Mathematics

Motivated by the problem of characterizing KMS states on the reduced C ??-algebras of étale groupoids, we show that the reduced norm on these algebras induces a C ??-norm on the group algebras of the isotropy groups. This C ??-norm coincides with the reduced norm for the transformation groupoids, but, as follows from examples of Higson-Lafforgue-Skandalis, it can be exotic already for groupoids of germs associated with group actions. We show that the norm is still the reduced one for some classes of graded groupoids, in particular, for the groupoids associated with partial actions of groups and the semidirect products of exact groups and groupoids with amenable isotropy groups.

We characterize noncommutative symmetric Banach spaces for which every bounded sequence admits either a convergent subsequence, or a 2 -co-lacunary subsequence. This extends the classical characterization, due to Räbiger.

The paper is devoted to 2-local automorphisms on A W ∗ -algebras. Using the technique of matrix algebras over a unital Banach algebra we prove that any 2-local automorphism on an arbitrary A W ∗ -algebra without finite type~I direct summands is a global automorphism.

We consider 2-positive almost order zero (disjointness preserving) maps on C*-algebras. Generalizing the argument of M. Choi for multiplicative domains, we give an internal characterization of almost order zero for 2-positive maps. It is also shown that complete positivity can be reduced to 2-positivity in the definition of decomposition rank for unital separable C*-algebras.

Let M be a σ -finite von Neumann algebra, equipped with a normal faithful state φ , and let A be maximal subdiagonal subalgebra of M . We prove a Beurling-Blecher-Labuschagne theorem for A -invariant subspaces of L p (M) when 1≤p<∞ . As application, we give a characterization of outer operators in Haagerup noncommutative H p -spaces associated with A .

We provide a Cuntz-Pimsner model for graph of groups C ∗ -algebras. This allows us to compute the K -theory of a range of examples and show that graph of groups C ∗ -algebras can be realised as Exel-Pardo algebras. We also make a preliminary investigation of whether the crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain trees satisfies Poincaré duality in KK -theory. By constructing a K -theory duality class we compute the K -homology of these crossed products.

A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We establish that the same holds for a Lie groupoid. Most of the heavy lifting is done by a lemma in the original work of Dixmier-Malliavin. We also need the technology of Lie algebroids and the corresponding notion of exponential map. As an application, we obtain a result on the arithmetic of ideals in the smooth convolution algebra of a Lie groupoid arising from functions vanishing to given order on an invariant submanifold of the unit space.

We prove the following two results for a given uniformly locally finite metric space with Yu's property A: 1) The group of outer automorphisms of its uniform Roe algebra is isomorphic to its group of bijective coarse equivalences modulo closeness. 2) The group of outer automorphisms of its Roe algebra is isomorphic to its group of coarse equivalences modulo closeness. The main difficulty lies in the latter. To prove that, we obtain several uniform approximability results for maps between Roe algebras and use them to obtain a theorem about the `uniqueness' of Cartan masas of Roe algebras. We finish the paper with several applications of the results above to concrete metric spaces.

We prove a generalized version of Renault's theorem for Cartan subalgebras. We show that the original assumptions of second countability and separability are not needed. This weakens the assumption of topological principality of the underlying groupoid to effectiveness.

Let j:Y→X be a continuous surjection of compact metric spaces. Whyburn proved that j is irreducible, meaning that j(F)⊊X for any proper closed subset F⊊Y , if and only if j is almost one-to-one, in the sense that {y∈Y: j −1 (j(y))=y} ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ =Y. In this note we prove the following generalization: There exists a unique minimal closed set K⊆Y such that j(K)=X if and only if {x∈X:card( j −1 (x))=1} ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ =X. Translated to the language of operator algebras, this says that if A⊆B is a unital inclusion of separable abelian C ∗ -algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extension property of Nagy-Reznikoff holds. More generally, we prove that a unital inclusion of (not necessarily separable) abelian C ∗ -algebras has a unique pseudo-expectation if and only if it is aperiodic (in the sense of Kwaśniewski-Meyer).