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On invariants of \(\mathrm C^{\ast}\)-algebras with the ideal property. (English) Zbl 1464.46065
A C*-algebra has the ideal property if each of its closed two-sided ideals is generated by projections. This class of C*-algebras includes both simple, unital C*-algebras, as well as C*-algebras of real rank zero.
The paper shows that for C*-algebras with the ideal property, the extended Elliott invariant and the Stevens invariant can be recovered from each other, which means that both invariants contain the same information.
Both invariants consist of the scaled, ordered \(K_0\)-group and the \(K_1\)-group. Additionally, they encode different aspects of tracial data: The extended Elliott invariant contains the non-cancellative cone of extended-valued lower-semicontinuous traces (as studied in [G. A. Elliott et al., Am. J. Math. 133, No. 4, 969–1005 (2011; Zbl 1236.46052)]) together with the natural pairing between \(K_0\) and the cone of traces. On the other hand, the Stevens invariant contains for each projection the cancellative cone of finite traces on the unital corner given by the projection, together with natural pairing maps between \(K_0\) and these cones of traces, and with natural restriction maps between the cones of traces.

MSC:
46L35 Classifications of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
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