Diagram Groupoids and von Neumann Algebras

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism ?? on the graph groupoid ${\mathbb{G}}$ of a given graph G, we define the diagram of G by the image ${\delta(\mathbb{G})}$ of ??, equipped with the inherited binary operation on ${\mathbb{G}}$ . We study the fundamental properties of the diagram ${\delta(\mathbb{G})}$ , and compare them with those of ${\mathbb{G}}$ . Similar to Cho (Acta Appl Math 95:95?C134, 2007), we construct the groupoid von Neumann algebra ${\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}$ , generated by ${\delta(\mathbb{G})}$ , and consider the operator algebraic properties of ${\mathcal{M}_{G}}$ . In particular, we show ${\mathcal{M}_{G}}$ is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.     

Coarse non-amenability and covers with small eigenvalues

Given a closed Riemannian manifold M and a (virtual) epimorphism ${\pi_1(M)\twoheadrightarrow \mathbb{F}_2}$ of the fundamental group onto a free group of rank 2, we construct a tower of finite sheeted regular covers ${\left\{M_n\right\}_{n=0}^{\infty}}$ of M such that ?? ₁(M _n ) ?? 0 as n ?? ??. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where ?? ₁(M) is supposed to surject onto an amenable group.     

Some estimates of the normal approximation for ��-mixing random variables

Let ?? _n be a ??-mixing sequence of real random variables such that $ \mathbb{E}{\xi_n} = 0 $ , and let Y be a standard normal random variable. Write S _n = ?? ₁ + · · · + ?? _n and consider the normalized sums Z _n = S _n /B _n , where $ B_n^2 = \mathbb{E}S_n^2 $ . Assume that a thrice differentiable function $ h:\mathbb{R} \to \mathbb{R} $ satisfies $ {\sup_{x \in \mathbb{R}}}\left| {{h^s}(x)} \right| < \infty $ . We obtain upper bounds for $ {\Delta_n} = \left| {\mathbb{E}h\left( {{Z_n}} \right) - \mathbb{E}h(Y)} \right| $ in terms of Lyapunov fractions with explicit constants (see Theorem 1). In a particular case, the obtained upper bound of ?? _n is of order O(n ^?1/2). We note that the ??-mixing coefficients ??(r) are defined between the ??past?? and ??future.?? To prove the results, we apply the Bentkus approach.     

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

A partial orthomorphism of ${\mathbb{Z}_{n}}$ is an injective map ${\sigma : S \rightarrow \mathbb{Z}_{n}}$ such that ${S \subseteq \mathbb{Z}_{n}}$ and ??(i)?Ci ? ??(j)? j (mod n) for distinct ${i, j \in S}$ . We say ?? has deficit d if ${|S| = n - d}$ . Let ??(n, d) be the number of partial orthomorphisms of ${\mathbb{Z}_{n}}$ of deficit d. Let ??(n, d) be the number of partial orthomorphisms ?? of ${\mathbb{Z}_n}$ of deficit d such that ??(i) ? {0, i} for all ${i \in S}$ . Then ??(n, d) =???(n, d)n ²/d ² when ${1\,\leqslant\,d < n}$ . Let R _{k, n} be the number of reduced k ×?n Latin rectangles. We show that $$R_{k, n} \equiv \chi (p, n - p)\frac{(n - p)!(n - p - 1)!^{2}}{(n - k)!}R_{k-p,\,n-p}\,\,\,\,(\rm {mod}\,p)$$ when p is a prime and ${n\,\geqslant\,k\,\geqslant\,p + 1}$ . In particular, this enables us to calculate some previously unknown congruences for R _{n, n} . We also develop techniques for computing ??(n, d) exactly. We show that for each a there exists??? _a such that, on each congruence class modulo??? _a , ??(n, n-a) is determined by a polynomial of degree 2a in n. We give these polynomials for ${1\,\leqslant\,a\,\leqslant 6}$ , and find an asymptotic formula for ??(n, n-a) as n ?? ??, for arbitrary fixed a.     

