Search

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ -summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module. [ABSTRACT FROM AUTHOR]

Abstract Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B (ℓ p) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space ℓ p of all p -summable sequences. The ℓ p -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among ℓ p -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B (ℓ 2) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network. [ABSTRACT FROM AUTHOR]

In this paper we study the isomorphism problem for reduced twisted group and groupoid L p -operator algebras. For a locally compact group G and a continuous 2-cocycle σ we define the reduced σ -twisted L p -operator algebra F λ p (G , σ). We show that if p ∈ (1 , ∞) ∖ { 2 } , then two such algebras are isometrically isomorphic if and only if the groups are topologically isomorphic and the continuous 2-cocyles are cohomologous. For a twist E over an étale groupoid G , we define the reduced twisted groupoid L p -operator algebra F λ p (G ; E). In the main result of this paper, we show that for p ∈ [ 1 , ∞) ∖ { 2 } if the groupoids are topologically principal, Hausdorff, étale and have a compact unit space, then two such algebras are isometrically isomorphic if and only if the groupoids are isomorphic and the twists are properly isomorphic. [ABSTRACT FROM AUTHOR]

In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the p ( x ) -Laplacian on nonsmooth domains and obtain sharp Calderón–Zygmund type estimates in the variable exponent setting. In a recent work of [12] , the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p ( x ) , see (1.3) and (1.4) . Here, we bridge this gap to obtain the end point case of the estimates obtained in [12] , see (1.5) . In order to do this, we have to obtain significantly improved a priori estimates below p ( x ) , which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature. [ABSTRACT FROM AUTHOR]

Let A and B be simple separable nuclear monotracial C⁎-algebras, and let α and β be strongly outer actions of a countable discrete amenable group Γ on A and B , respectively. In this paper, we show that α ⊗ id W on A ⊗ W and β ⊗ id W on B ⊗ W are cocycle conjugate where W is the Razak-Jacelon algebra. Also, we characterize such actions by using the fixed point subalgebras of Kirchberg's central sequence C⁎-algebras. [ABSTRACT FROM AUTHOR]

We address two errors made in our paper [7]. The most significant error is in Theorem 1.1. We repair this error, and show that the main result, Theorem 2.5 of [7] , is true. The second error is in one of our examples, Remark 2.4 (iv), and we partially resolve it. [ABSTRACT FROM AUTHOR]

Abstract One of the most fundamental operators studied in geometric analysis is the classical Laplace–Beltrami operator. On pseudo-Hermitian manifolds, higher Laplacians L m are defined for each positive integer m , where L 1 coincides with the Laplace–Beltrami operator. Despite their natural definition, these higher Laplacians have not yet been studied in detail. In this paper, we consider the setting of simple pseudo-Hermitian symmetric spaces, i.e., let X = G / H be a symmetric space for a real simple Lie group G , equipped with a G -invariant complex structure. We show that the higher Laplacians L 1 , L 3 , ... , L 2 r − 1 form a set of algebraically independent generators for the algebra D G (X) of G -invariant differential operators on X , where r denotes the rank of X. For higher rank, this is the first instance of a set of generators for D G (X) defined explicitly in purely geometric terms, and confirms a conjecture of Engliš and Peetre, originally stated in 1996 for the class of Hermitian symmetric spaces. [ABSTRACT FROM AUTHOR]

Abstract Let G be a noncompact connected Lie group, denote with ρ a right Haar measure and choose a family of linearly independent left-invariant vector fields X on G satisfying Hörmander's condition. Let χ be a positive character of G and consider the measure μ χ whose density with respect to ρ is χ. In this paper, we introduce Sobolev spaces L α p (μ χ) adapted to X and μ χ (1 < p < ∞ , α ≥ 0) and study embedding theorems and algebra properties of these spaces. As an application, we prove local well-posedness and regularity results of solutions of some nonlinear heat and Schrödinger equations on the group. [ABSTRACT FROM AUTHOR]

Abstract We study ground states of two-component Bose–Einstein condensates (BEC) with trapping potentials in R 2 , where the intraspecies interaction (− a 1 , − a 2) and the interspecies interaction − β are both attractive, i. e , a 1 , a 2 and β are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated L 2 -critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as β ↗ β ⁎ = a ⁎ + (a ⁎ − a 1) (a ⁎ − a 2) , where 0 < a i < a ⁎ : = ‖ w ‖ 2 2 (i = 1 , 2) is fixed and w is the unique positive solution of Δ w − w + w 3 = 0 in R 2. The semi-trivial limit behavior of ground states is tackled in the companion paper [12]. [ABSTRACT FROM AUTHOR]

