arXiv subjects

Hannes Thiel

Publications and source records attributed to Hannes Thiel.

At least 55 records · Page 3Linked to original sources

arXiv2020

Cuntz semigroups of ultraproduct C*-algebras

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C*-algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C*-algebras. As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C*-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.

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arXiv2020

Preduals and complementation of spaces of bounded linear operators

For Banach spaces X and Y, we establish a natural bijection between preduals of Y and preduals of L(X,Y) that respect the right L(X)-module structure. If X is reflexive, it follows that there is a unique predual making L(X) into a dual Banach algebra. This removes the condition that X have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement Y in its bidual and L(X)-linear projections that complement L(X,Y) in its bidual. It follows that Y is complemented in its bidual if and only if L(X,Y) is (either as a module or as a Banach space). Our results are new even in the well-studied case of isometric preduals.

math.FA↗

arXiv2019

Edwards' condition for quasitraces on C*-algebras

We prove that Cuntz semigroups of C*-algebras satisfy Edwards' condition with respect to every quasitrace. This condition is a key ingredient in the study of the realization problem of functions on the cone of quasitraces as ranks of positive elements. In the course of our investigation, we identify additional structure of the Cuntz semigroup of an arbitrary C*-algebra and of the cone of quasitraces.

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arXiv2019

Ranks of operators in simple C*-algebras with stable rank one

Let $A$ be a separable, unital, simple C*-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of $A$ is realized as the rank of an operator in the stabilization of $A$. Assuming moreover that $A$ has locally finite nuclear dimension, we deduce that $A$ is $\mathcal{Z}$-stable if and only if it has strict comparison of positive elements. In particular, the Toms-Winter conjecture holds for separable, unital, simple, approximately subhomogeneous C*-algebras with stable rank one.

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arXiv2019

Inductive limits of semiprojective C*-algebras

We prove closure properties for the class of C*-algebras that are inductive limits of semiprojective C*-algebras. Most importantly, we show that this class is closed under shape domination, and so in particular under shape and homotopy equivalence. It follows that the considered class is quite large. It contains for instance the stable suspension of any nuclear C*-algebra satisfying the UCT and with torsion-free $K_0$-group. In particular, the stabilized C*-algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives.

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arXiv2018

Abstract bivariant Cuntz semigroups

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[[S,T]]$ playing the role of morphisms from $S$ to $T$. Applied to C$^*$-algebras $A$ and $B$, the semigroup $[[\mathrm{Cu}(A),\mathrm{Cu}(B)]]$ should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We also show that order-zero maps between C$^*$-algebras naturally define elements in the respective bivariant Cuntz semigroup.

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arXiv2018

Isomorphisms of Algebras of Convolution Operators

For $p,q\in [1,\infty)$, we study the isomorphism problem for the $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its group von Neumann algebra, we show that this is the case for the algebras $\mathrm{CV}_p(G)$ of $p$-convolvers and $\mathrm{PM}_p(G)$ of $p$-pseudomeasures, for $p\neq 2$. More generally, we show that if $\mathrm{CV}_p(G)$ is isometrically isomorphic to $\mathrm{CV}_q(H)$, with $p,q\neq 2$, then $G$ must be isomorphic to $H$ and $p$ and $q$ are either equal or conjugate. This implies that there is no $L^p$-version of Connes' uniqueness of the hyperfinite II$_1$-factor. Similar results apply to the algebra $\mathrm{PF}_p(G)$ of $p$-pseudofunctions, generalizing a classical result of Wendel. We also show that other $L^p$-rigidity results for groups can be easily recovered and extended using our main theorem. Our results answer questions originally formulated in the work of Herz in the 70's. Moreover, our methods reveal new information about the Banach algebras in question. As a non-trivial application, we verify the reflexivity conjecture for all Banach algebras lying between $\mathrm{PF}_p(G)$ and $\mathrm{CV}_p(G)$: if any such algebra is reflexive and amenable, then $G$ is finite.

