Twisted first homology groups of the automorphism group of a free group

The automorphism group and outer automorphism group of a free group F_n of rank n act on the abelianized group H of F_n and the dual group H^* of H. The twisted first homology groups of and with coefficients in H and H^* are calculated.     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

Separable states and positive maps

Using the natural duality between linear functionals on tensor products of C^∗-algebras with the trace class operators on a Hilbert space H and linear maps of the C^∗-algebra into B(H), we study the relationship between separability, entanglement and the Peres condition of states and positivity properties of the linear maps.     

Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C^∗-algebra, an Exel-Laca algebra, and an ultragraph C^∗-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C^∗-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C^∗-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C^∗-algebra of a row-finite graph with no sinks.     

On fractional powers of generators of fractional resolvent families

We show that if −A generates a bounded α-times resolvent family for some α∈(0,2], then −Aβ generates an analytic γ-times resolvent family for and γ∈(0,2). And a generalized subordination principle is derived. In particular, if −A generates a bounded α-times resolvent family for some α∈(1,2], then −A1/α generates an analytic C₀-semigroup. Such relations are applied to study the solutions of Cauchy problems of fractional order and first order.     

The Rohlin property for automorphisms of the Jiang-Su algebra

For projectionless C^∗-algebras absorbing the Jiang-Su algebra tensorially, we study a kind of the Rohlin property for automorphisms. We show that the crossed products obtained by automorphisms with this Rohlin property also absorb the Jiang-Su algebra tensorially under a mild technical condition on the C^∗-algebras. In particular, for the Jiang-Su algebra we show the uniqueness up to outer conjugacy of the automorphism with this Rohlin property.     

Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals

We establish several generalisations of Urysohn's lemma in the setting of JB^∗-triples which provide full answers to Problems 1.12 and 1.13 in Fernández-Polo and Peralta (2007) [22]. These results extend the previous generalisations obtained by C.A. Akemann, G.K. Pedersen and L.G. Brown in the setting of C^∗-algebras. A generalised Kadison's transitivity theorem is established for finite sums of pairwise orthogonal compact tripotents in JBW^∗-triples. We introduce the notion of positively open tripotent in the bidual of a JB^∗-triple as an extension of a concept which was already considered in the setting of ternary rings of operators. We investigate the connections appearing between positively open tripotents and hereditary inner ideals.     

Groupoids and pseudodifferential calculus on manifolds with corners

We associate to any manifold with corners (even with non-embedded hyperfaces) a (non-Hausdorff) longitudinally smooth Lie groupoid, on which we define a pseudodifferential calculus. This calculus generalizes the b-calculus of Melrose, defined for manifolds with embedded corners. The groupoid of a manifold with corners is shown to be unique up to equivalence for manifolds with corners of same codimension. Using tools from the theory of C∗-algebras of groupoids, we also obtain new proofs for the study of b-calculus.     

On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP-hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pairs is fixed, but remains NP-hard in directed acyclic graphs and APX-hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP-hard in unweighted cacti of bounded degree and bounded path-width.     

On the topology of the Kasparov groups and its applications

In this paper we establish a direct connection between stable approximate unitary equivalence for *-homomorphisms and the topology of the KK-groups which avoids entirely C^*-algebra extension theory and does not require nuclearity assumptions. To this purpose we show that a topology on the Kasparov groups can be defined in terms of approximate unitary equivalence for Cuntz pairs and that this topology coincides with both Pimsner's topology and the Brown-Salinas topology. We study the generalized Rørdam group , and prove that if a separable exact residually finite dimensional C^*-algebra satisfies the universal coefficient theorem in KK-theory, then it embeds in the UHF algebra of type 2^∞. In particular such an embedding exists for the C^*-algebra of a second countable amenable locally compact maximally almost periodic group.     

Projective modules over noncommutative tori are multi-window Gabor frames for modulation spaces

In the present investigation we link noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. We are therefore in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to display the deeper properties of Gabor frames. Furthermore, we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this investigation is twofold: (i) an interpretation of projective modules over noncommutative tori in terms of Gabor analysis and (ii) to show that the Morita-Rieffel equivalence between noncommutative tori is the natural framework for the duality theory of Gabor frames. More concretely, we interpret generators of projective modules over noncommutative tori as the Gabor atoms of multi-window Gabor frames for modulation spaces. Moreover, we show that this implies the existence of good multi-window Gabor frames for modulation spaces with Gabor atoms in e.g. Feichtinger's algebra or in Schwartz space.     