Abstract: This is the final one in the series of papers where we introduce and study the -algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated -algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as -algebras associated with topological graphs. [Copyright &y& Elsevier]

Abstract: Let be a class of unital C*-algebras. The class of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in is studied (Lin considered the case that consists of finite-dimensional C*-algebras or the tensor products of such with ). In particular, the question is considered whether, for any simple separable , there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in , such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.) [Copyright &y& Elsevier]

Abstract: In this paper we consider automorphisms of the domains of closed *-derivations of C*-algebras and show that they extend to automorphisms of C*-algebras, so we call them diffeomorphisms. The diffeomorphisms generate transformations of the sets of closed *-derivations of C*-algebras. In this paper we study the subgroups of diffeomorphisms that define “bounded” shifts of derivations and the subgroups of the stabilizers of derivations. [Copyright &y& Elsevier]

Abstract: The paper studies unbounded reflexive *-derivations of C*-algebras of bounded operators on Hilbert spaces whose domains are weekly dense in and contain compact operators. It describes a one-to-one correspondence between these derivations and pairs , where are symmetric densely operators on and are J-orthogonal -reflexive lattices of subspaces in the deficiency spaces of . The domains of these *-derivations are associated with some non-selfadjoint reflexive algebras of bounded operators on . The paper analyzes the structure of the lattices of invariant subspaces of and of the normalizers of -the largest Lie subalgebras of such that are their Lie ideals. [Copyright &y& Elsevier]

Let G be a locally compact amenable group, B(G) its Fourier–Stieltjes algebra and I be a closed ideal of it. In this paper we prove the following result: The ideal I has a unit element iff it is principal. This is the noncommutative version of the Glicksberg–Host–Parreau Theorem. The paper also contains an abstract version of this theorem. [Copyright &y& Elsevier]

In this paper, we discuss the positivity of the Hermitian form (,)π introduced by Li in Invent. Math. 27 (1989) 237–255. Let (G1,G2) be a type I dual pair with G1 the smaller group. Let π be an irreducible unitary representation in the semistable range of θ(MG1,MG2) (see Communications in Contemporary Mathematics, Vol. 2, 2000, pp. 255–283). We prove that the invariant Hermitian form (,)π is positive semidefinite under certain restrictions on the size of G2 and a mild growth condition on the matrix coefficients of π. Therefore, if (,)π does not vanish, θ(MG1,MG2)(π) is unitary.Theta correspondence over R was established by Howe in (J. Amer. Math. Soc. 2 (1989) 535–552). Li showed that theta correspondence preserves unitarity for dual pairs in stable range. Our results generalize the results of Li for type I classical groups (Invent. Math. 27 (1989) 237). The main result in this paper can be used to construct irreducible unitary representations of classical groups of type I. [Copyright &y& Elsevier]

The theory of regularity structures [9] sets up an abstract framework of modelled distributions generalising the usual Hölder functions and allowing one to give a meaning to several ill-posed stochastic PDEs. A key result in that theory is the so-called reconstruction theorem: it defines a continuous linear operator that maps spaces of modelled distributions into the usual space of distributions. In the present paper, we extend the scope of this theorem to analogues to the whole class of Besov spaces B p , q γ with non-integer regularity indices. We then show that these spaces behave very much like their classical counterparts by obtaining the corresponding embedding theorems and Schauder-type estimates. [ABSTRACT FROM AUTHOR]

