arXiv subjects

Hannes Thiel

Publications and source records attributed to Hannes Thiel.

58 records · Page 4Linked to original sources

arXiv2013

The topological dimension of type I C*-algebras

While there is only one natural dimension concept for separable, metric spaces, the theory of dimension in noncommutative topology ramifies into different important concepts. To accommodate this, we introduce the abstract notion of a noncommutative dimension theory by proposing a natural set of axioms. These axioms are inspired by properties of commutative dimension theory, and they are for instance satisfied by the real and stable rank, the decomposition rank and the nuclear dimension. We add another theory to this list by showing that the topological dimension, as introduced by Brown and Pedersen, is a noncommutative dimension theory of type I C*-algebras. We also give estimates of the real and stable rank of a type I C*-algebra in terms of its topological dimension.

math.OA↗

arXiv2012

The generator problem for Z-stable C*-algebras

The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable, Z-stable C*-algebras A is singly generated, which means that there exists an element x in A that is not contained in any proper sub-C*-algebra of A. To give applications of our result, we observe that Z can be embedded into the reduced group C*-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with reduced group C*-algebras are singly generated. In particular, the tensor product of two reduced free group C*-algebras is singly generated.

math.OA↗

arXiv2011

A characterization of semiprojectivity for commutative C*-algebras

Given a compact, metric space X, we show that the commutative C*-algebra C(X) is semiprojective if and only if X is an absolute neighborhood retract of dimension at most one. This confirms a conjecture of Blackadar. Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As further application of our findings we verify two conjectures of Loring and Blackadar in the commutative case, and we give a partial answer to the question, when a commutative C*-algebra is weakly (semi-)projective.

math.OA↗

arXiv2010

The Cuntz Semigroup and Comparison of Open Projections

We show that a number of naturally occurring comparison relations on positive elements in a C*-algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the C*-algebra. In particular we show that Cuntz comparison of positive elements corresponds to a comparison relation on open projections, that we call Cuntz comparison, and which is defined in terms of-and is weaker than-a comparison notion defined by Peligrad and Zsidó. The latter corresponds to a well-known comparison relation on positive elements defined by Blackadar. We show that Murray-von Neumann comparison of open projections corresponds to tracial comparison of the corresponding positive elements of the C*-algebra. We use these findings to give a new picture of the Cuntz semigroup.

math.OA↗