Geodesic
Representations of non-commutative Banach algebras by continuous functions
Algebra i analiz, Tome 3 (1991) no. 4, pp. 171-185.

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A main result of the Gelfand theory states that any commutative semi-simple Banach algebra is isomorphic to an algebra of continuous complex-valued functions on a Hausdorff compact. In the present paper we extend this result to a class of non-commutative algebras. We introduce and compare several concepts of continuity of functions taking values in Banach algebras which differ from point to point and show that they, in a certain sense, coincide. Finally, we give applications to algebras of singular integral and convolution operators and to algebras of approximation methods. 
@article{AA_1991_3_4_a7,
     author = {S. Roch and B. Silbermann},
     title = {Representations of non-commutative {Banach} algebras by continuous functions},
     journal = {Algebra i analiz},
     pages = {171--185},
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     volume = {3},
     number = {4},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_1991_3_4_a7/}
}
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S. Roch; B. Silbermann. Representations of non-commutative Banach algebras by continuous functions. Algebra i analiz, Tome 3 (1991) no. 4, pp. 171-185. http://geodesic.mathdoc.fr/item/AA_1991_3_4_a7/