The Morse–Sard theorem requires that a mapping v : R n → R m is of class C k , k > max ⁡ ( n − m , 0 ) . In 1957 Dubovitskiĭ generalized this result by proving that almost all level sets for a C k mapping have H s -negligible intersection with its critical set, where s = max ⁡ ( n − m − k + 1 , 0 ) . Here the critical set, or m -critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) < m } . Another generalization was obtained independently by Dubovitskiĭ and Federer in 1966, namely for C k mappings v : R n → R d and integers m ≤ d they proved that the set of m -critical values v ( Z v , m ) is H q ∘ -negligible for q ∘ = m − 1 + n − m + 1 k . They also established the sharpness of these results within the C k category. Here we prove that Dubovitskiĭ's theorem can be generalized to the case of continuous mappings of the Sobolev–Lorentz class W p , 1 k ( R n , R d ) , p = n k (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not to be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin N -property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a bridge theorem that includes all the above results as particular cases. As a limiting case in this bridge theorem we also establish a new coarea type formula: if E ⊂ { x ∈ R n : rank ∇ v ( x ) ≤ m } , then ∫ E J m v ( x ) d x = ∫ R d H n − m ( E ∩ v − 1 ( y ) ) d H m ( y ) . The mapping v is R d -valued, with arbitrary d , and the formula is obtained without any restrictions on the image v ( R n ) (such as m -rectifiability or σ -finiteness with respect to the m -Hausdorff measure). These last results are new also for smooth mappings, but are presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J. Bourgain (2013, 2015). [ABSTRACT FROM AUTHOR]

In this paper we study Johnson–Schechtman inequalities for noncommutative martingales. More precisely, disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space E ( M ) of a finite von Neumann algebra M without making any assumption on the Boyd indices of E . We show that we can obtain Johnson–Schechtman inequalities for arbitrary martingale difference sequences and that, in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables, the inequalities are one-sided except when E = L 2 ( 0 , 1 ) . As an application, we partly resolve a problem stated by Randrianantoanina and Wu in [46] . We also show that we can obtain sharp Φ-moment analogues for Orlicz functions satisfying p -convexity and q -concavity for 1 ≤ p ≤ 2 , q = 2 and p = 2 , 2 < q < ∞ . This is new even for the classical case. We also extend and strengthen the noncommutative Burkholder–Gundy inequalities in symmetric spaces and in the Φ-moment case. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of weak and strong convergence of derivations, and of the flows associated to them, when dealing with a sequence of metric measure structures ( X , d , m n ) , m n weakly convergent to m . In particular, under curvature assumptions, either only on the limit metric structure ( X , d , m ) or on the whole sequence of metric measure spaces, we provide several stability results. [ABSTRACT FROM AUTHOR]

It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms λ u of the Cuntz algebra O n such that λ u ( F n ) ⊆ F n but u ∉ F n , and a question was raised if for such a u there must always exist a unitary v ∈ F n with λ u | F n = λ v | F n . In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms λ u for which the relative commutant λ u ( F n ) ′ ∩ F n is finite dimensional. [ABSTRACT FROM AUTHOR]

In this paper, we first study the relationship between the existence of twisted Kähler–Ricci solitons and the properness of modified twisted K -energy. Approximating by a sequence of smooth twisted Kähler–Ricci solitons, we obtain an existence result of conical Kähler–Ricci solitons. [ABSTRACT FROM AUTHOR]

For the moment map m : P V n → i u (n) for the action of GL (n) on V n = Λ 3 (C n) ⁎ ⊗ C n , we will study the critical points of F n = ‖ m ‖ 2 : P V n → R in this paper. We first prove that [ μ ] ∈ P V n is a critical point if and only if m ([ μ ]) = c μ I + D μ for some c μ ∈ R and D μ ∈ Der (μ) , and that there exists a constant c > 0 such that the eigenvalues of c D μ are integers prime to each other and some eigenvalues can be strictly negative. Then we give a description of the maxima and minima of F n : L n → R , where L n is the projectivization of all n -dimensional 3-Lie algebras. Furthermore, the structure of the critical points of F n is discussed. Finally, as an application, we show that every three-dimensional 3-Lie algebra is isomorphic to a critical point of F 3 ; and there exists a curve of nonisomorphic four-dimensional 3-Lie algebras which are not isomorphic to any critical point of F 4. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on C ⁎ -algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes. [ABSTRACT FROM AUTHOR]

An error is identified and corrected in the construction of a non- Z -stable, stably projectionless, simple, nuclear C ⁎ -algebra carried out in a paper by the second author. [ABSTRACT FROM AUTHOR]

