Quasitraces on exact real C*-algebras are traces

Abstract

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.