\) for all elements \({a_{1},\dots ,a_{n} \in A}\). If the norm of \({{(\varphi,n)}^{\vee}}\) is small, then φ is approximately n-multiplicative linear functional and it is of interest whether or not \({\|{(\varphi,n)}^{\vee}\|}\) being small implies that φ is near to an n-multiplicative linear functional. If this property holds for a Banach algebra A, then A is an n-AMNM algebra (approximately n-multiplicative linear functionals are near n-multiplicative linear functionals). We show that some properties of AMNM (2-AMNM) algebras are also valid for n-AMNM algebras. For example, we give some alternative definitions of n-AMNM. We also prove some theorems on the hereditary properties of n-AMNM condition and we use an equivalent condition for the n-AMNM property on certain Banach algebras when the Gelfand and norm topologies coincide on the character space of the algebra. We also give some examples which are n-AMNM and finally, exhibit an example which is not n-AMNM.
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H. Shayanpour was partially supported by the Center of Excellence for Mathematics, University of Shahrekord.
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Shayanpour, H., Ansari-Piri, E., Heidarpour, Z. et al. Approximately n-Multiplicative Functionals on Banach Algebras. Mediterr. J. Math. 13, 1907–1920 (2016). https://doi.org/10.1007/s00009-015-0567-6
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DOI: https://doi.org/10.1007/s00009-015-0567-6