Nonlinear $$\ast$$-Lie-type derivations on von Neumann algebras

Let \({\mathcal{H}}\) be a complex Hilbert space, \({\mathcal{B(H)}}\) be the algebra of all bounded linear operators on \({\mathcal{H}}\) and \({\mathcal{A} \subseteq \mathcal{B(H)}}\) be a von Neumann algebra without nonzero central abelian projections. Let \({p_n(x_1,x_2 ,\ldots ,x_n)}\) be the commutator polynomial defined by n indeterminates \({x_1, \ldots , x_n}\) and their skew Lie products. It is shown that a mapping \({\delta \colon \mathcal{A} \longrightarrow \mathcal{B(H)}}\) satisfies

$$\delta(p_n(A_1, A_2 ,\ldots , A_n))=\sum_{k=1}^np_n(A_1 ,\ldots , A_{k-1}, \delta(A_k), A_{k+1} ,\ldots , A_n)$$

for all \({A_1, A_2 ,\ldots , A_n \in \mathcal{A}}\) if and only if \({\delta}\) is an additive *-derivation. This gives a positive answer to Conjecture 4.2 of [14].

     

Continued fraction normality is not preserved along arithmetic progressions

It is well known that if \({0.a_1a_2a_3\ldots}\) is the base-\({b}\) expansion of a number normal to base-\({b}\), then the numbers \({0.a_ka_{m+k}a_{2m+k}\ldots}\) for \({m\ge 2}\), \({k\ge 1}\) are all normal to base-\({b}\) as well. In contrast, given a continued fraction expansion \({\langle a_1,a_2,a_3,\ldots\rangle}\) that is normal (now with respect to the continued fraction expansion), we show that for any integers \({m\ge 2}\), \({k\ge 1}\), the continued fraction \({\langle a_k, a_{m+k},a_{2m+k},a_{3m+k},\ldots\rangle}\) will never be normal.     

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

We prove a Beurling-Blecher-Labuschagne theorem for \({H^\infty}\)-invariant spaces of \({L^p(\mathcal{M},\tau)}\) when \({0 < p \leq\infty}\), using Arveson’s non-commutative Hardy space \({H^\infty}\) in relation to a von Neumann algebra \({\mathcal{M}}\) with a semifinite, faithful, normal tracial weight \({\tau}\). Using the main result, we are able to completely characterize all \({H^\infty}\)-invariant subspaces of \({L^p(\mathcal{M} \rtimes_\alpha \mathbb{Z},\tau)}\), where \({\mathcal{M} \rtimes_\alpha \mathbb{Z} }\) is a crossed product of a semifinite von Neumann algebra \({\mathcal{M}}\) by the integer group \({\mathbb{Z}}\), and \({H^\infty}\) is a non-selfadjoint crossed product of \({\mathcal{M}}\) by \({\mathbb{Z}^+}\). As an example, we characterize all \({H^\infty}\)-invariant subspaces of the Schatten p-class \({S^p(\mathcal{H})}\), where \({H^\infty}\) is the lower triangular subalgebra of \({B(\mathcal{H})}\), for each \({0 < p \leq\infty}\).     

Mutually prime sequences of inner functions and the ranks of subspaces over the bidisk

Let \({\{\varphi_n(z)\}_{n\ge0}}\) be a sequence of inner functions satisfying that \({\zeta_n(z):=\varphi_n(z)/\varphi_{n+1}(z)\in H^\infty(z)}\) for every n ≥ 0 and \({\{\varphi_n(z)\}_{n\ge0}}\) have no nonconstant common inner divisors. Associated with it, we have a Rudin type invariant subspace \({\mathcal{M}}\) of \({H^2(\mathbb{D}^2)}\) . We write \({\mathcal{N}= H^2(\mathbb{D}^2)\ominus\mathcal{M}}\) . If \({\{\zeta_n(z)\}_{n\ge0}}\) ia a mutually prime sequence, then we shall prove that \({rank_{\{T^\ast_z,T^\ast_w\}} \mathcal{N}=1}\) and \({rank_{\{\mathcal{F}^\ast_z\}}(\mathcal{M}\ominus w\mathcal{M})=1}\) , where \({\mathcal{F}_z}\) is the fringe operator on \({\mathcal{M}\ominus w\mathcal{M}}\) .     