In this paper, we study closed convex hulls of unitary orbits in various C ⁎ -algebras. For unital C ⁎ -algebras with real rank zero and a faithful tracial state determining equivalence of projections, a notion of majorization describes the closed convex hulls of unitary orbits for self-adjoint operators. Other notions of majorization are examined in these C ⁎ -algebras. Combining these ideas with the Dixmier property, we demonstrate unital, infinite dimensional C ⁎ -algebras of real rank zero and strict comparison of projections with respect to a faithful tracial state must be simple and have a unique tracial state. Also, closed convex hulls of unitary orbits of self-adjoint operators are fully described in unital, simple, purely infinite C ⁎ -algebras. [ABSTRACT FROM AUTHOR]

As recently pointed out by Gabe, a fundamental paper by Elliott and Kucerovsky concerning the absorption theory for C ⁎ -algebras contains an error, and as a consequence we must report that Lemma 4.5 in [3] is not true as stated. In this corrigendum, we prove an adjusted statement and explain why the error has no consequences to the main results of [3] . In particular, it is noted that all the authors' claims concerning Morita equivalence or stable isomorphism of graph C ⁎ -algebras remain correct as stated. [ABSTRACT FROM AUTHOR]

We continue our investigation, from [10] , of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a C ⁎ -algebra is purely infinite if and only if any of its ultrapowers are. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a "Cuntz-like" Banach ⁎-algebra which is purely infinite, but whose ultrapowers are not even simple, and hence not purely infinite. This algebra is a naturally occurring analogue of the Cuntz algebra, and of the L p -analogues introduced by Phillips. However, our proof of being purely infinite is combinatorial, but direct, and so differs from existing proofs. We show that there are non-zero traces on our algebra, which in particular implies that our algebra is not isomorphic to any of the L p -analogues of the Cuntz algebra. [ABSTRACT FROM AUTHOR]

The generalized state space S H (A) of all unital completely positive (UCP) maps on a unital C ⁎ -algebra A taking values in the algebra B (H) of all bounded operators on a Hilbert space H , is a C ⁎ -convex set. In this paper, we establish a connection between C ⁎ -extreme points of S H (A) and a factorization property of certain algebras associated to the UCP maps. In particular, the factorization property of some nest algebras is used to give a complete characterization of those C ⁎ -extreme maps which are direct sums of pure UCP maps. This significantly extends a result of Farenick and Zhou (1998) [14] from finite to infinite dimensional Hilbert spaces. Also it is shown that normal C ⁎ -extreme maps on type I factors are direct sums of normal pure UCP maps if and only if an associated algebra is reflexive. Further, a Krein-Milman type theorem is established for C ⁎ -convexity of the set S H (A) equipped with bounded weak topology, whenever A is a separable C ⁎ -algebra or it is a type I factor. As an application, we provide a new proof of a classical factorization result on operator-valued Hardy algebras. [ABSTRACT FROM AUTHOR]

Abstract: Through a new powerful potential-theoretic analysis, this paper is devoted to discovering the geometrically equivalent isocapacity forms of Chou–Wang's Sobolev type inequality and Tian–Wang's Moser–Trudinger type inequality for the fully nonlinear Hessian operators. [Copyright &y& Elsevier]

Abstract: The goal of this paper is to generalize most of the moment formulae obtained in [12]. More precisely, we consider a general point process μ, and show that the quantities relevant to our problem are the so-called Papangelou intensities. When the Papangelou intensities of μ are well-defined, we show some general formulae to recover the moment of order n of the stochastic integral of the point process. We will use these extended results to introduce a divergence operator and study a random transformation of the point process. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the problem of characterizing Hilbert space operators which are expressible as a sum of (finitely many) orthogonal projections. We obtain a special operator matrix representation and some necessary/sufficient conditions for an infinite-dimensional operator to be expressible as a sum of projections. We prove that a positive operator with essential norm strictly greater than one is always a sum of projections, and if an injective operator of the form , where K is compact, is a sum of projections, then either or K is of trace class with trace K a nonnegative integer. We also consider sums of those projections which have a fixed rank. The closure of the set of sums of projections is also characterized. [Copyright &y& Elsevier]

Abstract: It is known that a graph -algebra is approximately finite dimensional (AF) if and only if the graph E has no loops. In this paper we consider the question of when a labeled graph -algebra is AF. A notion of loop in a labeled space is defined when is the smallest one among the accommodating sets that are closed under relative complements and it is proved that if a labeled graph -algebra is AF, the labeled space has no loops. A sufficient condition for a labeled space to give rise to an AF algebra is also given. For graph -algebras , this sufficient condition is also a necessary one. Besides, we discuss other equivalent conditions for a graph -algebra to be AF in the setting of labeled graphs and prove that these conditions are not always equivalent by invoking various examples. [Copyright &y& Elsevier]

