Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

On polynomial congruences

We deal with functions which fulfil the condition \({\Delta_h^{n+1} \varphi(x)\in\mathbb{Z}}\) for all x, h taken from some linear space V. We derive necessary and sufficient conditions for such a function to be decent in the following sense: there exist functions \({f\colon V\rightarrow \mathbb{R},\ g\colon V \rightarrow \mathbb{Z}}\) such that \({\varphi = f + g}\) and \({\Delta_h^{n+1}f(x)=0}\) for all \({x, h\in V}\).     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

Heat kernel approach for sup-norm bounds for cusp forms of integral and half-integral weight

In this article, using the heat kernel approach from Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), we derive sup-norm bounds for cusp forms of integral and half-integral weight. Let \({\Gamma\subset \mathrm{PSL}_{2}(\mathbb{R})}\) be a cocompact Fuchsian subgroup of first kind. For \({k \in \frac{1}{2} \mathbb{Z}}\) (or \({k \in 2\mathbb{Z}}\)), let \({S^{k}_{\nu}(\Gamma)}\) denote the complex vector space of cusp forms of weight-k and nebentypus \({\nu^{2k}}\) (\({\nu^{k\slash 2}}\), if \({k \in 2\mathbb{Z}}\)) with respect to \({\Gamma}\), where \({\nu}\) is a unitary character. Let \({\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}\) denote an orthonormal basis of \({S^{k}_{\nu}(\Gamma)}\). In this article, we show that as \({k \rightarrow \infty,}\) the sup-norm for \({\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}}\) is bounded by O(k), where the implied constant is independent of \({\Gamma}\). Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when \({\Gamma}\) is cofinite.     

Nonarchimedean Green functions and dynamics on projective space

Let \({\varphi: \mathbb{P}^N_K\to\mathbb{P}^N_K}\) be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function \({\hat{g}_\varphi}\) associated to \({\varphi}\) is Hölder continuous on \({\mathbb{P}^N(K)}\) and that the Fatou set \({\mathcal{F}(\varphi)}\) of \({\varphi}\) is equal to the set of points at which \({\hat{g}_\Phi}\) is locally constant. Further, \({\hat{g}_\varphi}\) vanishes precisely on the set of points P such that \({\varphi}\) has good reduction at every point in the forward orbit \({\mathcal{O}_\varphi(P)}\) of P. We also prove that the iterates of \({\varphi}\) are locally uniformly Lipschitz on \({\mathcal{F}(\varphi)}\) .     

Efficient construction of 2-chains representing a basis of $H_{2}(\overline {\Omega }, \partial {\Omega }; \mathbb {Z})$

We present an efficient algorithm for the construction of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of \(\overline {\Omega }\) the idea is to find, first g different 1-boundaries of \(\overline {\Omega }\) with supports contained in ?Ω whose homology classes in \(\mathbb {R}^{3} \setminus {\Omega }\) form a basis of \(H_{1}(\mathbb {R}^{3} \setminus {\Omega };\mathbb {Z})\), and then to construct a set of 2-chains in \(\overline {\Omega }\) having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in \(\overline {\Omega }\) modulo ?Ω form a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\). We devise a simple procedure for the construction of the required set of 1-boundaries of \(\overline {\Omega }\) that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of \(H_{2}(\overline {\Omega },\partial {\Omega };\mathbb {Z})\) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms.     

On the additivity of block designs

We show that symmetric block designs \({\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})\) can be embedded in a suitable commutative group \({\mathfrak {G}}_{\mathcal {D}}\) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of \({\mathrm {PG}}(d,2)\) and \({\mathrm {AG}}(d,3)\). In both cases, the blocks can be characterized as the only k-subsets of \(\mathcal {P}\) whose elements sum to zero. It follows that the group of automorphisms of any such design \(\mathcal {D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal {D}\) that leave \(\mathcal {P}\) invariant. In some special cases, the group \({\mathfrak {G}}_\mathcal {D}\) can be determined uniquely by the parameters of \(\mathcal {D}\). For instance, if \(\mathcal {D}\) is a 2-\((v,k,\lambda )\) symmetric design of prime order p not dividing k, then \({\mathfrak {G}}_\mathcal {D}\) is (essentially) isomorphic to \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of \(\mathcal {B}\) can be characterized also as the v intersections of \(\mathcal {P}\) with v suitable hyperplanes of \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\).     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let \(\mathrm{SM}_{2n}(S^1,\mathbb {R})\) be a set of stable Morse functions of an oriented circle such that the number of singular points is \(2n\in \mathbb {N}\) and the order of singular values satisfies the particular condition. For an orthogonal projection \(\pi :\mathbb {R}^2\rightarrow \mathbb {R}\), let \({\tilde{f}}_0\) and \({\tilde{f}}_1:S^1\rightarrow \mathbb {R}^2\) be embedding lifts of f. If there is an ambient isotopy \(\tilde{\varphi }_t:\mathbb {R}^2\rightarrow \mathbb {R}^2\) \((t\in [0,1])\) such that \({\pi \circ \tilde{\varphi }}_t(y_1,y_2)=y_1\) and \(\tilde{\varphi }_1\circ {\tilde{f}}_0={\tilde{f}}_1\), we say that \({\tilde{f}}_0\) and \({\tilde{f}}_1\) are height isotopic. We define a function \(I:\mathrm{SM}_{2n}(S^1,\mathbb {R})\rightarrow \mathbb {N}\) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals \(2^{n-1}\).     

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 \({0 < p \leq\infty}\).     

