New structure for Villadsen algebras of the first type
Abstract
In this paper we establish two main results concerning the comparison theory of positive elements in approximately homogeneous C*-algebras with diagonal maps. The first is an existence theorem which interfaces with the rank problem; given a Villadsen algebra A, we construct a convex cone that induces an embedding of a semigroup into Cu(A). This semigroup is identified with the set of strictly positive, lower-semicontinuous affine maps on the trace simplex of A. The second is a construction of an AH algebra that produces an obstacle to classifying the approximate unitary orbits of positive elements in terms of their spectral data. Our counterexample sheds light on a critical topological obstruction when attempting to understand equivalence classes of *-homomorphisms from C0((0,1]) to an arbitrary C*-algebra of stable rank one.
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BibTeX
@misc{cano2026structure,
title = {{New structure for Villadsen algebras of the first type}},
author = {Alejandro Cano and Andrew Toms},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38142},
eprint = {2610.38142},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38142},
}
RIS
TY - UNPB
TI - New structure for Villadsen algebras of the first type
AU - Alejandro Cano
AU - Andrew Toms
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38142
UR - https://arxiv.org/abs/2610.38142
AB - In this paper we establish two main results concerning the comparison theory of positive elements in approximately homogeneous C^*-algebras with diagonal maps. The first is an existence theorem which interfaces with the rank problem; given a Villadsen algebra A, we construct a convex cone that induces an embedding of a semigroup into Cu(A). This semigroup is identified with the set of strictly positive, lower-semicontinuous affine maps on the trace simplex of A. The second is a construction of an AH algebra that produces an obstacle to classifying the approximate unitary orbits of positive elements in terms of their spectral data. Our counterexample sheds light on a critical topological obstruction when attempting to understand equivalence classes of *-homomorphisms from C_0((0,1]) to an arbitrary C^*-algebra of stable rank one.
ER -