Leonel Robert

The cone of lower semicontinuous traces on a C*-algebra

The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra.

An algebraic approach to the radius of comparison

The radius of comparison is an invariant for unital C∗-algebras which extends the theory of covering dimension to noncommutative spaces. We extend its definition to general C∗-algebras, and give an algebraic (as opposed to functional-theoretic) reformulation. This yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces. We then give several applications of these results. New examples of C∗-algebras with finite radius of comparison are given, and the question of when the Cuntz classes of finitely generated Hilbert modules form a hereditary subset of the Cuntz semigroup is addressed. Most interestingly, perhaps, we treat the question of when a full hereditary subalgebra B of a stable C∗-algebra A is itself stable, giving a characterization in terms of the radius of comparison. We also use the radius of comparison to quantify the least n for which a C∗-algebra D without bounded 2-quasitraces or unital quotients has the property that Mn(D) is stable.

Divisibility properties for C*-algebras

We consider three notions of divisibility in the Cuntz semigroup of a C*-algebra, and show how they reflect properties of the C*-algebra. We develop methods to construct (simple and non-simple) C*-algebras with specific divisibility behaviour. As a byproduct of our investigations, we show that there exists a sequence

HILBERT C -MODULES OVER A COMMUTATIVE C -ALGEBRA

(A_n)

Nuclear dimension and Z-stability of non-simple C*-algebras

of simple unital infinite dimensional C*-algebras such that the product

Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family

\prod_{n=1}^\infty A_n

REMARKS ON -STABLE PROJECTIONLESS C*-ALGEBRAS

has a character.

Finite sections method for Hessenberg matrices

This paper studies the problems of embedding and isomorphism for countably generated Hilbert C ∗ -modules over commutative C ∗ -algebras. When the fiber dimensions differ sufficiently, relative to the dimension of the spectrum, we show that there is an embedding between the modules. This result continues to hold over recursive subhomogeneous C ∗ -algebras. For certain modules, including all modules over C0(X )w hen dimX 3, isomorphism and embedding are determined by the restrictions to the sets where the fiber dimensions are constant. These considerations yield results for the Cuntz semigroup, including a computation of the Cuntz semigroup for C0(X )w hen dimX 3, in terms of cohomological data about X.

Sums of commutators in pure C*-algebras

We investigate the interplay of the following regularity properties for non-simple C*-algebras: finite nuclear dimension, Z-stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic regularity in the Cuntz semigroup, provided that known type I obstructions are avoided. We demonstrate how finite nuclear dimension can be used to study the structure of the central sequence algebra, by factorizing the identity map on the central sequence algebra, in a manner resembling the factorization arising in the definition of nuclear dimension. Results about the central sequence algebra are used to attack the conjecture that finite nuclear dimension implies Z-stability, for sufficiently non-type I, separable C*-algebras. We prove this conjecture in the following cases: (i) the C*-algebra has no purely infinite subquotients and its primitive ideal space has a basis of compact open sets, (ii) the C*-algebra has no purely infinite quotients and its primitive ideal space is Hausdorff. In particular, this covers C*-algebras with finite decomposition rank and real rank zero. Our results hold more generally for C*-algebras with locally finite nuclear dimension which are (M,N)-pure (a regularity condition of the Cuntz semigroup).

Nuclear dimension and -stability of non-simple *-algebras

In this paper we revisit once again, see [ShSu], a family of expanding circle endomorphisms. We consider a family {Bθ} of Blaschke products acting on the unit circle, T, in the complex plane obtained by composing a given Blashke product B with the rotations about zero given by mulitplication by θ ∈ T. While the initial map B may have a fixed sink on T there is always an open set of θ for which Bθ is an expanding map. We prove a lower bound for the average measure theoretic entropy of this family of maps in terms of R ln|B′(z)|dz .