The heat semigroup on sectorial domains,highly singular initial values and applications

We show that the heat semigroup is well defined on the Banach space \({\mathcal{X}_{m,\gamma} = \{ \psi:\Omega_m \to \mathbb{R} ;\; |x|^{\gamma +2m}(\prod_{i=1}^m x_i)^{-1}\psi(x) \in L^\infty(\Omega_m)\},}\) \({0 < \gamma < N}\), where \({\Omega_m=\{(x_1,\, x_2,\, \ldots,\, x_N) \in \mathbb{R}^{N};\; x_i > 0,\, 1\leq i\leq m\},}\) \({1\leq m\leq N}\). We then investigate the large time behavior of solutions of the heat equation \({u_{t}-\Delta u=0}\) for t > 0 and \({x \in \Omega_m.}\) Using certain notions from dynamical systems, we show that the large time behavior is related to the spatial asymptotic behavior of its initial value. Since the space \({\mathcal{X}_{m, \gamma}}\) contains highly singular initial data, which can be extended to all of \({\mathbb{R}^{N}}\) by antisymmetry, we also obtain new results on the complexity in the asymptotic behavior of solutions for the heat equation on the whole space.     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Semantical conditions for the definability of functions and relations

Let \({\mathcal{L}\subseteq \mathcal{L}^\prime}\) be first order languages, let \({R \in \mathcal{L}^\prime- \mathcal{L}}\) be a relation symbol, and let \({\mathcal{K}}\) be a class of \({\mathcal{L}^\prime}\)-structures. In this paper, we present semantical conditions equivalent to the existence of an \({\mathcal{L}}\)-formula \({\varphi(\vec{x})}\) such that \({\mathcal{K}\vDash \varphi(\vec{x}) \leftrightarrow R(\vec{x})}\), where \({\varphi}\) has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential Horn, etc.). For each of these definability results for relations, we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley’s theorem for quasiprimal algebras, and the Baker–Pixley Theorem for finite algebras with a majority term.     

Virtual Bound Levels in a Gap of the Essential Spectrum of the Weakly Perturbed Periodic Schrödinger Operator

In the space \({L_{2}(\mathbf{R}^{d}) (d \le 3)}\) we consider the Schrödinger operator \({H_{\gamma}=-{\Delta}+ V(\mathbf{x})\cdot+\gamma W(\mathbf{x})\cdot}\), where \({V(\mathbf{x})=V(x_{1}, x_{2}, \dots, x_{d})}\) is a periodic function with respect to all the variables, \({\gamma}\) is a small real coupling constant and the perturbation \({W(\mathbf{x})}\) tends to zero sufficiently fast as \({|\mathbf{x}|\rightarrow\infty}\). We study so called virtual bound levels of the operator \({H_\gamma}\), i.e., those eigenvalues of \({H_\gamma}\) which are born at the moment \({\gamma=0}\) in a gap \({(\lambda_-,\,\lambda_+)}\) of the spectrum of the unperturbed operator \({H_0=-\Delta+ V(\mathbf{x})\cdot}\) from an edge of this gap while \({\gamma}\) increases or decreases. We assume that the dispersion function of H₀, branching from an edge of \({(\lambda_-,\lambda_+)}\), is non-degenerate in the Morse sense at its extremal set. For a definite perturbation \({(W(\mathbf{x})\ge 0)}\) we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as \({\gamma\rightarrow 0}\). For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap \({(\lambda_-,\,\lambda_+)}\) are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., \({\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset}\) for a small enough \({|\gamma|}\).     

<Emphasis Type="Italic">k</Emphasis>-Extreme Points in Symmetric Spaces of Measurable Operators

Let \({\mathcal{M}}\) be a semifinite von Neumann algebra with a faithful, normal, semifinite trace \({\tau}\) and E be a strongly symmetric Banach function space on \({[0,\tau({\bf 1}))}\) . We show that an operator x in the unit sphere of \({E(\mathcal{M}, \tau)}\) is k-extreme, \({k \in {\mathbb{N}}}\) , whenever its singular value function \({\mu(x)}\) is k-extreme and one of the following conditions hold (i) \({\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}\) or (ii) \({n(x)\mathcal{M}n(x^*) = 0}\) and \({|x| \geq \mu(\infty, x)s(x)}\) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever \({\mathcal{M}}\) is non-atomic. The global k-rotundity property follows, that is if \({\mathcal{M}}\) is non-atomic then E is k-rotund if and only if \(E(\mathcal{M}, \tau)\) is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement \({\mu(f)}\) is k-extreme and \({|f| \geq \mu(\infty,f)}\) . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit \({\Omega(g),\,g \in L_1 + L_{\infty}}\) , or \({\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}\) , if and only if \({\mu(f) = \mu(g)}\) and \({|f| \geq \mu(\infty, f)}\) . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.     

