An Introduction to C*-Algebras and the Classification Program

Abstract

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Smooth lattice orbits of nilpotent groups and strict comparison of projections

“…[119,Corollary 7.3], and the third is due to Rørdam, cf. [108,Corollary 4.6] and [114,Theorem 15.4.6]; to be a bit more precise, this implication can be deduced from [108, Theorem 4.5] by arguing in the same way as in the proof of [107, Theorem 5.2 (a)], taking into account Blackadar and Handelman's characterization of lower semi-continuous dimension functions, cf. [18], and Haagerup's result that quasitraces on exact C * -algebras, hence on nuclear C * -algebras, are traces, cf.…”

“…If τ is a tracial state on A, we define τ : We note that this property implies that A has strict comparison of projections, in the following extended sense (compared to [16,FCQ2,p. 22] and [114,Definition 11.3.8]):…”

“…[ We next mention some conditions ensuring that strict comparison of positive elements hold whenever A belongs to a certain class of C * -algebras. As we will not work explicitly with any of the properties involved, we do not recall the lengthy definitions and simply refer the reader to Strung's book [114] for undefined terminology in the following theorem and in the comments related to it. Theorem 5.1 ([107,108,119]).…”

“…This approach is reminiscent of the method [12] towards Theorem 1.2 using von Neumann algebras. However, in contrast to the setting of von Neumann algebras, the comparison theory for projections in C * -algebras is remarkably subtle, see, e.g., [16,114] and the references therein. Among others, this is caused by the fact that C * -algebras, in contrast to von Neumann algebras, might not possess sufficiently many projections for a satisfactory comparison theory.…”

“…An additional result of independent interest, at least to operator algebraists, is the following theorem, which provides new examples of classifiable C * -algebras in the sense of the classification program for simple separable nuclear C * -algebras (see e.g. [114,Chapter 18] and references therein).…”