×

Unitary orbits of selfadjoints in some \(C^*\)-algebras. (English) Zbl 0812.46057

The basic setting is a \(C^*\)-algebra \(A\) with unit. The authors consider the natural metric space associate to the following semi-metric \(D\) on the self-adjoint elements of \(A\): \[ D(a,b)= \inf\{\| uau^*- b\|: u\text{ unitary in }A\}. \] They show that for a very large class of algebras (the details are too technical to reproduce here) with suitable trace this metric space is isometric to a subset of the space of all compactly supported Borel probability measures on the line, provided with a suitable metric. Using the familiar identification of such measures with their distribution functions, a third description of this space as a family of non-decreasing, bounded, left-continuous functions, provided with the supremum norm is obtained. This isometry in question is quite natural. To each self-adjoint element \(a\) one associates the measure which is obtained by composing the functional calculus induced by \(a\) with the trace whose existence is postulated above.
Reviewer: J.B.Cooper (Linz)

MSC:

46L05 General theory of \(C^*\)-algebras
PDFBibTeX XMLCite