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- W4248871808 abstract "Suppose that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=application/x-tex>b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are elements of a complex unital Banach algebra such that the spectra of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=a> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding=application/x-tex>a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=application/x-tex>b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=2 pi i> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π<!-- π --></mml:mi> <mml:mi>i</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>2pi i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-congruence-free. E.M.E. Wermuth has shown that then <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=e Superscript a Baseline e Superscript b Baseline equals e Superscript b Baseline e Superscript a Baseline implies that a b equals b a period> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mi>a</mml:mi> </mml:msup> <mml:mspace width=1em /> <mml:mtext>implies that</mml:mtext> <mml:mspace width=1em /> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> <mml:mi>a</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>e^a e^b = e^b e^a quad text {implies that} quad ab = ba.</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth’s result." @default.
- W4248871808 created "2022-05-12" @default.
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- W4248871808 date "1999-01-28" @default.
- W4248871808 modified "2023-09-30" @default.
- W4248871808 title "Remarks on commuting exponentials in Banach algebras" @default.
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- W4248871808 doi "https://doi.org/10.1090/s0002-9939-99-04701-2" @default.
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