Sunder, V. S.; Thomsen, Klaus Unitary orbits of selfadjoints in some \(C^*\)-algebras. (English) Zbl 0812.46057 Houston J. Math. 18, No. 1, 127-137 (1992). 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) Cited in 2 ReviewsCited in 8 Documents MSC: 46L05 General theory of \(C^*\)-algebras Keywords:unitary orbits of selfadjoints; \(C^*\)-algebra with unit PDFBibTeX XMLCite \textit{V. S. Sunder} and \textit{K. Thomsen}, Houston J. Math. 18, No. 1, 127--137 (1992; Zbl 0812.46057)