Comparison theory and smooth minimal C*-dynamics

TL;DR

This paper establishes key classification and structural results for minimal diffeomorphism C*-algebras, including comparison properties, K-theory classification, and a conjecture resolution, advancing the understanding of their invariants and diversity.

Contribution

It proves Blackadar's comparison property for minimal diffeomorphism C*-algebras, classifies their Hilbert modules and orbit closures, and constructs many non-Morita-equivalent algebras with identical invariants.

Findings

C*-algebra of minimal diffeomorphism satisfies comparison property

Classification of Hilbert modules via K-theory and traces

Construction of uncountably many non-Morita-equivalent algebras

Abstract

We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar's Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff's uncountable family of II_1 factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras.