Domains of infinitesimal generators of automorphism flows

We present a systematic characterization of the domain of a generator of a one parameter group on certain C∗-subalgebras of via finite-dimensional estimates. Our approach yields an example of a densely defined closed symmetric derivation on a C∗-subalgebras of whose domain is not closed with respect to the C¹-functional calculus. This completes and complements the earlier example of McIntosh (J. Funct. Anal. 30 (1977) 264). Our methods are partly based on the theory of adjoint C₀-semigroups.     

A unique decomposition result for HT factors with torsion free core

We prove that II₁ factors M have a unique (up to unitary conjugacy) cross-product type decomposition around “core subfactors” N⊂M satisfying the property HT of [S. Popa, On a class of type II₁ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006) 809-899] and a certain “torsion freeness” condition. In particular, this shows that isomorphism of factors of the form Lαi(Z²)?Fni, i=1,2, for Fni⊂SL(2,Z) free groups of rank ni and αj=e2πitj, tj∉Q, implies n₁=n₂.     

Strong characterizing sequences of countable groups

András Biró and Vera Sós prove that for any subgroup G of T generated freely by finitely many generators there is a sequence A⊆N such that for all β∈T we have (‖.‖ denotes the distance to the nearest integer)

     

Simple compact quantum groups I

The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu(Q) for Q∈GL(n,C) satisfying , n?2; (b) The quantum automorphism groups Aaut(B,τ) of finite-dimensional C^∗-algebras B endowed with the canonical trace τ when dim(B)?4, including the quantum permutation groups Aaut(Xn) on n points (n?4); (c) The standard deformations Kq of simple compact Lie groups K and their twists , as well as Rieffel's deformation KJ.     

Parseval frames for ICC groups

We analyze Parseval frames generated by the action of an ICC group on a Hilbert space. We parametrize the set of all such Parseval frames by operators in the commutant of the corresponding representation. We characterize when two such frames are strongly disjoint. We prove an undersampling result showing that if the representation has a Parseval frame of equal norm vectors of norm , the Hilbert space is spanned by an orthonormal basis generated by a subgroup. As applications we obtain some sufficient conditions under which a unitary representation admits a Parseval frame which is spanned by a Riesz sequences generated by a subgroup. In particular, every subrepresentation of the left-regular representation of a free group has this property.     

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.     

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.     

Degree conditions for group connectivity

Let G be a 2-edge-connected simple graph on n≥13 vertices and A an (additive) abelian group with |A|≥4. In this paper, we prove that if for every uv∉E(G), max{d(u),d(v)}≥n/4, then either G is A-connected or G can be reduced to one of K_2,3,C₄ and C₅ by repeatedly contracting proper A-connected subgraphs, where C_k is a cycle of length k. We also show that the bound n≥13 is the best possible.     

Estimating parameters of a multiple autoregressive model by the modified maximum likelihood method

We consider a multiple autoregressive model with non-normal error distributions, the latter being more prevalent in practice than the usually assumed normal distribution. Since the maximum likelihood equations have convergence problems (Puthenpura and Sinha, 1986) [11], we work out modified maximum likelihood equations by expressing the maximum likelihood equations in terms of ordered residuals and linearizing intractable nonlinear functions (Tiku and Suresh, 1992) [8]. The solutions, called modified maximum estimators, are explicit functions of sample observations and therefore easy to compute. They are under some very general regularity conditions asymptotically unbiased and efficient (Vaughan and Tiku, 2000) [4]. We show that for small sample sizes, they have negligible bias and are considerably more efficient than the traditional least squares estimators. We show that our estimators are robust to plausible deviations from an assumed distribution and are therefore enormously advantageous as compared to the least squares estimators. We give a real life example.     

Maximal ergodic theorems for some group actions

We prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative Lp-space. We also obtain the corresponding individual ergodic theorems. Applying these results to actions of a free group on a von Neumann algebra, we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of Nevo-Stein.     

Further improved recursions for a class of compound Poisson distributions

In the present paper we develop more efficient recursive formulae for the evaluation of the t-order cumulative function Γ^th(x) and the t-order tail probability Λ^th(x) of the class of compound Poisson distributions in the case where the derivative of the probability generating function of the claim amounts can be written as a ratio of two polynomials. These efficient recursions can be applied for the exact evaluation of the probability function (given by De Pril [De Pril, N., 1986a. Improved recursions for some compound Poisson distributions. Insurance Math. Econom. 5, 129-132]), distribution function, tail probability, stop-loss premiums and t-order moments of stop-loss transforms of compound Poisson distributions. Also, efficient recursive algorithms are given for the evaluation of higher-order moments and r-order factorial moments about any point for this class of compound Poisson distributions. Finally, several examples of discrete claim size distributions belonging to this class are also given.     

A cohomological description of property (T) for quantum groups

We prove a Delorme-Guichardet type theorem for discrete quantum groups expressing property (T) of the quantum group in question in terms of its first cohomology groups. As an application, we show that the first L²-Betti number of a discrete property (T) quantum group vanishes.     

Divergence points of self-similar measures satisfying the OSC

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K. It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of Rd, which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.     

Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems

By computing the E-critical groups at θ and infinity of the corresponding functional of Hamiltonian systems, we proved the existence of nontrivial periodic solutions for the systems which may be resonant at θ and infinity under some new conditions. Some results in the literature are extended and some new type of theorems are proved. The main tool is the E-Morse theory developed by Kryszewski and Szulkin.     

Pseudocompact groups: progress and problems

Several months ago the speaker and Jan van Mill gave a proof of this result [W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Abstracts Amer. Math. Soc. 27 (2006) 78 (Abstract #1014-22-958); W.W. Comfort, J. van Mill, Extremal pseudocompact abelian groups are compact metrizable, Proc. Amer. Math. Soc. 135 (2007) 4039-4044]: A pseudocompact abelian group of uncountable weight admits both a proper dense pseudocompact subgroup and a strictly larger pseudocompact group topology.This presentation will describe both the necessary new details of the argument and the historical development (useful tools, special cases). Among those who contributed essentially are: K.A. Ross (1964, 1966); T. Soundararajan (1982); L.C. Robertson (1982, 1988); J. van Mill (1989); J. van Mill and H. Gladdines (1994); J. Galindo (2002).Several related unsolved problems will be cited.     

Optimal stopping for non-linear expectations—Part II

Relying on the stochastic analysis tools developed in Bayraktar and Yao (2011) [1], we solve the optimal stopping problems for non-linear expectations.     

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.     

A power penalty approach to a Nonlinear Complementarity Problem

We propose a novel power penalty approach to a Nonlinear Complementarity Problem (NCP) in which the NCP is approximated by a nonlinear equation containing a power penalty term. We show that the solution to the penalty equation converges to that of the NCP at an exponential rate when the function involved is continuous and ξ-monotone. A higher convergence rate is also obtained when the function becomes Lipschitz continuous. Numerical results are presented to confirm the theoretical findings.     

Strong rigidity of generalized Bernoulli actions and computations of their symmetry groups

We study equivalence relations and II₁ factors associated with (quotients of) generalized Bernoulli actions of Kazhdan groups. Specific families of these actions are entirely classified up to isomorphism of II₁ factors. This yields explicit computations of outer automorphism and fundamental groups. In particular, every finitely presented group is concretely realized as the outer automorphism group of a continuous family of non stably isomorphic II₁ factors.