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The Gelfand Transform
Let be a Banach algebra over . Let be the space of all multiplicative linear functionals in , endowed with the weak-* topology. Let denote the algebra of complex valued continuous functions in .
The Gelfand transform is the mapping
where is defined by
The Gelfand transform is a continuous homomorphism from to .
Theorem - Let denote the algebra of complex valued continuous functions in , that vanish at infinity. The image of the Gelfand transform is contained in .
The Gelfand transform is a very useful tool in the study of commutative Banach algebras and, particularly, commutative $C^*$-algebras.
Classification of commutative -algebras: Gelfand-Naimark theorems
The following results are called the Gelfand-Naimark theorems. They classify all commutative -algebras and all commutative -algebras with identity element.
Theorem 1 - Let be a -algebra over . Then is *-isomorphic to for some locally compact Hausdorff space . Moreover, the Gelfand transform is a *-isomorphism between and .
Theorem 2 - Let be a unital -algebra over . Then is *-isomorphic to for some compact Hausdorff space . Moreover, the Gelfand transform is a *-isomorphism between and .
The above theorems can be substantially improved. In fact, there is an equivalence between the category of commutative -algebras and the category of locally compact Hausdorff spaces. For more information and details about this, see the entry about the general Gelfand-Naimark theorem.
Gelfand transform is owned by Rui Palma, Roger Lipsett, bci1.
Mathematics Subject Classification
46H05 General theory of topological algebras46J05 General theory of commutative topological algebras
46J40 Structure, classification of commutative topological algebras
46L05 General theory of <n0:math xmlns:n0="http://www.w3.org/1998/Math/MathML" alttext="$C^*$"><n0:msup><n0:mi>C</n0:mi><n0:mi>⁎</n0:mi></n0:msup></n0:math>-algebras
46L35 Classifications of <n0:math xmlns:n0="http://www.w3.org/1998/Math/MathML" alttext="$C^*$"><n0:msup><n0:mi>C</n0:mi><n0:mi>⁎</n0:mi></n0:msup></n0:math>-algebras

Comments
looks good, but bad luck about the MSC codes...
Topic
It would be good to get a full list of the MSC codes that don’t work well. Wondering what query would produce that.