Applications of Operator Space Theory to Nest Algebra Bimodules

Recently, Blecher and Kashyap have generalized the notion of W ^*-modules over von Neumann algebras to the setting where the operator algebras are σ closed algebras of operators on a Hilbert space. They call these modules weak* rigged modules. We characterize the weak* rigged modules over nest algebras. We prove that Y is a right weak* rigged module over a nest algebra Alg(M){\rm{Alg}(\mathcal M)} if and only if there exists a completely isometric normal representation F{\Phi } of Y and a nest algebra Alg(N){\rm{Alg}(\mathcal N)} such that Alg(N) F(Y)Alg(M) ì F(Y){\rm{Alg}(\mathcal N) \Phi (Y)\rm{Alg}(\mathcal M)\subset \Phi (Y)} while F(Y){\Phi (Y)} is implemented by a continuous nest homomorphism from M{\mathcal M} onto N{\mathcal N} . We describe some properties which are preserved by continuous CSL homomorphisms.     

Constructions of Large Graphs on Surfaces

We consider the degree/diameter problem for graphs embedded in a surface, namely, given a surface Σ and integers Δ and k, determine the maximum order N(Δ,k,Σ) of a graph embeddable in Σ with maximum degree Δ and diameter k. We introduce a number of constructions which produce many new largest known planar and toroidal graphs. We record all these graphs in the available tables of largest known graphs. Given a surface Σ of Euler genus g and an odd diameter k, the current best asymptotic lower bound for N(Δ,k,Σ) is given by $$\sqrt{\frac{3}{8}}g \Delta^{\lfloor k/2 \rfloor}.$$ Our constructions produce new graphs of order $$\left\{\begin{array}{ll}6 \Delta^{\lfloor k/2 \rfloor} \qquad \qquad \qquad \qquad {\rm if \Sigma\;is\;the\;Klein\;bottle} \\ \left(\frac{7}{2} + \sqrt{6g + \frac{1}{4}}\right) \Delta^{\lfloor k/2 \rfloor} \quad {\rm otherwise},\end{array}\right.$$ thus improving the former value.     

Stable systolic category of manifolds and the cup-length

It follows from a theorem of Gromov that the stable systolic category cat_stsys M{\rm cat}_{\rm stsys} M of a closed manifold M is bounded from below by cl_\mathbbQ M{\rm cl}_{\mathbb{Q}} M, the rational cup-length of M [Ka07]. We study the inequality in the opposite direction. In particular, combining our results with Gromov’s theorem, we prove the equality cat_stsys M = cl_\mathbbQ M{\rm cat}_{\rm stsys} M = {\rm cl}_{\mathbb{Q}} M for simply connected manifolds of dimension ≤ 7.     

Parametrices of mixed elliptic problems

Mixed elliptic problems for differential operators A in a domain Ω with smooth boundary Y are studied in theform $$ Au = f \;\; {\rm in} \;\; {\rm \Omega} \, , \quad T_{\pm}u = g_{\pm} \;\; {\rm on} \;\; Y_{\pm} \, , $$ where Y_± ? Y are manifolds with a common boundary Z, such that Y_– ∪ Y₊ = Y and Y_– ∩ Y₊ = Z, with boundary conditions T_± on Y_± (with smooth coefficients up to Z from the respective side) satisfying the Shapiro–Lopatinskij condition. We consider such problems in standard Sobolev spaces and characterise natural extra conditions on the interface Z with an analogue of Shapiro–Lopatinskij ellipticity for an associated transmission problem on the boundary; then the extended operator is Fredholm. The transmission operators on the boundary with respect to Z belong to a complete pseudo‐differential calculus, a modification of the algebra of boundary value problems without the transmission property. We construct parametrices of elliptic elements in that calculus, and we obtain parametrices of the original mixed problems under additional conditions on the interface. We consider the Zaremba problem and other mixed problems for the Laplace operator, determine the number of extra conditions and calculate the index of associated Fredholm operators. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Similarity Between Finitary and Artinian-Finitary Groups

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is and the Artinian-finitary automorphism group of M over R is We investigate further the surprisingly close relationship between these two types of automorphism groups. Their group theoretic properties seem practically identical.     