An A∞-structure on the Cohomology Ring of the Symmetric Group S p with Coefficients in 𝔽 p

Let p be a prime. Let ?? _p S _p be the group algebra of the symmetric group over the finite field with p elements ?? _p . Let ?? _p be the trivial ?? _p S _p -module. We choose a projective resolution PRes?? _p of the module ?? _p and equip the Yoneda algebra \(\mathrm{Ext}^{\ast }_{\mathbb{F}_{p} S_{p}}\left( \mathbb{F}_{p}, \mathbb{F}_{p}\right)\) with an A_∞-structure such that \(\mathrm{Ext}^{\ast }_{\mathbb{F}_{p} S_{p}}\left( \mathbb{F}_{p}, \mathbb{F}_{p}\right)\) becomes a minimal model in the sense of Kadeishvili of the dg-algebra \(\mathrm{Hom}^{\ast }_{\mathbb{F}_{p} S_{p}}\left(PRes \mathbb{F}_{p}, PRes \mathbb{F}_{p}\right)\) .     

Noncommutative Multivariable Operator Theory

In this paper, we study noncommutative domains ${\mathbb{D}_f^\varphi(\mathcal{H}) \subset B(\mathcal{H})^n}$ generated by positive regular free holomorphic functions f and certain classes of n-tuples ${\varphi = (\varphi_1, \ldots, \varphi_n)}$ of formal power series in noncommutative indeterminates Z ₁, . . . , Z _n . Noncommutative Poisson transforms are employed to show that each abstract domain ${\mathbb{D}_f^\varphi}$ has a universal model consisting of multiplication operators (M _Z1, . . . , M _{Z n} ) acting on a Hilbert space of formal power series. We provide a Beurling type characterization of all joint invariant subspaces under M _Z1, . . . , M _{Z n} and show that all pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ are compressions of ${M_{Z_1} \otimes I, \ldots, M_{Z_n} \otimes I}$ to their coinvariant subspaces. We show that the eigenvectors of ${M_{Z_1}^*, \ldots, M_{Z_n}^*}$ are precisely the noncommutative Poisson kernels ${\Gamma_\lambda}$ associated with the elements ${\lambda}$ of the scalar domain ${\mathbb{D}_{f,<}^\varphi(\mathbb{C}) \subset \mathbb{C}^n}$ . These are used to solve the Nevanlinna-Pick interpolation problem for the noncommutative Hardy algebra ${H^\infty(\mathbb{D}_f^\varphi)}$ . We introduce the characteristic function of an n-tuple ${T=(T_1, \ldots , T_n) \in \mathbb{D}_f^\varphi(\mathcal{H})}$ , present a model for pure n-tuples of operators in the noncommutative domain ${\mathbb{D}_f^\varphi(\mathcal{H})}$ in terms of characteristic functions, and show that the characteristic function is a complete unitary invariant for pure n-tuples of operators in ${\mathbb{D}_f^\varphi(\mathcal{H})}$ .     

On twin and anti-twin words in the support of the free Lie algebra

Let ${\mathcal{L}}_{K}(A)$ be the free Lie algebra on a finite alphabet A over a commutative ring K with unity. For a word u in the free monoid A ^? let $\tilde{u}$ denote its reversal. Two words in A ^? are called twin (resp. anti-twin) if they appear with equal (resp. opposite) coefficients in each Lie polynomial. Let l denote the left-normed Lie bracketing and ?? be its adjoint map with respect to the canonical scalar product on the free associative algebra K??A??. Studying the kernel of ?? and using several techniques from combinatorics on words and the shuffle algebra , we show that, when K is of characteristic zero, two words u and v of common length n that lie in the support of ${\mathcal{L}}_{K}(A)$ ??i.e., they are neither powers a ⁿ of letters a??A with exponent n>1 nor palindromes of even length??are twin (resp. anti-twin) if and only if u=v or $u = \tilde{v}$ and n is odd (resp. $u =\tilde{v}$ and n is even).     