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arXiv2018

Preduals for spaces of operators involving Hilbert spaces and trace-class operators

Continuing the study of preduals of spaces $\mathcal{L}(H,Y)$ of bounded, linear maps, we consider the situation that $H$ is a Hilbert space. We establish a natural correspondence between isometric preduals of $\mathcal{L}(H,Y)$ and isometric preduals of $Y$. The main ingredient is a Tomiyama-type result which shows that every contractive projection that complements $\mathcal{L}(H,Y)$ in its bidual is automatically a right $\mathcal{L}(H)$-module map. As an application, we show that isometric preduals of $\mathcal{L}(\mathcal{S}_1)$, the algebra of operators on the space of trace-class operators, correspond to isometric preduals of $\mathcal{S}_1$ itself (and there is an abundance of them). On the other hand, the compact operators are the unique predual of $\mathcal{S}_1$ making its multiplication separately weak* continuous.

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arXiv2018

Extending representations of Banach algebras to their biduals

We show that a representation of a Banach algebra $A$ on a Banach space $X$ can be extended to a canonical representation of $A^{**}$ on $X$ if and only if certain orbit maps $A\to X$ are weakly compact. When this is the case, we show that the essential space of the representation is complemented if $A$ has a bounded left approximate identity. This provides a tool to disregard the difference between degenerate and nondegenerate representations. Our results have interesting consequences both in C*-algebras and in abstract harmonic analysis. For example, a C*-algebra $A$ has an isometric representation on an $L^p$-space, for $p\in[1,\infty)\setminus\{2\}$, if and only if $A$ is commutative. Moreover, the $L^p$-operator algebra of a locally compact group is universal with respect to arbitrary representations on $L^p$-spaces.

math.FA↗

arXiv2018

Semiprojectivity and semiinjectivity in different categories

Projectivity and injectivity are fundamental notions in category theory. We consider natural weakenings termed semiprojectivity and semiinjectivity, and study these concepts in different categories. For example, in the category of metric spaces, (semi)injective objects are precisely the absolute (neighborhood) retracts. We show that the trivial group is the only semiinjective group, while every free product of a finitely presented group and a free group is semiprojective. To a compact, metric space $X$ we associate the commutative C*-algebra $C(X)$. This association is contravariant, whence semiinjectivity of $X$ is related to semiprojectivity of $C(X)$. Together with Adam Sørensen, we showed that $C(X)$ is semiprojective in the category of all C*-algebras if and only if $X$ is an absolute neighborhood retract with dimension at most one.

math.CT↗

arXiv2017

Inductive limits of projective C*-algebras

We show that a separable C*-algebra is an inductive limits of projective C*-algebras if and only if it has trivial shape, that is, if it is shape equivalent to the zero C*-algebra. In particular, every contractible C*-algebra is an inductive limit of projectives, and one may assume that the connecting morphisms are surjective. Interestingly, an example of Dadarlat shows that trivial shape does not pass to full hereditary sub-C*-algebra. It then follows that the same fails for projectivity. To obtain these results, we develop criteria for inductive limit decompositions, and we discuss the relation with different concepts of approximation. As a main application of our findings we show that a C*-algebra is (weakly) projective if and only if it is (weakly) semiprojective and has trivial shape. It follows that a \ca{} is projective if and only if it is contractible and semiprojective. This confirms a conjecture of Loring.

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arXiv2016

Representations of $p$-convolution algebras on $L^q$-spaces

For a nontrivial locally compact group $G$, and $p\in [1,\infty)$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\in [1,\infty)$, these Banach algebras can be represented on an $L^q$-space if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $|\frac 1p - \frac 12|=|\frac 1q - \frac 12|$. This result can be interpreted as follows: for $p,q\in [1,\infty)$, the $L^p$- and $L^q$-representation theories of a group are incomparable, except in the trivial cases when they are equivalent. As an application, we show that, for distinct $p,q\in [1,\infty)$, if the $L^p$ and $L^q$ crossed products of a topological dynamical system are isomorphic, then $\frac 1p + \frac 1q=1$. In order to prove this, we study the following relevant aspects of $L^p$-crossed products: existence of approximate identities, duality with respect to $p$, and existence of canonical isometric maps from group algebras into their multiplier algebras.

math.FA↗

arXiv2016

Quotients of Banach algebras acting on $L^p$-spaces

We show that the class of Banach algebras that can be isometrically represented on an $L^p$-space, for $p\neq 2$, is not closed under quotients. This answers a question asked by Le Merdy 20 years ago. Our methods are heavily reliant on our earlier study of Banach algebras generated by invertible isometries of $L^p$-spaces.