The Elliott conjecture for Villadsen algebras of the first type

We study the class of simple C^∗-algebras introduced by Villadsen in his pioneering work on perforated ordered K-theory. We establish six equivalent characterisations of the proper subclass which satisfies the strong form of Elliott's classification conjecture: two C^∗-algebraic (Z-stability and approximate divisibility), one K-theoretic (strict comparison of positive elements), and three topological (finite decomposition rank, slow dimension growth, and bounded dimension growth). The equivalence of Z-stability and strict comparison constitutes a stably finite version of Kirchberg's characterisation of purely infinite C^∗-algebras. The other equivalences confirm, for Villadsen's algebras, heretofore conjectural relationships between various notions of good behaviour for nuclear C^∗-algebras.     

The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. To this end, we associate with such a current an equivariant cyclic cohomology class of Connes' C^∗-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points.     

Sequence characterization of Riordan arrays

In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup.     

Remark on uniqueness of mild solutions to the Navier-Stokes equations

We investigate a limiting uniqueness criterion to the Navier-Stokes equations. We prove that the mild solution is unique under the class , where bmo^-1 is the “critical” space including Lⁿ. As an application of uniqueness theorem, we also consider the local well-posedness of Navier-Stokes equations in bmo^-1.     

Partition and composition matrices

This paper introduces two matrix analogues for set partitions. A composition matrix on a finite set X is an upper triangular matrix whose entries partition X, and for which there are no rows or columns containing only empty sets. A partition matrix is a composition matrix in which an order is placed on where entries may appear relative to one-another.We show that partition matrices are in one-to-one correspondence with inversion tables. Non-decreasing inversion tables are shown to correspond to partition matrices with a row ordering relation. Partition matrices which are s-diagonal are classified in terms of inversion tables. Bidiagonal partition matrices are enumerated using the transfer-matrix method and are equinumerous with permutations which are sortable by two pop-stacks in parallel.We show that composition matrices on X are in one-to-one correspondence with (2+2)-free posets on X. Also, composition matrices whose rows satisfy a column-ordering relation are shown to be in one-to-one correspondence with parking functions. Finally, we show that pairs of ascent sequences and permutations are in one-to-one correspondence with (2+2)-free posets whose elements are the cycles of a permutation, and use this relation to give an expression for the number of (2+2)-free posets on {1,…,n}.     

Binary relations and reduced hypergroups

Different partial hypergroupoids are associated with binary relations defined on a set H. In this paper we find sufficient and necessary conditions for these hypergroupoids in order to be reduced hypergroups. Given two binary relations ρ and σ on H we investigate when the hypergroups associated with the relations ρ∩σ, ρ∪σ and ρσ are reduced. We also determine when the cartesian product of two hypergroupoids associated with a binary relation is a reduced hypergroup.     

3-Factor-criticality in domination critical graphs

A graph G is said to be k-γ-critical if the size of any minimum dominating set of vertices is k, but if any edge is added to G the resulting graph can be dominated with k-1 vertices. The structure of k-γ-critical graphs remains far from completely understood when k?3.A graph G is factor-critical if G-v has a perfect matching for every vertex v∈V(G) and is bicritical if G-u-v has a perfect matching for every pair of distinct vertices u,v∈V(G). More generally, a graph is said to be k-factor-critical if G-S has a perfect matching for every set S of k vertices in G. In three previous papers [N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs, Discrete Math. 272 (2003) 5-15; N. Ananchuen, M.D. Plummer, Matching properties in domination critical graphs, Discrete Math. 277 (2004) 1-13; N. Ananchuen, M.D. Plummer, Some results related to the toughness of 3-domination-critical graphs. II. Utilitas Math. 70 (2006) 11-32], we explored the toughness of 3-γ-critical graphs and some of their matching properties. In particular, we obtained some properties which are sufficient for a 3-γ-critical graph to be factor-critical and, respectively, bicritical. In the present work, we obtain similar results for k-factor-critical graphs when k=3.