Non-white Wishart ensembles

We consider non-white Wishart ensembles , where X is a p×N random matrix with i.i.d. complex standard Gaussian entries and Σ is a covariance matrix, with fixed eigenvalues, close to the identity matrix. We prove that the largest eigenvalue of such random matrix ensembles exhibits a universal behavior in the large-N limit, provided Σ is “close enough” to the identity matrix. If not, we identify the limiting distribution of the largest eigenvalues, focusing on the case where the largest eigenvalues almost surely exit the support of the limiting Marchenko-Pastur's distribution.     

The Similarity Between Finitary and Artinian-Finitary Groups

Let M be a module over the commutative ring R. The finitary automorphism group of M over R is FAut_RM = {g ? Aut_RM :M(g-1) is R-Noetherian}{\rm FAut}_RM =\{g\in{\rm Aut}_RM :M(g-1)\ {\rm is}\ R\hbox{-}{\rm Noetherian}\} and the Artinian-finitary automorphism group of M over R is F₁Aut_RM = {g ? Aut_RM : M(g-1) is R-Artinian}.{\rm F}_1{\rm Aut}_RM = \{g\in{\rm Aut}_RM : M(g-1)\ {\rm is}\ R\hbox{-}{\rm Artinian}\}. We investigate further the surprisingly close relationship between these two types of automorphism groups. Their group theoretic properties seem practically identical.     

Starke Kotorsionsmoduln

Suppose that $(R, m)$ is a noetherian local ring and that E is the injective hull of the residue class field $R/m$. Suppose that M is an R-module, $M^0 = {\mbox{\rm Hom}}_R (M, E)$ is the Matlis dual of M and ${\mbox{\rm Coass}(M)} = {\mbox{\rm Ass} (M^0)}$. M is called cotorsion if every prime ideal ${\frak p} \in {\mbox{\rm Coass}}(M)$ is regular; it is called strongly cotorsion if $\cap {\rm Coass}(M)$ is regular. In the first part, we completely describe the structure of the strongly cotorsion modules over R, use this to determine the coassociated prime ideals of the bidual $M^{00}$, and give in the second part criteria for a cotorsion module being strongly cotorsion. Received: 7 March 2002     

Ergodic actions of mapping class groups on moduli spaces of representations of non-orientable surfaces

Let M be a non-orientable surface with Euler characteristic χ(M) ≤ −2. We consider the moduli space of flat SU(2)-connections, or equivalently the space of conjugacy classes of representations

The Generating Function for the Airy Point Process and a System of Coupled Painlevé II Equations

For a wide class of Hermitian random matrices, the limit distribution of the eigenvalues close to the largest one is governed by the Airy point process. In such ensembles, the limit distribution of the k th largest eigenvalue is given in terms of the Airy kernel Fredholm determinant or in terms of Tracy–Widom formulas involving solutions of the Painlevé II equation. Limit distributions for quantities involving two or more near‐extreme eigenvalues, such as the gap between the k th and the ℓth largest eigenvalue or the sum of the k largest eigenvalues, can be expressed in terms of Fredholm determinants of an Airy kernel with several discontinuities. We establish simple Tracy–Widom type expressions for these Fredholm determinants, which involve solutions to systems of coupled Painlevé II equations, and we investigate the asymptotic behavior of these solutions.     

