Strict comparison and selflessness
It is shown that an infinite dimensional, simple, unital, monotracial C*-algebra with strict comparison with respect to its trace is selfless.
math.OA↗
arXiv subjects
Leonel Robert
Publications and source records attributed to Leonel Robert.
Bang--bang representation of $3\times 3$ embeddable stochastic matrices
We prove that every embeddable $3\times3$ row-stochastic matrix is a product of at most seven elementary row-stochastic matrices. Frydman's example shows that this bound is sharp. The proof combines Frydman's six-factor criterion and spiral theorem with a reduction to a three-parameter critical family and an analysis of the membership certificates that persist along it.
math.PR↗
Selfless C*-correspondences, operator valued C*-probability spaces and completely positive maps
We develop a general theory of selflessness for C*-correspondences, with several applications. Specializing this theory to completely positive maps, in particular to conditional expectations, gives rise to a novel notion of relative selflessness. Among applications, the machinery developed here yields new examples of selfless C*-algebras, for instance among minimal tensor products and reduced crossed products, new examples of MF C*-algebras arising as reduced amalgamated free products, a conceptually new proof of Kirchberg's $\mathcal{O}_{\infty}$-absorption theorem, and the lack of ``phantom'' traces on ultrapowers.
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Selfless Inclusions of C*-Algebras
We introduce and study a natural notion of selflessness for inclusions of C*-probability spaces, which in particular implies that all intermediate C*-algebras are selfless in the sense of Robert. We identify natural sources of selfless inclusions in the realms of Z-stable and free product C*-algebras. As an application of this, we prove selflessness for a new family of C*-probability spaces outside the regime of free products and group C*-algebras. These include the reduced free unitary compact quantum groups.
math.OA↗
Products of commutators in simple algebras
Let $A$ be a finite-dimensional simple algebra that is not a field. We show that every $a\in A$ can be written as $a=(bc-cb)(de-ed)$ for some $b,c,d,e\in A$. This is not always true for infinite-dimensional simple algebras. In fact, for any $m\in \mathbb N$ we provide an example of an infinite-dimensional simple unital $C^*$-algebra $A$ in which $1$ cannot be written as $\sum_{i=1}^m x_i(a_ib_i-b_ia_i)y_i$ for some $x_i,a_i,b_i,y_i\in A$.
math.RA↗
Selfless reduced free product $C^*$-algebras
We study selflessness in the general setting of reduced free products of $C^*$-algebras. Towards this end, we develop a suitable theory of rapid decay for filtrations in arbitrary $C^*$-probability spaces. We provide several natural examples and permanence properties of this phenomenon. By using this framework in combination with von Neumann algebraic techniques involving approximate forms of orthogonality, we are able to prove selflessness for general families of reduced free product $C^*$-algebras. As an instance of our results, we prove selflessness and thus strict comparison for the canonical $C^*$-algebras generated by Voiculescu's free semicircular systems. Our results also provide new examples of purely infinite reduced free products.
math.OA↗
Selfless C*-algebras
The aim of this note is to advertise a class of simple C*-algebras which includes noteworthy examples such as the Jiang-Su C*-algebra, the infinite dimensional UHF C*-algebras, the reduced group C*-algebra of the free group in infinitely many generators, and the Cuntz algebras.
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Decreasing paths of polygons
We call a continuous path of polygons decreasing if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by a decreasing path (attainability problem), and we show that this can be done by a finite application of "pull-in" moves (bang-bang problem). In the case of triangles, this problems was investigated by Goodman, Johansen, Ramsey, and Frydman among others, in connection with the embeddability problem for non-homogeneous Markov processes.
math.MG↗
C*-algebras of stable rank one and their Cuntz semigroups
The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: We answer affirmatively, for the class of stable rank one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the Global Glimm Halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.
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Distinguishing C*-algebras by their unitary groups
We obtain partial affirmative answers to the question whether isomorphism of the unitary groups of two C*-algebras, either as topological groups or as discrete groups, implies isomorphism of the C*-algebras as real C*-algebras.
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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.
math.OA↗
Simplicity, bounded normal generation, and automatic continuity of groups of unitaries
We show that the commutator subgroup of the group of unitaries connected to the identity in a simple unital C*-algebra is simple modulo its center. We then go on to investigate the role of regularity properties in the structure of the special unitary group of a C*-algebra. Under mild assumptions, we show that this group has the invariant automatic continuity property and a unique polish group topology. Strengthening our assumptions in the case of simple C*-algebras, we show that the special unitary group modulo its center has bounded normal generation. These results apply to all simple purely infinite C*-algebras and too all simple nuclear C*-algebras in the "classifiable class". We show with counterexamples how our conclusions may in general fail if no regularity conditions are imposed on the C*-algebra.
math.OA↗
Local variants of the Dixmier property and weak centrality for C*-algebras
We study variants of the Dixmier property that apply to elements of a unital C*-algebra, rather than to the C*-algebra itself. By a Dixmier element in a C*-algebra we understand one that can be averaged into a central element by means of a sequence of unitary mixing operators. Examples include all self-commutators and all quasinilpotent elements. We do a parallel study of an element-wise version of weak centrality, where the averaging to the centre is done using unital completely positive elementary operators (as in Magajna's characterization of weak centrality). We also obtain complete descriptions of more tractable sets of elements, where the corresponding averaging can be done arbitrarily close to the centre. This is achieved through several "spectral conditions", involving numerical ranges and tracial states.
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C*-algebras where every element is a limit of products of positive elements
We obtain a characterization of the unital C*-algebras with the property that every element is a limit of products of positive elements, thereby answering a question of Murphy and Phillips.
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Simultaneous averaging to zero by unitary mixing operators
We show that if every element a vector subspace of a C*-algebra can be averaged to zero by means of unitary mixing operators, then all the elements of the subspace can be simultaneously averaged to zero by a net of unitary mixing operators. Moreover, such subspaces admit a simple description in terms of commutators and kernels of states on the C*-algebra. We apply this result to center-valued expectations in C*-algebras with the Dixmier property.
math.OA↗
Cones of traces arising from AF C*-algebras
We characterize the topological non-cancellative cones that are expressible as projective limits of finite powers of $[0,\infty]$. These are also the cones of lower semicontinuous extended-valued traces on AF C*-algebras. Our main result may be regarded as a generalization of the fact that any Choquet simplex is a projective limit of finite dimensional simplices. To obtain our main result, we first establish a duality between certain non-cancellative topological cones and Cuntz semigroups with real multiplication. This duality extends the duality between compact convex sets and complete order unit vector spaces to a non-cancellative setting.
math.OA↗
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.
math.OA↗
A revised augmented Cuntz semigroup
We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of 1-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: We show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.
math.OA↗