The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals must be given as integration against an absolutely continuous measure on the circle. We develop generalizations of this result to the multivariate non-commutative setting, upon reinterpreting the classical result. In one direction, we show that the GNS representation naturally associated to an analytic functional on the Cuntz algebra cannot have any singular summand. Following a different interpretation, we seek weak-⁎ continuous extensions of analytic functionals on the free disc operator system. In contrast with the classical setting, such extensions do not always exist, and we identify the obstruction precisely in terms of the so-called universal structure projection. We also apply our ideas to commutative algebras of multipliers on some complete Nevanlinna–Pick function spaces. [ABSTRACT FROM AUTHOR]

Abstract: Macroscopic observables in a quantum spin system are given by sequences of spatial means of local elements , , in a UHF algebra. One of their properties is that they commute asymptotically, as n goes to infinity. It is not true that any given set of asymptotically commuting matrices can be approximated by commuting ones in the norm topology. In this paper, we show that for macroscopic observables, this is true. [Copyright &y& Elsevier]

Abstract: Guionnet et al. gave a construction of a II1 factor associated to a subfactor planar algebra. In this paper we define an unshaded planar algebra. To any unshaded planar algebra we associate a finite von Neumann algebra . We prove that is a II1 factor that contains a generic maximal abelian subalgebra called the cup subalgebra. [Copyright &y& Elsevier]

Abstract: In this paper, we deal with the existence of solutions to the nonuniformly elliptic equation of the form in when behaves like when and satisfies the Ambrosetti–Rabinowitz condition. In particular, in the case of N-Laplacian, i.e., , we obtain multiplicity of weak solutions of (0.1). Moreover, we can get the nontriviality of the solution in this case when . Finally, we show that the main results remain true if one replaces the Ambrosetti–Rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity, see Section 7 for more details. [Copyright &y& Elsevier]

Abstract: In this paper, we prove a Galois correspondence for compact group actions on C⁎-algebras in the presence of a commuting minimal action. Namely, we show that there is a one-to-one correspondence between the C⁎-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group. [Copyright &y& Elsevier]

Abstract: The purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems. After giving an axiomatic definition of tensor products in this category, we examine in detail several particular examples of tensor products, including a minimal, maximal, maximal commuting, maximal injective and some asymmetric tensor products. We characterize these tensor products in terms of their universal properties and give descriptions of their positive cones. We also characterize the corresponding tensor products of operator spaces induced by a certain canonical inclusion of an operator space into an operator system. We examine notions of nuclearity for our tensor products which, on the category of -algebras, reduce to the classical notion. We exhibit an operator system which is not completely order isomorphic to a -algebra yet has the property that for every -algebra A, the minimal and maximal tensor product of and A are equal. [Copyright &y& Elsevier]

Abstract: For any abstract subfactor planar algebra P, there exists a finite index extremal subfactor with P as its standard invariant. In this paper, we classify the automorphism group of a bipartite graph planar algebra, and obtain subfactor planar subalgebras by taking fixed points under groups of automorphisms. This construction provides both new examples of subfactors and new descriptions of the planar algebras of previously known examples. [ABSTRACT FROM AUTHOR]

Abstract: Let A be an AH algebra, that is, A is the inductive limit -algebra of with , where are compact metric spaces, and are positive integers, and are projections. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that . (This condition can be relaxed to a certain condition called very slow dimension growth.) In this article, we prove that if we further assume that is torsion free, then A is an approximate circle algebra (or an algebra), that is, A can be written as the inductive limit of where . One of the main technical results of this article, called the decomposition theorem, is proved for the general case, i.e., without the assumption that is torsion free. This decomposition theorem will play an essential role in the proof of a general reduction theorem, where the condition that is torsion free is dropped, in the subsequent paper Gong et al. (preprint) —of course, in that case, in addition to space , we will also need the spaces , , and , as in Gong (2002) . [Copyright &y& Elsevier]