Packing circuits in matroids

The purpose of this paper is to characterize all matroids M that satisfy the following minimax relation: for any nonnegative integral weight function w defined on E(M),

On the annihilators and attached primes of top local cohomology modules

Let ${\mathfrak{a}}$ be an ideal of a commutative Noetherian ring R and M a finitely generated R-module. It is shown that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))= {\rm Ann}_R(M/T_R(\mathfrak{a}, M))}$ , where ${T_R(\mathfrak{a}, M)}$ is the largest submodule of M such that ${{\rm cd}(\mathfrak{a}, T_R(\mathfrak{a}, M)) < {\rm cd}(\mathfrak{a}, M)}$ . Several applications of this result are given. Among other things, it is shown that there exists an ideal ${\mathfrak{b}}$ of R such that ${{\rm Ann}_R(H_{\mathfrak{a}}^{{\rm dim} M}(M))={\rm Ann}_R(M/H_{\mathfrak{b}}^{0}(M))}$ . Using this, we show that if ${ H_{\mathfrak{a}}^{{\rm dim} R}(R)=0}$ , then ${{{\rm Att}_R} H^{{\rm dim} R-1}_{\mathfrak a}(R)= \{\mathfrak{p} \in {\rm Spec} R | \,{\rm cd}(\mathfrak{a}, R/\mathfrak{p}) = {\rm dim} R-1\}.}$ These generalize the main results of Bahmanpour et al. (see [2, Theorem 2.6]), Hellus (see [7, Theorem 2.3]), and Lynch (see [10, Theorem 2.4]).     

Integration of the lifting formulas and the cyclic homology of the algebras of differential operators

We integrate the Lifting cocycles Y_2n+1, Y_2n+3, Y_2n+5,? ([Sh1,2]) \Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,([\rm Sh1,2]) on the Lie algebra Dif_n of holomorphic differential operators on an n-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle l \lambda on an n-dimensional complex manifold M in the sense of Gelfand--Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin--Tsygan theorem [FT1]:¶¶ H^·_Lie(\frak g\frak l^fin_￥(Dif_n);\Bbb C) = ù^·(Y_2n+1, Y_2n+3, Y_2n+5,? ) H^\bullet_{\rm Lie}({\frak g}{\frak l}^{\rm fin}_\infty({\rm Dif}_n);{\Bbb C}) = \wedge^\bullet(\Psi_{2n+1}, \Psi_{2n+3}, \Psi_{2n+5},\ldots\,) .     

σk-Scalar curvature and eigenvalues of the Dirac operator

On a four-dimensional closed spin manifold (M ⁴, g), the eigenvalues of the Dirac operator can be estimated from below by the total σ₂-scalar curvature of M ⁴ as follows: Equality implies that (M ⁴, g) is a round sphere and the corresponding eigenspinors are Killing spinors.Dedicated to Professor Wang Guangyin on the occasion of his 80th birthday.     

Finiteness properties of minimax and coatomic local cohomology modules

Let R be a noetherian ring, \mathfraka{\mathfrak{a}} an ideal of R, and M an R-module. We prove that for a finite module M, if Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is minimax for all i ≥ r ≥ 1, then Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is coatomic for i ≤ r (M finite) then Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.     

Lp theory for the multidimensional aggregation equation

We consider well‐posedness of the aggregation equation ∂_tu + div(uv) = 0, v = −▿K * u with initial data in \input amssym ${\cal P}_2 {\rm (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ in dimensions 2 and higher. We consider radially symmetric kernels where the singularity at the origin is of order |x|^α, α > 2 − d, and prove local well‐posedness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ for sufficiently large p < p_s. In the special case of K(x) = |x|, the exponent p_s = d/(d = 1) is sharp for local well‐posedness in that solutions can instantaneously concentrate mass for initial data in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ with p < p_s. We also give an Osgood condition on the potential K(x) that guarantees global existence and uniqueness in \input amssym ${\cal P}_2 { (\Bbb R}^d {\rm )} \cap L^p ({\Bbb R}^d )$ . © 2010 Wiley Periodicals, Inc.     

Cohomological dimension,self-linking,and systolic geometry

Given a closed manifold M, we prove the upper bound of