Is it true that for simple

vittorecostao1 2022-06-24 Answered
Is it true that for simple C -algebras, meaning that they don't have non-trivial two-sided ideals, it holds that they are necessarily non-commutative? And why?
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Answers (2)

Nola Rivera
Answered 2022-06-25 Author has 21 answers
This is false. C is commutative and simple.
But this is the only simple commutative C -algebra. This is because if A is not isomorphic to C, then it has a non-zero, non-invertible element x by the Gelfand-Mazur theorem. If A is also commutative, then A x is a proper ideal.

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Theresa Archer
Answered 2022-06-26 Author has 2 answers
For a different point of view, a commutative C -algebra A is isomorphic, via the Gelfand transform, to C 0 ( X ) for a locally compact Hausdorff space X. If X is a singleton, then A = C. Otherwise, for nontrivial closed Y X, then set
J = { f C 0 ( X ) :   f | J = 0 }
is a closed two-sided ideal.

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