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Devin Anderson 2022-06-27 Answered
Let ( 2 , | | . 2 ) with the coordinatewise product.Prove that ( 2 , | | . 2 ) is a commutative and semisimple Banach algebra
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Answers (1)

timmeraared
Answered 2022-06-28 Author has 22 answers
The space 2 is not isometrically isomorphic to any C -algebra. It is reflexive, and any reflexive C -algebra is finite-dimensional.
Clearly 2 is a commutative Banach algebra. To prove that 2 is semi-simple, we must show that the character space Γ of 2 separates the points of 2 . This is easy enough: For every k N the map δ k : 2 C is linear, multiplicative and non-zero. Moreover, δ k ( f ) = 0 for all k N implies f = 0.
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