Localization and Locality for Resistance Forms

For a resistance form ${(X, \mathcal{D}(\varepsilon),\varepsilon)}For a resistance form (X, D(e),e){(X, \mathcal{D}(\varepsilon),\varepsilon)} and a point x₀ ? X{x_0 \in X} as boundary, on the space X₀:=X \{x₀}{X_0:=X {\setminus}\{x_0\}} we consider the Dirichlet space D_x0:={f ? D(e) | f(x₀)=0}{\mathcal{D}_{x_0}:=\{f\in\mathcal{D}(\varepsilon)\, |\, f(x_0)=0\}} and we develop a good potential theory. For any finely open subset D of X ₀ we consider a localized resistance form (D_{X0 \ D},e_D{\mathcal{D}_{X_0 {\setminus} D},\varepsilon_{D}}) where D_{X0 \ D}:={f ? D_x0 | f=0{\mathcal{D}_{X_0 {\setminus} D}:=\{f\in\mathcal{D}_{x_0}\, |\, f=0} on X₀ \ D}, e_D(f,g):=e(f,g){X_0 {\setminus} D\},\, \varepsilon_D(f,g):=\varepsilon(f,g)} for all f,g ? D_{X0 \ D}{f,g\in\mathcal{D}_{X_0 {\setminus} D}}. The main result is the equivalence between the local property of the resistance form and the sheaf property for the excessive elements on finely open sets.     

Averaging of directional derivatives in vertices of nonobtuse regular triangulations

For a shape-regular triangulation ${\mathcal{T}_h}For a shape-regular triangulation _h{\mathcal{T}_h} without obtuse angles of a bounded polygonal domain W ì ?²{\Omega\subset\Re^2} , let L_h{\mathcal L_h} be the space of continuous functions linear on the triangles from T_h{\mathcal{T}_h} and Π _h the interpolation operator from C([`(W)]){C(\overline\Omega)} to L<sub>h</sub>{\mathcal L_h} . This paper is devoted to the following classical problem: Find a second-order approximation of the derivative ?u/?z(a){\partial u/\partial z(a)} in a direction z of a function u ? C<sup>3</sup>([`(W)]){u\in C^3(\overline\Omega)} in a vertex a in the form of a linear combination of the constant directional derivatives ?P_h(u)/?z{\partial \Pi_h(u)/\partial z} on the triangles surrounding a. An effective procedure for such an approximation is presented, its error is proved to be of the size O(h ²), an operator W_h: L_h?L_h×L_h{\mbox{W}_h: \mathcal L_h\longrightarrow\mathcal L_h\times\mathcal L_h} relating a second-order approximation W _h [Π _h (u)] of ?u{\nabla u} to every u ? C³([`(W)]){u\in C^3(\overline\Omega)} is constructed and shown to be a so-called recovery operator. The accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically.     

On centralizers of reflexive algebras

Let ${\mathcal{L}}$ be a subspace lattice on a complex Banach space X and δ be a linear mapping from ${alg\mathcal{L}}$ into B(X) such that for every ${A \in alg\mathcal{L}, 2\delta(A^2)=\delta(A)A + A\delta(A)}$ or ${\delta(A^3) = A\delta(A)A}$ . We show that if one of the following holds (1) ${\vee\{L : L \in \mathcal{J}(\mathcal{L})\}=X}$ , (2) ${\wedge\{L_-: L \in \mathcal{J}(\mathcal{L})\}=(0)}$ and X is reflexive, then δ is a centralizer. We also show that if ${\mathcal{L}}$ is a CSL and δ is a linear mapping from ${alg\mathcal{L}}$ into itself, then δ is a centralizer.     

A Noncommutative Version of <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and Characterizations of Subdiagonal Algebras

Let M{\mathcal M} be a σ-finite von Neumann algebra and \mathfrak A{\mathfrak A} a maximal subdiagonal algebra of M{\mathcal M} with respect to a faithful normal conditional expectation F{\Phi} . Based on Haagerup’s noncommutative L ^p space L^p(M){L^p(\mathcal M)} associated with M{\mathcal M} , we give a noncommutative version of H ^p space relative to \mathfrak A{\mathfrak A} . If h ₀ is the image of a faithful normal state j{\varphi} in L¹(M){L^1(\mathcal M)} such that j°F = j{\varphi\circ \Phi=\varphi} , then it is shown that the closure of {\mathfrak Ah₀^\frac1p}{\{\mathfrak Ah_0^{\frac1p}\}} in L^p(M){L^p(\mathcal M)} for 1 ≤ p < ∞ is independent of the choice of the state preserving F{\Phi} . Moreover, several characterizations for a subalgebra of the von Neumann algebra M{\mathcal M} to be a maximal subdiagonal algebra are given.     

