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Gelfand Theorem


If X is a locally compact T2-space, then the set C_ degrees(X) of all continuous complex valued functions on X vanishing at infinity (i.e., for each epsilon>0, the set {x in X:|f(x)|>=epsilon} is compact) equipped with the supremum norm ||f||=sup{|f(x)|:x in X} is a commutative C^*-algebra.

The Gelfand theorem states that each commutative C^*-algebra A is of the form C_ degrees(X) where X is the maximal ideal space) of A. C_ degrees(X) is unital iff X is compact.


See also

C-*-Algebra, T2-Space

This entry contributed by Mohammad Sal Moslehian

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References

Murphy, G. J. C-*-Algebras and Operator Theory. New York: Academic Press, 1990.

Referenced on Wolfram|Alpha

Gelfand Theorem

Cite this as:

Moslehian, Mohammad Sal. "Gelfand Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/GelfandTheorem.html

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