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Hannes Thiel

Publications and source records attributed to Hannes Thiel.

At least 37 records · Page 2Linked to original sources
arXiv2024

Products of commutators in matrix rings

Let $R$ be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when $R$ is commutative, is provided. If, however, $R$ has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if $R$ is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If $R$ is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of $a$ is greater than $2$.

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arXiv2024

Rings and C*-algebras generated by commutators

We show that a unital ring is generated by its commutators as an ideal if and only if there exists a natural number $N$ such that every element is a sum of $N$ products of pairs of commutators. We show that one can take $N \leq 2$ for matrix rings, and that one may choose $N \leq 3$ for rings that contain a direct sum of matrix rings -- this in particular applies to C*-algebras that are properly infinite or have real rank zero. For Jiang-Su-stable C*-algebras, we show that $N\leq 6$ can be arranged. For arbitrary rings, we show that every element in the commutator ideal admits a power that is a sum of products of commutators. We prove that a C*-algebra cannot be a radical extension over a proper ideal, and we use this to deduce that a C*-algebra is generated by its commutators as a not necessarily closed ideal if and only if every element is a finite sum of products of pairs of commutators.

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arXiv2024

Real rank of extensions of C*-algebras

Given a closed ideal $I$ in a C*-algebra $A$, we develop techniques to bound the real rank of $A$ in terms of the real ranks of $I$ and $A/I$. Building on work of Brown, Lin and Zhang, we obtain complete solutions if $I$ belongs to any of the following classes: (1) C*-algebras with real rank zero, stable rank one and vanishing $K_1$-group; (2) simple, purely infinite C*-algebras; (3) simple, $\mathcal{Z}$-stable C*-algebras with real rank zero; (4) separable, stable C*-algebras with an approximate unit of projections and the Corona Factorization Property.

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arXiv2024

Real rank of some multiplier algebras

We show that there exists a separable, nuclear C*-algebra with real rank zero and trivial K-theory such that its multiplier and corona algebra have real rank one. This disproves two conjectures of Brown and Pedersen. We also compute the real rank of the stable multiplier algebra and the stable corona algebra of countably decomposable type $\mathrm{I}_\infty$ and type $\mathrm{II}_\infty$ factors. Together with results of Zhang this completes the computation of the real rank for stable multiplier and corona algebras of countably decomposable factors.

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arXiv2023

Ranks of soft operators in nowhere scattered C*-algebras

We show that for C*-algebras with the Global Glimm Property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-C*-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a C*-algebra is determined by the soft part of its Cuntz semigroup. Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

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arXiv2023

Prime ideals in C*-algebras and applications to Lie theory

We show that every proper, dense ideal in a C*-algebra is contained in a prime ideal. It follows that a subset generates a C*-algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal. This allows us to transfer Lie theory results from prime rings to C*-algebras. For example, if a C*-algebra $A$ is generated by its commutator subspace $[A,A]$ as a ring, then $[[A,A],[A,A]] = [A,A]$. Further, given Lie ideals $K$ and $L$ in $A$, then $[K,L]$ generates $A$ as a not necessarily closed ideal if and only if $[K,K]$ and $[L,L]$ do, and moreover this implies that $[K,L]=[A,A]$. We also discover new properties of the subspace generated by square-zero elements and relate it to the commutator subspace of a C*-algebra.

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arXiv2023

Zero-product balanced algebras

We say that an algebra is zero-product balanced if $ab\otimes c$ and $a\otimes bc$ agree modulo tensors of elements with zero-product. This is closely related to but more general than the notion of a zero-product determined algebra of Brešar, Grašič and Ortega. Every surjective, zero-product preserving map from a zero-product balanced algebra is automatically a weighted epimorphism, and this implies that zero-product balanced algebras are determined by their linear and zero-product structure. Further, the commutator subspace of a zero-product balanced algebra can be described in terms of square-zero elements. We show that a semiprime, commutative algebra is zero-product balanced if and only if it is generated by idempotents. It follows that every commutative, zero-product balanced algebra is spanned by nilpotent and idempotent elements. We deduce a dichotomy for unital, zero-product balanced algebras: They either admit a character or are generated by nilpotents.

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arXiv2023

Soft operators in C*-algebras

We say that a C*-algebra is soft if it has no nonzero unital quotients, and we connect this property to the Hjelmborg-Rørdam condition for stability and to property (S) of Ortega-Perera-Rørdam. We further say that an operator in a C*-algebra is soft if its associated hereditary subalgebra is, and we provide useful spectral characterizations of this concept. Of particular interest are C*-algebras that have an abundance of soft elements in the sense that every hereditary subalgebra contains an almost full soft element. We show that this property is implied by the Global Glimm Property, and that every C*-algebra with an abundance of soft elements is nowhere scattered. This sheds new light on the long-standing Global Glimm Problem of whether every nowhere scattered C*-algebra has the Global Glimm Property.

