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Closed convex hulls of unitary orbits in certain simple real rank zero \(C^*\)-algebras. (English) Zbl 1398.46043

Given a unital \(C^*\)-algebra \(A\) and a self-adjoint element \(b\in A\), one considers the unitary orbit \(\mathcal{U}(b)\), given as the set of \(u^*bu\) with \(u\) ranging over all unitaries in \(A\). Further, one considers the convex hull \(\mathrm{conv}(\mathcal{U}(b))\). One can think of elements in \(\mathcal{U}(b)\) as “translates” of \(b\), which means that elements of \(\mathrm{conv}(\mathcal{U}(b))\) are “averages of translates” of \(b\).
Given another self-adjoint element \(a\), it is natural to consider the distances of \(a\) to \(\mathcal{U}(b)\) and to \(\mathrm{conv}(\mathcal{U}(b))\) as measures for the similarity of \(a\) and \(b\). It is therefore an interesting problem to compute these distances, for instance, in terms of tracial data of \(a\) and \(b\). Of particular relevance is the case of distance zero.
For von Neumann algebras, both problems have been well-studied and the case of distance zero was settled completely, see [D. Sherman, J. Reine Angew. Math. 605, 95–132 (2007; Zbl 1120.46042)] and [F. Hiai and Y. Nakamura, Trans. Am. Math. Soc. 323, No. 1, 1–38 (1991; Zbl 0743.46057)].
The precise definition of the tracial invariant is too complicated to recall here. It suffices to say that, given a tracial state \(\tau\) on \(A\), there is a notion of tracial comparison \(a\prec_\tau b\) that only depends on the trace \(\tau\) applied to spectral projections of \(a\) and \(b\) (Definition 2.4).
If \(M\) is a finite factor and \(a,b\in M\) are self-adjoint, then the following facts are known:
(1) \(\mathrm{dist}(a,\mathcal{U}(b))=0\) if and only if \(a\prec_\tau b\) and \(b\prec_\tau a\).
(2) \(\mathrm{dist}(a,\mathrm{conv}(\mathcal{U}(b)))=0\) if and only if \(a\prec_\tau b\).
It is an area of active research to study to what extent these statements can be transferred from the setting of (finite) factors to that of (finite) simple \(C^*\)-algebras. For instance, in [B. Jacelon et al., J. Funct. Anal. 269, No. 10, 3304–3315 (2015; Zbl 1342.46051)], an analogue of Statement (1) has been obtained for the class of simple, unital, exact, \(\mathcal{Z}\)-stable \(C^*\)-algebras.
The main result of the paper under review (Theorem 5.3) transfers Statement (2) to certain simple \(C^*\)-algebras: If \(A\) is a unital, separable, simple, non-elementary \(C^*\)-algebra with nonempty tracial state space \(T(A)\), with real rank zero and stable rank one, and with strict comparison of projections by traces, then self-adjoint elements \(a,b\) satisfy \(\mathrm{dist}(a,\mathrm{conv}(\mathcal{U}(b)))=0\) if and only if \(a\prec_\tau b\) for every \(\tau\in T(A)\).
This theorem generalizes an earlier result of the second author (Theorem 4.1 in [P. Skoufranis, J. Funct. Anal. 270, No. 4, 1319–1360 (2016; Zbl 1346.46050)]) by removing the assumption of unique trace and by relaxing the condition of strong comparison of projections by traces to the much more common condition of strict comparison of projections by traces.
Lastly, it should be mentioned that a \(C^*\)-algebra is said to have the Dixmier property if, for every self-adjoint element \(b\), there exists an element in the centre that has distance zero to \(\mathrm{conv}(\mathcal{U}(b))\). It is known that every von Neumann algebra has the Dixmier property. The Dixmier property for \(C^*\)-algebras has been studied in [R. Archbold et al., J. Funct. Anal. 273, No. 8, 2655–2718 (2017; Zbl 1373.81238)].

MSC:

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