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arXiv2015

Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(A\otimes B)$ with $Cu(A)\otimes_{Cu}Cu(B)$ for certain classes of C*-algebras. As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic. We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.

math.OA↗

arXiv2015

Group algebras acting on $L^p$-spaces

For $p\in [1,\infty)$ we study representations of a locally compact group $G$ on $L^p$-spaces and $QSL^p$-spaces. The universal completions $F^p(G)$ and $F^p_{\mathrm{QS}}(G)$ of $L^1(G)$ with respect to these classes of representations (which were first considered by Phillips and Runde, respectively), can be regarded as analogs of the full group \ca{} of $G$ (which is the case $p=2$). We study these completions of $L^1(G)$ in relation to the algebra $F^p_λ(G)$ of $p$-pseudofunctions. We prove a characterization of group amenability in terms of certain canonical maps between these universal Banach algebras. In particular, $G$ is amenable if and only if $F^p_{\mathrm{QS}}(G)=F^p(G)=F^p_λ(G)$. One of our main results is that for $1\leq p< q\leq 2$, there is a canonical map $γ_{p,q}\colon F^p(G)\to F^q(G)$ which is contractive and has dense range. When $G$ is amenable, $γ_{p,q}$ is injective, and it is never surjective unless $G$ is finite. We use the maps $γ_{p,q}$ to show that when $G$ is discrete, all (or one) of the universal completions of $L^1(G)$ are amenable as a Banach algebras if and only if $G$ is amenable. Finally, we exhibit a family of examples showing that the characterizations of group amenability mentioned above cannot be extended to $L^p$-operator crossed products of topological spaces.

math.FA↗

arXiv2014

Banach algebras generated by an invertible isometry of an $L^p$-space

We provide a complete description of those Banach algebras that are generated by an invertible isometry of an $L^p$-space together with its inverse. Examples include the algebra $PF_p(\mathbb{Z})$ of $p$-pseudofunctions on $\mathbb{Z}$, the commutative $C^*$-algebra $C(S^1)$ and all of its quotients, as well as uncountably many `exotic' Banach algebras. We associate to each isometry of an $L^p$-space, a spectral invariant called `spectral configuration', which contains considerably more information than its spectrum as an operator. It is shown that the spectral configuration describes the isometric isomorphism type of the Banach algebra that the isometry generates together with its inverse. It follows from our analysis that these algebras are semisimple. With the exception of $PF_p(\mathbb{Z})$, they are all closed under continuous functional calculus, and their Gelfand transform is an isomorphism. As an application of our results, we show that Banach algebras that act on $L^1$-spaces are not closed under quotients. This answers the case $p=1$ of a question asked by Le Merdy 20 years ago.

math.FA↗

arXiv2014

Functoriality of group algebras acting on $L^p$-spaces

We continue our study of group algebras acting on $L^p$-spaces, particularly of algebras of $p$-pseudofunctions of locally compact groups. We focus on the functoriality properties of these objects. We show that $p$-pseudofunctions are functorial with respect to homomorphisms that are either injective, or whose kernel is amenable and has finite index. We also show that the universal completion of the group algebra with respect to representations on $L^p$-spaces, is functorial with respect to quotient maps. As an application, we show that the algebras of $p$- and $q$-pseudofunctions on $\mathbb{Z}$ are (abstractly) isometrically isomorphic as Banach algebras if and only if $p$ and $q$ are either equal or conjugate.

math.FA↗

arXiv2014

Semiprojectivity with and without a group action

The equivariant version of semiprojectivity was recently introduced by the first author. We study properties of this notion, in particular its relation to ordinary semiprojectivity of the crossed product and of the algebra itself. We show that equivariant semiprojectivity is preserved when the action is restricted to a cocompact subgroup. Thus, if a second countable compact group acts semiprojectively on a C*-algebra $A$, then $A$ must be semiprojective. This fails for noncompact groups: we construct a semiprojective action of the integers on a nonsemiprojective C*-algebra. We also study equivariant projectivity and obtain analogous results, however with fewer restrictions on the subgroup. For example, if a discrete group acts projectively on a C*-algebra $A$, then $A$ must be projective. This is in contrast to the semiprojective case. We show that the crossed product by a semiprojective action of a finite group on a unital C*-algebra is a semiprojective C*-algebra. We give examples to show that this does not generalize to all compact groups.

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