Surfaces of Rotation with Constant Extrinsic Curvature in a Conformally Flat 3-Space

In this work, we relate the extrinsic curvature of surfaces with respect to the Euclidean metric and any metrics that are conformal to the Euclidean metric. We introduce the space ${\mathbb{E}_3}$ ??the 3-dimensional real vector space equipped with a conformally flat metric that is a solution of the Einstein equation. We characterize the surfaces of rotation with constant extrinsic curvature in the space ${\mathbb{E}_3}$ . We obtain a one-parameter family of two-sheeted hyperboloids that are complete surfaces with zero extrinsic curvature in ${\mathbb{E}_3}$ . Moreover, we obtain a one-parameter family of cones and show that there exists another one-parameter family of complete surfaces of rotation with zero extrinsic curvature in ${\mathbb{E}_3}$ . Moreover, we show that there exist complete surfaces with constant negative extrinsic curvature in ${\mathbb{E}_3}$ . As an application we prove that there exist complete surfaces with Gaussian curvature K ?? ? ?? < 0, in contrast with Efimov??s Theorem for the Euclidean space, and Schlenker??s Theorem for the hyperbolic space.     

Orbit Spaces of Free Involutions on the Product of Two Projective Spaces

Let X be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on X and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration ${X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}}$ . We also give an application of our result to show that if X has the mod 2 cohomology algebra of the product of two real projective spaces (respectively, complex projective spaces), then there does not exist any ${\mathbb{Z}_2}$ -equivariant map from ${\mathbb{S}^k \to X}$ for k ≥ 2 (respectively, k ≥ 3), where ${\mathbb{S}^k}$ is equipped with the antipodal involution.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

Fixed points of automorphisms preserving the length of words in free solvable groups

Let ?? be an automorphism of prime order p of the free group F _n . Suppose ?? has no fixed points and preserves the length of words. By ?? :=??? ^(m) we denote the automorphism of the free solvable group ${F_{n}/F_n^{(m)} }$ induced by ??. We show that every fixed point of ?? has the form ${cc^{\sigma} \ldots c^{\sigma^{p-1}}}$ , where ${c\in F_n^{(m-1)}/F_n^{(m)}}$ . This is a generalization of some known results, including the Macedo??ska?CSolitar Theorem [10].     

On Extending {\mathcal{A}}-Modules Through the Coefficients

In classical linear algebra, extending the ring of scalars of a free module gives rise to a new free module containing an isomorphic copy of the former and satisfying a certain universal property. Also, given two free modules on the same ring of scalars and a morphism between them, enlarging the ring of scalars results in obtaining a new morphism having the nice property that it coincides with the initial map on the isomorphic copy of the initial free module in the new one. We investigate these problems in the category of free ${\mathcal{A}}$ -modules, where ${\mathcal{A}}$ is an ${\mathbb{R}}$ -algebra sheaf. Complexification of free ${\mathcal{A}}$ -modules, which is defined to be the process of obtaining new free ${\mathcal{A}}$ -modules by enlarging the ${\mathbb{R}}$ -algebra sheaf ${\mathcal{A}}$ to a ${\mathbb{C}}$ -algebra sheaf, denoted ${\mathcal{A}_\mathbb{C}}$ , is an important particular case (see Proposition 2.1, Proposition 3.1). Attention, on the one hand, is drawn on the sub- ${_{\mathbb{R}}\mathcal{A}}$ -sheaf of almost complex structures on the sheaf ${{_\mathbb{R}}\mathcal{A}^{2n}}$ , the underlying ${\mathbb{R}}$ -algebra sheaf of a ${\mathbb{C}}$ -algebra sheaf ${\mathcal{A}}$ , and on the other hand, on the complexification of the functor ${\mathcal{H}om_\mathcal {A}}$ , with ${\mathcal{A}}$ an ${\mathbb{R}}$ -algebra sheaf.     

