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rmd1228887e 2022-07-10 Answered
Let A be a commutative Banach algebra. Let χ 1 and χ 2 be characters of A.
I am having some difficulty seeing why the following statement is true:
If ker χ 1 = ker χ 2 , then since χ 1 ( 1 ) = χ 2 ( 1 ) = 1, we have that χ 1 = χ 2 .
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Answers (1)

Miles Mueller
Answered 2022-07-11 Author has 11 answers
Suppose a is an element of the algebra, and χ 1 ( a ) = λ. Then a λ 1 ker χ 1 = ker χ 2 so
0 = χ 2 ( a λ 1 ) = χ 2 ( a ) λ
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