Duals of Weighted Exponential Systems

The paper considers the basis and frame properties of the system of weighted exponentials \({\mathcal{E}}(g,\mathbb{Z}\backslash F) = \{e^{2\pi i n x} g(x)\}_{n\in\mathbb{Z}\backslash F}\) in \(L^{2}({\mathbb{T}})\), where \(g \in L^{2}({\mathbb{T}}) \backslash\{0\}\) and F??. It is shown that many of the frame properties of \({\mathcal {E}}(g,\mathbb{Z}\backslash F)\) are affected by the cardinalities of F and the behavior of the zeros of g.     

Fuglede–Kadison Determinants and Sofic Entropy

We relate Fuglede–Kadison determinants to entropy of finitely-presented algebraic actions in essentially complete generality. We show that if \({f\in M_{m,n}(\mathbb{Z}(\Gamma))}\) is injective as a left multiplication operator on \({\ell^{2}(\Gamma)^{\oplus n},}\) then the topological entropy of the action of \({\Gamma}\) on the dual of \({\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus m}f}\) is at most the logarithm of the positive Fuglede–Kadison determinant of f, with equality if m = n. We also prove that when m = n the measure-theoretic entropy of the action of \({\Gamma}\) on the dual of \({\mathbb{Z}(\Gamma)^{\oplus n}/\mathbb{Z}(\Gamma)^{\oplus n}f}\) is the logarithm of the Fuglede–Kadison determinant of f. This work completely settles the connection between entropy of principal algebraic actions and Fuglede–Kadison determinants in the generality in which dynamical entropy is defined. Our main Theorem partially generalizes results of Li-Thom from amenable groups to sofic groups. Moreover, we show that the obvious full generalization of the Li-Thom theorem for amenable groups is false for general sofic groups. Lastly, we undertake a study of when the Yuzvinskiǐ addition formula fails for a non-amenable sofic group \({\Gamma}\), showing it always fails if \({\Gamma}\) contains a nonabelian free group, and relating it to the possible values of L ²-torsion in general.     

Integer Decomposition Property of Free Sums of Convex Polytopes

Let \({\mathcal{P} \subset \mathbb{R}^{d}}\) and \({\mathcal{Q} \subset \mathbb{R}^{e}}\) be integral convex polytopes of dimension d and e which contain the origin of \({\mathbb{R}^{d}}\) and \({\mathbb{R}^{e}}\), respectively. We say that an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^{d}}\) possesses the integer decomposition property if, for each \({n\geq1}\) and for each \({\gamma \in n\mathcal{P}\cap\mathbb{Z}^{d}}\), there exist \({\gamma^{(1)}, . . . , \gamma^{(n)}}\) belonging to \({\mathcal{P}\cap\mathbb{Z}^{d}}\) such that \({\gamma = \gamma^{(1)} +. . .+\gamma^{(n)}}\). In the present paper, under some assumptions, the necessary and sufficient condition for the free sum of \({\mathcal{P}}\) and \({\mathcal{Q}}\) to possess the integer decomposition property will be presented.     

The fractional Neumann and Robin type boundary conditions for the regional fractional <Emphasis Type="Italic">p</Emphasis>-Laplacian

Let \({p \in (1,\infty)}\), \({s \in (0,1)}\) and \({\Omega \subset {\mathbb{R}^{N}}}\) a bounded open set with boundary \({\partial\Omega}\) of class C ^1,1. In the first part of the article we prove an integration by parts formula for the fractional p-Laplace operator \({(-\Delta)_{p}^{s}}\) defined on \({\Omega \subset {\mathbb{R}^{N}}}\) and acting on functions that do not necessarily vanish at the boundary \({\partial\Omega}\). In the second part of the article we use the above mentioned integration by parts formula to clarify the fractional Neumann and Robin boundary conditions associated with the fractional p-Laplacian on open sets.     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

On von Neumann regular elements in <Emphasis Type="Italic">f</Emphasis>-rings

Let B be an Archimedean reduced f-ring. A positive element \({\omega}\) in B is said to satisfy the property \({(\ast)}\) if for every f-ring A with identity e and every \({\ell}\)-group homomorphism \({\gamma : A \rightarrow B}\) with \({\gamma(e) = \omega}\), there exists a unique \({\ell}\)-ring homomorphism \({\rho: B \rightarrow B}\) such that \({\gamma = \omega \rho}\) and \({\rho(e)^{\perp \perp} = \omega^{\perp \perp}}\). Boulabiar and Hager proved that any (positive) von Neumann regular element in B satisfies the property \({(\ast)}\) and proved that the converse holds in the C(X)-case. In this regard, they asked about this converse in the general case. Our main purpose in this note is to prove, via a counter-example, that the converse in question fails in general. In addition, we shall take the opportunity to extend the direct result obtained by Boulabiar and Hager, and to get the C(X)-case we were talking about in an easier way.     

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 \({h(\Omega_{a,b})=1}\).     

Symbol <Emphasis Type="Italic">p</Emphasis>-algebras of prime degree and their <Emphasis Type="Italic">p</Emphasis>-central subspaces

We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is \({p+1}\). We do it by proving the following number theoretic fact: let \({\{s_1,\dots,s_{p+1}\}}\) be \({p+1}\) distinct nonzero elements in the additive group \({G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}\), then every nonzero element \({g \in G}\) can be expressed as \({d_1 s_1+\dots+d_{p+1} s_{p+1}}\) for some non-negative integers \({d_1,\dots,d_{p+1}}\) with \({d_1+\dots+d_{p+1}\leq p-1}\).