ACTA issues

Spectrality in $C^*$-algebras

János Krisfóf

Acta Sci. Math. (Szeged) 62:3-4(1997), 467-483

Abstract. In this paper an attempt is made to generalize the classical spectral theorem for normal elements in $C^*$-algebras. Three types of $C^*$-algebras will be introduced which are characterised by a particular spectral property. A relatively simple and clear line of reasoning leads to a very strong form of the spectral theorem for ultraspectral $C^*$-algebras. The essential part of the proof is based on the measure theoretical representation theorem of Riesz and an extension theorem for weakly continuous linear operators in locally convex spaces. The proof is independent of the classical spectral theorem of Hilbert for bounded normal operators in Hilbert spaces. With regard to the simplicity of the conditions defining ultraspectrality, the result throws new light upon the classical spectral theorem revealing its proper reasons. Moreover, it is observed that the spectral property imposed on an ultraspectral $C^*$-algebra ensures that its set of projection is a $\sigma $-complete lattice with respect to the natural ordering. This is a new remarkable relation between spectrality and general probabilistic aspects of $C^*$-algebras.

AMS Subject Classification (1991): 46L05, 46A22

Received September 4, 1995. (Registered under 6129/2009.)