Abstract: In this paper we will give a thorough study of the notion of property (T) for -algebras (as introduced by M.B. Bekka) as well as a slightly stronger version of it, called “strong property (T)” (which is also an analogue of the corresponding concept in the case of discrete groups and type II1-factors). More precisely, we will give some interesting equivalent formulations as well as some permanence properties for both property (T) and strong property (T). We will also relate them to certain (T)-type properties of the unitary group of the underlying -algebra. [Copyright &y& Elsevier]

Abstract: In this paper we propose a representation theorem for local operator spaces which extends Ruan''s representation theorem for operator spaces. Based upon this result, we introduce local operator systems which are locally convex versions of the operator systems and prove Stinespring theorem for local operator systems. A local operator -algebra is an example of a local operator system. Finally, we investigate the injectivity in both local operator space and local operator system senses, and prove locally convex version of the known result by Choi and Effros, that an injective local operator system possesses unique multinormed -algebra structure with respect to the original involution and matrix topology. [Copyright &y& Elsevier]

Abstract: In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi [B. Clair, S. Mokhtari-Sharghi, Zeta functions of discrete groups acting on trees, J. Algebra 237 (2001) 591–620] on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised in [R.I. Grigorchuk, A. Żuk, The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps, in: V.A. Kaimanovich, et al. (Eds.), Proc. Workshop, Random Walks and Geometry, Vienna, 2001, de Gruyter, Berlin, 2004, pp. 141–180] by Grigorchuk and Żuk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs. [Copyright &y& Elsevier]

Abstract: In this paper, we define an equivariant higher index map from to if a torsion-free discrete group Γ acts on a manifold X properly, where is the norm closure of all locally compact, Γ-invariant operators with finite propagation. When Γ acts on X properly and cocompactly, this equivariant higher index map coincides with the Baum–Connes map [P. Baum, A. Connes, K-theory for discrete groups, in: D. Evens, M. Takesaki (Eds.), Operator Algebras and Applications, Cambridge Univ. Press, Cambridge, 1989, pp. 1–20; P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group -algebras, in: -Algebras: 1943–1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291]. When Γ is trivial, this equivariant higher index map is the coarse Baum–Connes map [J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Amer. Math. Soc. 104 (497) (1993); J. Roe, Index Theory, Coarse Geometry, and the Topology of Manifolds, CBMS Reg. Conf. Ser. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1996]. If X is a simply-connected complete Riemannian manifold with nonpositive sectional curvature and Γ is a torsion-free discrete group acting on X properly and isometrically, we prove that the equivariant higher index map is injective. [Copyright &y& Elsevier]

Abstract: The paper establishes, for a wide class of locally compact groupoids Γ, the E-theoretic descent functor at the -algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. Section 2 shows that Γ-actions on a -algebra B, where X is the unit space of Γ, can be usefully formulated in terms of an action on the associated bundle . Section 3 shows that the functor is continuous and exact, and uses the disintegration theory of J. Renault. Section 4 establishes the existence of the descent functor under a very mild condition on Γ, the main technical difficulty involved being that of finding a Γ-algebra that plays the role of in the group case. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to show that, for a large class of band-dominated operators on , with U being a complex Banach space, the injectivity of all limit operators of A already implies their invertibility and the uniform boundedness of their inverses. The latter property is known to be equivalent to the invertibility at infinity of A, which, on the other hand, is often equivalent to the Fredholmness of A. As a consequence, for operators A in the Wiener algebra, we can characterize the essential spectrum of A on , regardless of , as the union of point spectra of its limit operators considered as acting on . [Copyright &y& Elsevier]

Abstract: This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames. [Copyright &y& Elsevier]

Abstract: Let be a finite von Neumann algebra acting on a Hilbert space and be a transitive algebra containing . In this paper we prove that if is 2-fold transitive, then is strongly dense in . This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975) 271–283]) is 2-fold transitive, then is strongly dense in . Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., and , are studied. Brown measures of certain operators in are explicitly computed. [Copyright &y& Elsevier]

Abstract: In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms . Then the full automorphism group is considered as an extension of the open subgroup of diffeomorphisms of M preserving the equivalence class of under pull-backs, by the gauge group . We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac–Moody algebras and groups. [Copyright &y& Elsevier]

Abstract: In this paper we study uniqueness properties of solutions of the so-called k-generalized Korteweg–de Vries equations. Our goal is to obtain sufficient conditions on the behavior of the difference of two solutions of (1.1) at two different times and which guarantee that . [Copyright &y& Elsevier]