Von Neumann’s theorem on Abelian families of operators

Abstract

The short proof below resembles but differs from von Neumann's original proof [2; 3]. Let W be the uniformly closed algebra generated by the set {I, An, A*, n, m= 1, 2, . }. Then, from the general theory of Banach algebras [I ] we see that S and C(9) are isometrically isomorphic and that the maximal ideal space 9Y of S is a compact metric space, since W is separable. Hence there is a mappingf: S-*9J of the Cantor set S onto W9. For t in [0, 1], let At(M) be the characteristic function of the set f([0, t)GS) C9M. Each of these sets, as the union of the compact sets f([o, t/n]CS) is a Borel, hence since 9I is metric, a Baire set. In accordance with the isometric isomorphism between the set of bounded Baire functions on SW and a super-ring of 2{ [i, 26 F, 26 G], Et(M) corresponds to a projection E(t), and clearly {E(t) I 0?t? 1 is a resolution of the identity. For B in 91, let b(t) be defined as follows: