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