Operator-valued H ∞-calculus in Inter- and Extrapolation Spaces

We generalize a Hilbert space result by Auscher, McIntosh and Nahmod to arbitrary Banach spaces X and to not densely defined injective sectorial operators A. A convenient tool proves to be a certain universal extrapolation space associated with A. We characterize the real interpolation space ( X,D( A^a ) ?R( A^a ) )_q,p{\left( {X,\mathcal{D}{\left( {A^{\alpha } } \right)} \cap \mathcal{R}{\left( {A^{\alpha } } \right)}} \right)}_{{\theta ,p}} as

On the structure of convex sets with symmetries

Asymmetry of a compact convex body L ì Rⁿ{\mathcal L \subset {\bf R}^n} viewed from an interior point O{\mathcal O} can be measured by considering how far L{\mathcal L} is from its inscribed simplices that contain O{\mathcal O}. This leads to a measure of symmetry s(L, O){\sigma(\mathcal L, \mathcal O)}. The interior of L{\mathcal L} naturally splits into regular and singular sets, where the singular set consists of points O{\mathcal O} with largest possible s(L, O){\sigma(\mathcal L, \mathcal O)}. In general, to calculate the singular set of a compact convex body is difficult. In this paper we determine a large subset of the singular set in centrally symmetric compact convex bodies truncated by hyperplane cuts. As a function of the interior point O{\mathcal O}, s(L, .){\sigma(\mathcal L, .)} is concave on the regular set. It is natural to ask to what extent does concavity of s(L, .){\sigma(\mathcal L, .)} extend to the whole (interior) of L{\mathcal L}. It has been shown earlier that in dimension two, s(L, .){\sigma(\mathcal L, .)} is concave on L{\mathcal L}. In this paper, we show that in dimensions greater than two, for a centrally symmetric compact convex body L{\mathcal L}, s(L, .){\sigma(\mathcal L, .)} is a non-concave function provided that L{\mathcal L} has a codimension one simplicial intersection. This is the case, for example, for the n-dimensional cube, n ≥ 3. This non-concavity result relies on the fact that a centrally symmetric compact convex body has no regular points.     

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.     

On the Right (Left) Invertible Completions for Operator Matrices

Let ${\mathcal {H}_{1}}Let H₁{\mathcal {H}_{1}} and H₂{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H₁), B ? B(H₂){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H₂, H₁){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H₁, H₂){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained.     

On Representations of Integers in Thin Subgroups of {{\rm SL}_2({\mathbb {Z}})}

Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}Let G < SL(2, \mathbb Z){\Gamma < {\rm SL}(2, {\mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀, w₀ ? \mathbb Z²  \  {0}{v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}. We consider the set S{\mathcal {S}} of all integers occurring in áv₀g, w₀?{\langle v_{0}\gamma, w_{0}\rangle}, for g ? G{\gamma \in \Gamma} and the usual inner product on \mathbb R²{\mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{\mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set \mathfrak E(N){\mathfrak {E}(N)} of integers |n| < N which are locally admissible (n ? S   (mod  q)   for all   q 3 1){(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)} but fail to be globally represented, n ? S{n \notin \mathcal {S}}, has a power savings, |\mathfrak E(N)| << N^1-e₀{|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}} for some ${\varepsilon_{0} > 0}${\varepsilon_{0} > 0}, as N → ∞.     

Linear Maps Preserving Operators of Local Spectral Radius Zero

Let X be a complex Banach space and let B(X){\mathcal{B}(X)} be the space of all bounded linear operators on X. For x ? X{x \in X} and T ? B(X){T \in \mathcal{B}(X)}, let r_T(x) = limsup_{n ? ￥} || Tⁿx|| ^1/n{r_{T}(x) =\limsup_{n \rightarrow \infty} \| T^{n}x\| ^{1/n}} denote the local spectral radius of T at x. We prove that if j: B(X) ? B(X){\varphi : \mathcal{B}(X) \rightarrow \mathcal{B}(X)} is linear and surjective such that for every x ? X{x \in X} we have r _T (x) = 0 if and only if r_j(T)(x) = 0{r_{\varphi(T)}(x) = 0}, there exists then a nonzero complex number c such that j(T) = cT{\varphi(T) = cT} for all T ? B(X){T \in \mathcal{B}(X) }. We also prove that if Y is a complex Banach space and j:B(X) ? B(Y){\varphi :\mathcal{B}(X) \rightarrow \mathcal{B}(Y)} is linear and invertible for which there exists B ? B(Y, X){B \in \mathcal{B}(Y, X)} such that for y ? Y{y \in Y} we have r _T (By) = 0 if and only if r_{j( T)} (y)=0{ r_{\varphi ( T) }(y)=0}, then B is invertible and there exists a nonzero complex number c such that j(T) = cB^-1TB{\varphi(T) =cB^{-1}TB} for all T ? B(X){T \in \mathcal{B}(X)}.     

Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors

This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L²([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L²([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C¹([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and

Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces

The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space ${\Upsilon\colon\widehat X\to X}The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space U\colon[^(X)]? X{\Upsilon\colon\widehat X\to X} in which [^(Y)] o U^-1(Y){\widehat Y\equiv\Upsilon^{-1}(Y)} has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C ^∞ exhaustion function j{\varphi} on [^(X)]{\widehat X} which is strongly q-convex on [^(W)]=U^-1(W){\widehat\Omega=\Upsilon^{-1}(\Omega)} outside a uniform neighborhood of the q-dimensional compact irreducible components of [^(Y)]{\widehat Y}.     

