Constructing Menger manifold C*-diagonals in classifiable C*-algebras

TL;DR

This paper constructs and classifies C*-diagonals with connected spectra in all classifiable stably finite C*-algebras, revealing their topological structure as Menger curves or related spaces, and shows the abundance of non-conjugate examples.

Contribution

It provides a comprehensive construction and classification of C*-diagonals with connected spectra in classifiable C*-algebras, including explicit topological descriptions.

Findings

Constructed C*-diagonals with connected spectra in all classifiable stably finite C*-algebras.

Determined spectra up to homeomorphism for certain algebras, identifying Menger curve and related spaces.

Proved the existence of continuum many non-conjugate Menger manifold C*-diagonals in each algebra.

Abstract

We construct C*-diagonals with connected spectra in all classifiable stably finite C*-algebras which are unital or stably projectionless with continuous scale. For classifiable stably finite C*-algebras with torsion-free $K_0$ and trivial $K_1$, we further determine the spectra of the C*-diagonals up to homeomorphism. In the unital case, the underlying space turns out to be the Menger curve. In the stably projectionless case, the space is obtained by removing a non-locally-separating copy of the Cantor space from the Menger curve. We show that each of our classifiable C*-algebras has continuum many pairwise non-conjugate such Menger manifold C*-diagonals. Along the way, we also obtain a complete classification of C*-diagonals in all one-dimensional non-commutative CW complexes.