A Gelfand representation theory for C*-algebras

Abstract

Recent work by the author which was independently duplicated in part by Giles and Kummer has made it possible to generalize the Gelfand representation theorem for abelian C-*algebras to the non-abelian case. Let A be a C-algebra with unit. If A is abelian, it can be identified with the algebra of all continuous complex-valued functions on its maximal ideal space (with the hull-kernel topology). A less precise way of looking at this result would be to say that an abelian A is completely recoverable from the set of maximal ideals and a certain structure thereon (in this case, a topology). If we use the latter description as the basis for a theory applicable to non-abelian A, we find immediately that two changes are necessary. The set of maximal ideals is replaced by the set of maximal left ideals, and secondly, the structure defined thereon will not be a topology, though it will have many similar properties when viewed correctly. This paper shows how the C*-algebra is recovered from the maximal left ideals (with structure). © 1971, Pacific Journal of Mathematics.