{{C^m}} Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

We consider a broad class of linear Perron–Frobenius operators \({\Lambda:X \rightarrow X}\), where \({X}\) is a real Banach space of \({C^m}\) functions. We prove the existence of a strictly positive \({C^m}\) eigenvector \({v}\) with eigenvalue \({r=r(\Lambda) =}\) the spectral radius of \({\Lambda}\). We prove (see Theorem 6.5 in Sect. 6 of this paper) that \({r(\Lambda)}\) is an algebraically simple eigenvalue and that, if \({\sigma(\Lambda)}\) denotes the spectrum of the complexification of \({\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\}\subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}\), where \({r_* < r(\Lambda)}\). Furthermore, if \({u \in X}\) is any strictly positive function, \({(\frac 1r \Lambda)^k(u) \rightarrow s_u v}\) as \({k \rightarrow \infty}\), where \({s_u > 0}\) and convergence is in the norm topology on \({X}\). In applications to the computation of Hausdorff dimension, one is given a parametrized family \({\Lambda_s,s > s_*}\), of such operators and one wants to determine the (unique) value \({s_0}\) such that \({r(\Lambda_{s_0})=1}\). In another paper (Falk and Nussbaum in C\({^{\rm m}}\) Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector \({v_s}\) of \({\Lambda_s}\) can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for \({s_0}\).     

Generalized Joseph–Lundgren exponent and intersection properties for supercritical quasilinear elliptic equations

We study the solution \({u(r,\rho)}\) of the quasilinear elliptic problem

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'+|u|^{p-1}u=0, & 0 < r < \infty, \\u(0)=\rho > 0,\ u'(0)=0.\end{cases}$$

The usual Laplace, \({m}\)-Laplace, and \({k}\)-Hessian operators are included in the differential operator \({r^{-(\gamma-1)}(r^{\alpha}|u'|^{\beta-1}u')'}\). Under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and \({p}\), the equation has a singular positive solution \({u^*(r)}\) and the solution \({u(r,\rho)}\) is positive for \({r\ge 0}\). We study the intersection numbers between \({u(r,\rho)}\) and \({u^*(r)}\) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\). A generalized Joseph–Lundgren exponent \({p^*_{JL}}\) plays a crucial role. The main technique is a phase plane analysis. In particular, we use two changes of variables which transform the equation into two autonomous systems.

     

Conformally Covariant Differential Operators for the Diagonal Action of $$O(p,\,q)$$ on Real Quadrics

Let \(X=G/P\) be a real projective quadric, where \(G=O(p,\,q)\) and P is a parabolic subgroup of G. Let \((\pi _{\lambda ,\epsilon },\, \mathcal H_{\lambda ,\epsilon })_{ (\lambda ,\epsilon )\in {\mathbb {C}}\times \{\pm \}}\) be the family of (smooth) representations of G induced from the characters of P. For \((\lambda ,\, \epsilon ),\, (\mu ,\, \eta )\in {\mathbb {C}}\times \{\pm \},\) a differential operator \(\mathbf D_{(\mu ,\eta )}^\mathrm{reg}\) on \(X\times X,\) acting G-covariantly from \({\mathcal {H}}_{\lambda ,\epsilon } \otimes {\mathcal {H}}_{\mu , \eta }\) into \({\mathcal {H}}_{\lambda +1,-\epsilon } \otimes {\mathcal {H}}_{\mu +1, -\eta }\) is constructed.     