Clifford parallelism: old and new definitions, and their use

Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space ${{\rm PG}(3,\mathbb{R})}$ whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein??s definition). Our new access to the topic ??Clifford parallelism?? is free of complexification and involves Klein??s correspondence ?? of line geometry together with a bijective map ?? from all regular spreads of ${{\rm PG}(3,\mathbb{R})}$ onto those lines of ${{\rm PG}(5,\mathbb{R})}$ having no common point with the Klein quadric; a regular parallelism P of ${{\rm PG}(3,\mathbb{R})}$ is Clifford, if the spreads of P are mapped by ?? onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with ?? is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of ${{\rm PG}(3,\mathbb{R})}$ are Clifford (D3 = dimensionality definition). Submission of (D2) to ??^?1 yields a complexification free definition of a Clifford parallelism which uses only elements of ${{\rm PG}(3,\mathbb{R})}$ : A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group ${Aut_e({\bf P}_{\bf C})}$ of all automorphic collineations and dualities of the Clifford parallelism P _C and show ${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$ .     

Uniformly Quasiregular Maps on the Compactified Heisenberg Group

We show the existence of a non-injective uniformly quasiregular mapping acting on the one-point compactification $\bar{ {\mathbb{H}}}^{1}={\mathbb{H}}^{1}\cup\{\infty\}$ of the Heisenberg group ?¹ equipped with a sub-Riemannian metric. The corresponding statement for arbitrary quasiregular mappings acting on sphere ${\mathbb{S}}^{n} $ was proven by Martin (Conform. Geom. Dyn. 1:24?C27, 1997). Moreover, we construct uniformly quasiregular mappings on $\bar{ {\mathbb{H}}}^{1}$ with large-dimensional branch sets. We prove that for any uniformly quasiregular map g on $\bar{ {\mathbb{H}}}^{1}$ there exists a measurable CR structure ?? which is equivariant under the semigroup ?? generated by g. This is equivalent to the existence of an equivariant horizontal conformal structure.     

On a problem concerning the Banach algebra generated by the maps n\mapsto\lambda^{\binom{n}{k}}

In Jabbari and Namioka (Milan J. Math. 78:503?C522, 2010), the authors characterized the spectrum M(W) of the Weyl algebra W, i.e. the norm closure of the algebra generated by the family of functions $\{n\mapsto x^{n^{k}}; x\in\mathbb{T}, k\in\mathbb{N}\}$ , ( $\mathbb{T}$ the unit circle), with a closed subgroup of $E(\mathbb{T})^{\mathbb{N}}$ where $E(\mathbb{T})$ denotes the family of the endomorphisms of the multiplicative group $\mathbb{T}$ . But the size of M(W) in $E(\mathbb{T})^{\mathbb{N}}$ as well as the induced group operation were left as a problem. In this paper, we will give a solution to this problem.     