Actions of symbolic dynamical systems on C*-algebras. II. Simplicity of C*-symbolic crossed products and some examples

A C*-symbolic dynamical system ${(\mathcal{A}, \rho, \Sigma)}A C*-symbolic dynamical system (A, r, S){(\mathcal{A}, \rho, \Sigma)} consists of a unital C*-algebra A{\mathcal{A}} and a finite family { r_a }_{a ? S}{\{ \rho_\alpha \}_{\alpha \in \Sigma}} of endomorphisms ρ _α of A{\mathcal{A}} indexed by symbols α of Σ satisfying some conditions. The endomorphisms r_a, a ? S{\rho_\alpha, \alpha \in \Sigma } yield both a subshift Λ and a C*-algebra of a Hilbert C*-bimodule. The obtained C*-algebra is regarded as a crossed product of A{\mathcal{A}} by the subshift Λ. We will study simplicity condition of these C*-algebras. Some examples such as irrational rotation Cuntz–Krieger algebras will be studied.     

Homomorphisms with respect to a function

Let X be a realcompact space and H:C(X)?\mathbbR{H:C(X)\rightarrow\mathbb{R}} be an identity and order preserving group homomorphism. It is shown that H is an evaluation at some point of X if and only if there is j ? C(\mathbbR){\varphi\in C(\mathbb{R})} with ${\varphi(r)>\varphi(0)}${\varphi(r)>\varphi(0)} for all r ? \mathbbR-{0}{r\in\mathbb{R}-\{0\}} for which H°j = j°H{H\circ\varphi=\varphi\circ H} . This extends (and unifies) classical results by Hewitt and Shirota.     

Singularly perturbed 1D Cahn–Hilliard equation revisited

We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors ${\{{\mathcal M}_\epsilon\}, \epsilon\geq 0}We consider a singular perturbation of the one-dimensional Cahn–Hilliard equation subject to periodic boundary conditions. We construct a family of exponential attractors {M_e}, e 3 0{\{{\mathcal M}_\epsilon\}, \epsilon\geq 0} being the perturbation parameter, such that the map e? M_e{\epsilon \mapsto {\mathcal M}_\epsilon} is H?lder continuous. Besides, the continuity at e = 0{\epsilon=0} is obtained with respect to a metric independent of e.{\epsilon.} Continuity properties of global attractors and inertial manifolds are also examined.     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

Derived semidistributive lattices

For L a finite lattice, let ${\mathbb {C}(L) \subseteq L^2}$ denote the set of pairs γ = (γ ₀, γ ₁) such that ${\gamma_0 \prec \gamma_1}$ and order it as followsγ ≤ δ iff γ ₀ ≤ δ ₀, ${\gamma_{1} \nleq \delta_0,}$ and γ ₁ ≤ δ ₁. Let ${\mathbb {C}(L, \gamma)}$ denote the connected component of γ in this poset. Our main result states that, for any ${\gamma, \mathbb {C}(L, \gamma)}$ is a semidistributive lattice if L is semidistributive, and that ${\mathbb {C}(L, \gamma)}$ is a bounded lattice if L is bounded. Let ${\mathcal{S}_{n}}$ be the Permutohedron on n letters and let ${\mathcal{T}_{n}}$ be the Associahedron on n + 1 letters. Explicit computations show that ${\mathbb {C}(\mathcal{S}_{n}, \alpha) = \mathcal{S}_{n-1}}$ and ${\mathbb {C}(\mathcal {T}_n, \alpha) = \mathcal {T}_{n-1}}$ , up to isomorphism, whenever α1 is an atom of ${\mathcal{S}_{n}}$ or ${\mathcal{T}_{n}}$ . These results are consequences of new characterizations of finite join-semidistributive and of finite lower bounded lattices: (i) a finite lattice is join-semidistributive if and only if the projection sending ${\gamma \in \mathbb {C}(L)}$ to ${\gamma_0 \in L}$ creates pullbacks, (ii) a finite join-semidistributive lattice is lower bounded if and only if it has a strict facet labelling. Strict facet labellings, as defined here, are a generalization of the tools used by Caspard et al. to prove that lattices of finite Coxeter groups are bounded.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Balance in Stable Categories

We study when the stable category ${\mathcal A}/\langle{\mathcal T}\rangleWe study when the stable category A/áT?{\mathcal A}/\langle{\mathcal T}\rangle of an abelian category A{\mathcal A} modulo a full additive subcategory T{\mathcal T} is balanced and, in case T{\mathcal T} is functorially finite in A{\mathcal A}, we study a weak version of balance for A/áT?{\mathcal A}/\langle{\mathcal T}\rangle. Precise necessary and sufficient conditions are given in case T{\mathcal T} is either a Serre class or a class consisting of projective objects. The results in this second case apply very neatly to (generalizations of) hereditary abelian categories.