On the Convergence to Equilibrium of Unbounded Observables Under a Family of Intermittent Interval Maps

We consider a family \({\{T_{r}: [0, 1] \circlearrowleft \}_{r\in[0, 1]}}\) of Markov interval maps interpolating between the tent map \({T_{0}}\) and the Farey map \({T_{1}}\). Letting \({\mathcal{P}_{r}}\) denote the Perron–Frobenius operator of \({T_{r}}\), we show, for \({\beta \in [0, 1]}\) and \({\alpha \in (0, 1)}\), that the asymptotic behaviour of the iterates of \({\mathcal{P}_{r}}\) applied to observables with a singularity at \({\beta}\) of order \({\alpha}\) is dependent on the structure of the \({\omega}\)-limit set of \({\beta}\) with respect to \({T_{r}}\). The results presented here are some of the first to deal with convergence to equilibrium of observables with singularities.     

Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

A remark on the values of binary linear forms at prime arguments

Let λ₁, λ₂ be positive real numbers such that \({\frac{{\lambda_1}}{{\lambda_2}}}\) is irrational and algebraic. For any (C, c) well-spaced sequence \({\mathcal {V} = \{{v_i}\}_{i = 1}^\infty}\) and δ > 0 let \({E( {\mathcal {V},X,\delta})}\) denote the number of elements \({v \in \mathcal {V}, v \le X}\) for which the inequality

$| {\lambda_1 p_1 + \lambda_2 p_2 - v} | < X^{- \delta}$

is not solvable in primes p ₁, p ₂. In this paper it is proved that

$E( {\mathcal {V},X,\delta}) \ll X^{\frac{4}{5} + \delta + \varepsilon}$

for any \({\varepsilon > 0}\). This result constitutes an improvement upon that of Brüdern, Cook, and Perelli for the range \({\frac{2}{{15}} < \delta < \frac{1}{5}}\).

     

Behaviour at the non-positive integers of Dirichlet series associated to polynomials of several variables

In this paper, we relate the special values at a non-positive integer \({\underline{\mathbf{s}}=(s_{1},\ldots, s_{r})= -\underline{\mathbf{N}}= (-N_{1},\ldots, -N_{r})}\) obtained by meromorphic continuation of the multiple Dirichlet series \({{Z(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\sum_{\underline{m}\in {\mathbb{N}}^{*n}}{\frac{1}{\prod_{i=1}^{r}{P_{i}^{ s_{i}}(\underline{m})}}}}}\) to special values of the function \({Y(\underline{\mathbf{P}}, \underline{\mathbf{s}})=\int_{[1, +\infty[^{n}} {\prod_{i=1}^{r}{P_{i}^{- s_{i}}(\underline{\mathbf{x}})}\; d{\underline{\mathbf{x}}}}}\) where \({\underline{\mathbf{P}}=(P_{1},..., P_{r}),\; (r\geq 1)}\) are elliptic polynomials in “\({n}\) ” variables. We prove a simple relation between \({Z(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\) and \({Y(\underline{\mathbf{P}}_{\underline{\mathbf{a}}}, -\underline{\mathbf{N}})}\), such that for all \({\underline{\mathbf{a}} \in {\mathbb{R}}^{n}_{+}}\), we denote \({\underline{\mathbf{P}}_{\underline{\mathbf{a}}}:=(P_{1 \underline{\mathbf{a}}},\ldots, P_{r \underline{\mathbf{a}}})}\), where \({P_{i\;\underline{\mathbf{a}}}(\underline{\mathbf{x}}):= P_i(\underline{\mathbf{x}}+ \underline{\mathbf{a}})\; (1\leq i\leq r)}\) is the shifted polynomial.     

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let \(\mathcal S\) be an abelian group of automorphisms of a probability space \((X, {\mathcal A}, \mu )\) with a finite system of generators \((A_1, \ldots , A_d).\) Let \(A^{{\underline{\ell }}}\) denote \(A_1^{\ell _1} \ldots A_d^{\ell _d}\), for \({{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).\) If \((Z_k)\) is a random walk on \({\mathbb {Z}}^d\), one can study the asymptotic distribution of the sums \(\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}\) and \(\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f\), for a function f on X. In particular, given a random walk on commuting matrices in \(SL(\rho , {\mathbb {Z}})\) or in \({\mathcal M}^*(\rho , {\mathbb {Z}})\) acting on the torus \({\mathbb {T}}^\rho \), \(\rho \ge 1\), what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on \({\mathbb {T}}^\rho \) after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), \(\mathcal S\) a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.