In this paper, $n$-quasitraces on real C*-algebras are studied. It is proved that if $R$ is a real C*-algebra, then the natural extension of an $n$-quasitrace of $R$ to $R+iR$ is also an $n$-quasitrace, and conversely, the restriction of an $n$-quasitrace from $R+iR$ to $R$ is also an $n$-quasitrace. In 1982, Blackadar and Handelman proved that every quasitrace on an AW*-algebra is a $2$-quasitrace. In this paper, a real analogue of that result is obtained. However, in the general case (i.e., for C*-algebras), this result does not hold. Using Kirchberg's example -- where a unital C*-algebra and its quasitrace are constructed such that the quasitrace is not a 2-quasitrace (and therefore is not a trace) -- a similar example is constructed in the real case. The paper also studies the properties of 2-quasitrace on real C*-algebras. As is known, Kaplansky asked whether every (2-) quasitrace on a C*-algebra linear, i.e., a trace. This question remains open to this day. Haagerup has a positive answer to this question in the case where the C*-algebra is unital and exact. In this paper, a real analogue of Haagerup's result is proved.
| Mualliflar | Ramazonova, L., Rakhimov, A. |
|---|---|
| Jurnal | O‘zbekiston matematika jurnali |
| Nashr sanasi | 2026-03-25 |
| Jild | 70 |
| Son | 1 |
| Til | Ingliz |
| DOI | 10.29229/uzmj.2026-1-20 |
DOI: 10.29229/uzmj.2026-1-20 · Maqolaning asl sahifasi
C*-algebras, AW*-algebras, quasitraces on C*- and AW*-algebras
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