Constructing Menger Manifold C*-Diagonals in Classifiable C*-Algebras

Abstract

Abstract We initiate a detailed analysis of $C^{\ast }$-diagonals in classifiable $C^{\ast }$-algebras, answering natural questions concerning topological properties of their spectra and uniqueness questions. Firstly, we construct $C^{\ast }$-diagonals with connected spectra in all classifiable stably finite $C^{\ast }$-algebras, which are unital or stably projectionless with continuous scale. Secondly, for classifiable stably finite $C^{\ast }$-algebras with torsion-free $K_0$ and trivial $K_1$, we further determine the spectra of the $C^{\ast }$-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. Thirdly, we show that each of our classifiable $C^{\ast }$-algebras has continuum many pairwise non-conjugate such Menger manifold$C^{\ast }$-diagonals.