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arXiv2023

Traces on ultrapowers of C*-algebras

Using Cuntz semigroup techniques, we characterize when limit traces are dense in the space of all traces on a free ultrapower of a C*-algebra. More generally, we consider density of limit quasitraces on ultraproducts of C*-algebras. Quite unexpectedly, we obtain as an application that every simple C*-algebra that is (m,n)-pure in the sense of Winter is already pure. As another application, we provide a partial verification of the first Blackadar-Handelman conjecture on dimension functions. Crucial ingredients in our proof are new Hahn-Banach type separation theorems for noncancellative cones, which in particular apply to the cone of extended-valued traces on a C*-algebra.

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arXiv2022

The Global Glimm Property

It is known that a C*-algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a C*-algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is ideal-filtered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered). We show that ideal-filteredness and property (V) are automatic for C*-algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of C*-algebras.

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arXiv2022

Diffuse traces and Haar unitaries

We show that a tracial state on a unital C*-algebra admits a Haar unitary if and only if it is diffuse, if and only if it does not dominate a tracial functional that factors through a finite-dimensional quotient. It follows that a unital C*-algebra has no finite-dimensional representations if and only if each of its tracial states admits a Haar unitary. More generally, we study when nontracial states admit Haar unitaries. In particular, we show that every state on a unital, simple, infinite-dimensional C*-algebra admits a Haar unitary. We obtain applications to the structure of reduced free products. Notably, the tracial reduced free product of simple C*-algebras is always a simple C*-algebra of stable rank one.

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arXiv2022

Nowhere scattered C*-algebras

We say that a C*-algebra is nowhere scattered if none of its quotients contains a minimal open projection. We characterize this property in various ways, by topological properties of the spectrum, by divisibility properties in the Cuntz semigroup, by the existence of Haar unitaries for states, and by the absence of nonzero ideal-quotients that are elementary, scattered or type I. Under the additional assumption of real rank zero or stable rank one, we show that nowhere scatteredness implies even stronger divisibility properties of the Cuntz semigroup.

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arXiv2022

The generator rank of subhomogeneous C*-algebras

We compute the generator rank of a subhomgeneous C*-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every Z-stable ASH-algebra has generator rank one, which means that a generic element in such an algebra is a generator. This leads to a strong solution of the generator problem for classifiable, simple, nuclear C*-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear C*-algebras.

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arXiv2022

Weighted homomorphisms between C*-algebras

We show that a bounded, linear map between C*-algebras is a weighted $\ast$-homomorphism (the central compression of a $\ast$-homomorphism) if and only if it preserves zero-products, range-orthogonality, and domain-orthogonality. It follows that a self-adjoint, bounded, linear map is a weighted $\ast$-homomorphism if and only if it preserves zero-products. As an application we show that a linear map between C*-algebras is completely positive, order zero in the sense of Winter-Zacharias if and only if it is positive and preserves zero-products.

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arXiv2021

Covering dimension of Cuntz semigroups

We introduce a notion of covering dimension for Cuntz semigroups of C*-algebras. This dimension is always bounded by the nuclear dimension of the C*-algebra, and for subhomogeneous C*-algebras both dimensions agree. Cuntz semigroups of Z-stable C*-algebras have dimension at most one. Further, the Cuntz semigroup of a simple, Z-stable C*-algebra is zero-dimensional if and only if the C*-algebra has real rank zero or is stably projectionless.

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arXiv2021

Covering dimension of Cuntz semigroups II

We show that the dimension of the Cuntz semigroup of a C*-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-C*-algebras. This allows us to remove separability assumptions from previous results on the dimension of Cuntz semigroups. To obtain these results, we introduce a notion of approximation for abstract Cuntz semigroups that is compatible with the approximation of a C*-algebra by sub-C*-algebras. We show that many properties for Cuntz semigroups are preserved by approximation and satisfy a Löwenheim-Skolem condition.

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arXiv2020

The generator rank of C*-algebras

We show that every AF-algebra is generated by a single operator. This was previously unclear, since the invariant that assigns to a C*-algebra its minimal number of generators lacks natural permanence properties. In particular, it may increase when passing to ideals or inductive limits. To obtain a better behaved theory, we not only ask if a C*-algebra is generated by $n$ elements, but also if generating $n$-tuples are dense. This defines the generator rank, which we show has many natural permanence properties: it does not increase when passing to ideals, quotients or inductive limits.

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