The nuclear dimension of C*-algebras

We introduce the nuclear dimension of a C*-algebra; this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank. Our notion behaves well with respect to inductive limits, tensor products, hereditary subalgebras (hence ideals), quotients, and even extensions. It can be computed for many examples; in particular, it is finite for all UCT Kirchberg algebras. In fact, all classes of nuclear C*-algebras which have so far been successfully classified consist of examples with finite nuclear dimension, and it turns out that finite nuclear dimension implies many properties relevant for the classification program. Surprisingly, the concept is also linked to coarse geometry, since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform Roe algebra is dominated by the asymptotic dimension of the underlying space.Comment: 33 page

Non-closure of quantum correlation matrices and factorizable channels that require infinite dimensional ancilla

We show that there exist factorizable quantum channels in each dimension $\ge 11$ which do not admit a factorization through any finite dimensional von Neumann algebra, and do require ancillas of type II$_1$, thus witnessing new infinite-dimensional phenomena in quantum information theory. We show that the set of n by n matrices of correlations arising as second-order moments of projections in finite dimensional von Neumann algebras with a distinguished trace is non-closed, for all $n \ge 5$, and we use this to give a simplified proof of the recent result of Dykema, Paulsen and Prakash that the set of synchronous quantum correlations $C_q^s(5,2)$ is non-closed. Using a trick originating in work of Regev, Slofstra and Vidick, we further show that the set of correlation matrices arising from second-order moments of unitaries in finite dimensional von Neumann algebras with a distinguished trace is non-closed in each dimension $\ge 11$, from which we derive the first result above.Comment: 16 pages. An appendix by Narutaka Ozawa has been added. To appear in Comm. Math. Phy

A Dixmier type averaging property of automorphisms on a C∗C^*-algebra

In his study of the relative Dixmier property for inclusions of von Neumann algebras and of $C^*$-algebras, Popa considered a certain property of automorphisms on $C^*$-algebras, that we here call the strong averaging property. In this note we characterize when an automorphism on a $C^*$-algebra has the strong averaging property. In particular, automophisms on commutative $C^*$-algebras possess this property precisely when they are free. An automorphism on a unital separable simple $C^*$-algebra with at least one tracial state has the strong averaging property precisely when its extension to the finite part of the bi-dual of the $C^*$-algebra is properly outer, and in the simple, non-tracial case the strong averaging property is equivalent to being outer. To illustrate the usefulness of the strong averaging property we give three examples where we can provide simpler proofs of existing results on crossed product $C^*$-algebras, and we are also able to extend these results in different directions.Comment: 19 page

On the structure of simple C∗-algebras tensored with a UHF-algebra, II

AbstractLet D be a simple unital C∗-algebra, let B be a UHF-algebra, and put A = B ⊗ D. It is proved that if p, q ϵ A are projections and τ(p) < τ(q) for all quasitraces τ on A, then p ≲ q (in the sense of Murray and von Neumann). A more general result involving positive operators in A is also proved. If D has finitely many extremal quasi-traces, and the projections in D ⊗ K separate these, then it is proved that A has real rank zero. Finally it is proved that provided D is stably finite, then each positive state on K0(D) lifts to a quasi-trace

Elucidation of primary metabolic pathways in <i>Aspergillus </i>species: Orphaned research in characterizing orphan genes

Primary metabolism affects all phenotypical traits of filamentous fungi. Particular examples include reacting to extracellular stimuli, producing precursor molecules required for cell division and morphological changes as well as providing monomer building blocks for production of secondary metabolites and extracellular enzymes. In this review, all annotated genes from four Aspergillus species have been examined. In this process, it becomes evident that 80–96% of the genes (depending on the species) are still without verified function. A significant proportion of the genes with verified metabolic functions are assigned to secondary or extracellular metabolism, leaving only 2–4% of the annotated genes within primary metabolism. It is clear that primary metabolism has not received the same attention in the post-genomic area as many other research areas—despite its role at the very centre of cellular function. However, several methods can be employed to use the metabolic networks in tandem with comparative genomics to accelerate functional assignment of genes in primary metabolism. In particular, gaps in metabolic pathways can be used to assign functions to orphan genes. In this review, applications of this from the Aspergillus genes will be examined, and it is proposed that, where feasible, this should be a standard part of functional annotation of fungal genomes

Constructive Gelfand duality for C*-algebras

We present a constructive proof of Gelfand duality for C*-algebras by reducing the problem to Gelfand duality for real C*-algebras.Comment: 6page

Rank-two graphs whose C^*-algebras are direct limits of circle algebras

We describe a class of rank-2 graphs whose C^*-algebras are AT algebras. For a subclass which we call rank-2 Bratteli diagrams, we compute the K-theory of the C*-algebra. We identify rank-2 Bratteli diagrams whose C*-algebras are simple and have real-rank zero, and characterise the K-invariants achieved by such algebras. We give examples of rank-2 Bratteli diagrams whose C*-algebras contain as full corners the irrational rotation algebras and the Bunce-Deddens algebras.Comment: 41 pages, uses pictex for figure