On the Rabinowitz Floer homology of twisted cotangent bundles

Let (M, g) be a closed connected orientable Riemannian manifold of dimension n????2. Let ??:?=??? ₀?+??? ^* ?? denote a twisted symplectic form on T ^* M, where ${\sigma\in\Omega^{2}(M)}$ is a closed 2-form and ?? ₀ is the canonical symplectic structure ${dp\wedge dq}$ on T ^* M. Suppose that ?? is weakly exact and its pullback to the universal cover ${\widetilde{M}}$ admits a bounded primitive. Let ${H:T^{*}M\rightarrow\mathbb{R}}$ be a Hamiltonian of the form ${(q,p)\mapsto\frac{1}{2}\left|p\right|^{2}+U(q)}$ for ${U\in C^{\infty}(M,\mathbb{R})}$ . Let ?? _k :?=?H ^?1(k), and suppose that k?>?c(g, ??, U), where c(g, ??, U) denotes the Ma?é critical value. In this paper we compute the Rabinowitz Floer homology of such hypersurfaces. Under the stronger condition that k?>?c ₀(g, ??, U), where c ₀(g, ??, U) denotes the strict Ma?é critical value, Abbondandolo and Schwarz (J Topol Anal 1:307?C405, 2009) recently computed the Rabinowitz Floer homology of such hypersurfaces, by means of a short exact sequence of chain complexes involving the Rabinowitz Floer chain complex and the Morse (co)chain complex associated to the free time action functional. We extend their results to the weaker case k?>?c(g, ??, U), thus covering cases where ?? is not exact. As a consequence, we deduce that the hypersurface ?? _k is never (stably) displaceable for any k?>?c(g, ??, U). This removes the hypothesis of negative curvature in Cieliebak et?al. (Geom Topol 14:1765?C1870, 2010, Theorem 1.3) and thus answers a conjecture of Cieliebak, Frauenfelder and Paternain raised in Cieliebak et?al. (2010). Moreover, following Albers and Frauenfelder (2009; J Topol Anal 2:77?C98, 2010) we prove that for k?>?c(g, ??, U), any ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ has a leaf-wise intersection point in ?? _k , and that if in addition ${\dim\, H_{*}(\Lambda M;\mathbb{Z}_{2})=\infty}$ , dim M????2, and the metric g is chosen generically, then for a generic ${\psi\in\mbox{Ham}_{c}(T^{*}M,\omega)}$ there exist infinitely many such leaf-wise intersection points.     

Solutions with many mixed positive and negative interior spikes for a semilinear Neumann problem

We study the existence of sign-changing multiple interior spike solutions for the following Neumann problem $$\varepsilon^2\Delta v-v+f(v) = 0 \,\, {\rm in} \,\, \Omega, \quad \frac{\partial v}{\partial \nu} = 0 \,\, {\rm on} \,\, \partial \Omega,$$ where ?? is a smooth bounded domain of ${\mathbb {R}^N}$ , ?? is a small positive parameter, f is a superlinear, subcritical and odd nonlinearity. No symmetry on ?? is assumed. To our knowledge, only positive interior peak solutions have been obtained for this problem and it remains a question whether or not multiple interior peak solutions with mixed positive and negative peaks exist. In this paper we assume that ?? is a two-dimensional strictly convex domain and, provided that k is sufficiently large, we construct a (k?+?1)-peak solutions with k positive interior peaks aligned on a closed curve near ??? and 1 negative interior peak located in a more centered part of ??.     

On Bipartite Distance-Regular Graphs with Intersection Numbers c i = (q i ? 1)/(q ? 1)

Let q denote an integer at least two. Let ?? denote a bipartite distance-regular graph with diameter D ?? 3 and intersection numbers c _i = (q ⁱ ? 1)/(q ? 1), 1 ?? i ?? D. Let X denote the vertex set of ?? and let ${V = \mathbb{C}^X}$ denote the vector space over ${\mathbb{C}}$ consisting of column vectors whose coordinates are indexed by X and whose entries are in ${\mathbb{C}}$ . For ${z \in X}$ , let ${{\hat z}}$ denote the vector in V with a 1 in the z-coordinate and 0 in all other coordinates. Fix ${x, y \in X}$ such that ?(x, y) = 2, where ? denotes the path-length distance function. For 0 ?? i, j ?? D define ${w_{ij} = \sum {\hat z}}$ , where the sum is over all ${z \in X}$ such that ?(x, z) = i and ?(y, z) = j. We define W?=?span{w _ij | 0 ?? i, j ?? D}. In this paper we consider the space ${MW={\rm span} \{mw \mid m \in M, w \in W\}}$ , where M is the Bose?CMesner algebra of ??. We observe that MW is the minimal A-invariant subspace of V which contains W, where A is the adjacency matrix of ??. We give a basis for MW that is orthogonal with respect to the Hermitean dot product. We compute the square-norm of each basis vector. We compute the action of A on the basis. For the case in which ?? is the dual polar graph D _D (q) we show that the basis consists of the characteristic vectors of the orbits of the stabilizer of x and y in the automorphism group of ??.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).