Asymptotic unitary equivalence and classification of simple amenable <Emphasis Type="Italic">C</Emphasis><Superscript>∗</Superscript>-algebras

Let C and A be two unital separable amenable simple C ^?-algebras with tracial rank at most one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ? ₁,? ₂:C→A are two unital monomorphisms. We show that there is a continuous path of unitaries {u _t :t∈[0,∞)} of A such that

$\lim_{t\to\infty}u_t^*\varphi_1(c)u_t=\varphi_2(c)\quad\mbox{for all }c\in C$

if and only if [? ₁]=[? ₂] in \(KK(C,A),\varphi_{1}^{\ddag}=\varphi_{2}^{\ddag},(\varphi_{1})_{T}=(\varphi _{2})_{T}\) and a rotation related map \(\overline{R}_{\varphi_{1},\varphi_{2}}\) associated with ? ₁ and ? ₂ is zero.

Applying this result together with a result of W. Winter, we give a classification theorem for a class \({\mathcal{A}}\) of unital separable simple amenable C ^?-algebras which is strictly larger than the class of separable C ^?-algebras with tracial rank zero or one. Tensor products of two C ^?-algebras in \({\mathcal{A}}\) are again in \({\mathcal{A}}\). Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras for which the state space of K ₀ is the same as the tracial state space and also some unital simple ASH-algebras whose K ₀-group is ? and whose tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are \({\mathcal{Z}}\)-stable are isomorphic to ones with no dimension growth.     

Real Interpolation of Sobolev Spaces Associated to a Weight

We hereby study the interpolation property of Sobolev spaces of order 1 denoted by \(W^{1}_{p,V}\), arising from Schrödinger operators with positive potential. We show that for 1?≤?p ₁?p?p ₂?q ₀ with p?>?s ₀, \(W^{1}_{p,V}\) is a real interpolation space between \(W_{p_1,V}^{1}\) and \(W_{p_2,V}^{1}\) on some classes of manifolds and Lie groups. The constants s ₀, q ₀ depend on our hypotheses.     

On lie algebras of affine vector fields of ideal realizations of holomorphic linear connections

We study the properties of real realizations of holomorphic linear connections over associative commutative algebras \(\mathbb{A}\) _m with unity. The following statements are proved.If a holomorphic linear connection ? on M _n over \(\mathbb{A}\) _m (m ≥ 2) is torsion-free and R ≠ 0, then the dimension over ? of the Lie algebra of all affine vector fields of the space (M _mn ^? , ?^?) is no greater than (mn)² ? 2mn + 5, where m = dim_? \(\mathbb{A}\), \(n = dim_\mathbb{A} \) M _n , and ?^? is the real realization of the connection ?.Let ?^? =¹ ? ײ ? be the real realization of a holomorphic linear connection ? over the algebra of double numbers. If the Weyl tensor W = 0 and the components of the curvature tensor ¹ R ≠ 0, ² R ≠ 0, then the Lie algebra of infinitesimal affine transformations of the space (M _2n ^? , ?^?) is isomorphic to the direct sum of the Lie algebras of infinitesimal affine transformations of the spaces ( ^a M _n , ^a ?) (a = 1, 2).     

The Regularity Problems with Data in Hardy–Sobolev Spaces for Singular Schrödinger Equation in Lipschitz Domains

Let the nonnegative singular potential V belong to the reverse Hölder class \({\mathcal B}_n\) on \({\mathbb R}^n\), and let (n???1)/n?p?≤?2, we establish the solvability and derivative estimates for the solutions to the Neumann problem and the regularity problem of the Schrödinger equation ??Δu?+?Vu?=?0 in a connected Lipschitz domain Ω, with boundary data in the Hardy space \(H^p(\partial \Omega)\) and the modified Hardy–Sobolev space \(H_{1, V}^p(\partial \Omega)\) related to the potential V. To deal with the H ^p regularity problem, we construct a new characterization of the atomic decomposition for \(H_{1, V}^p(\partial \Omega)\) space. The invertibility of the boundary layer potentials on Hardy spaces and Hölder spaces are shown in this paper.     

Rigidity for Closed Totally Umbilical Hypersurfaces in Space Forms

In Perez (Thesis, 2011), Perez proved some L ² inequalities for closed convex hypersurfaces immersed in the Euclidean space ? ⁿ⁺¹, and more generally for closed hypersurfaces with non-negative Ricci curvature, immersed in an Einstein manifold. In this paper, we discuss the rigidity of these inequalities when the ambient manifold is ? ⁿ⁺¹, the hyperbolic space ? ⁿ⁺¹, or the closed hemisphere \(\mathbb{S}_{+}^{n+1}\) . We also obtain a generalization of Perez’s theorem to the hypersurfaces without the hypothesis of non-negative Ricci curvature.     

Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form \(k\left( {s,u} \right) = \sum {{a_n}} {n^{ - s - \overline u }}\), and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury–Arveson space H _d ² in d variables, where d can be any number in {1, 2,...,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H _d ² . Thus, a family of multiplier algebras of Dirichlet series is exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic as a reproducing kernel Hilbert space to H _d ² and when its multiplier algebra is isometrically isomorphic to Mult(H _d ² ).     

Approximate identities on some homogeneous Banach spaces

We study convergence of approximate identities on some complete semi-normed or normed spaces of locally L ^p functions where translations are isometries, namely Marcinkiewicz spaces \({\mathcal{M}^{p}}\) and Stepanoff spaces \({\mathcal{S}^p}\), 1 ≤ p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M ^p ) or even unbounded (\({M^{p}_{0}}\)). We construct a function f that belongs to these spaces and has the property that all approximate identities \({\phi_\varepsilon * f}\) converge to f pointwise but they never converge in norm.     

A Compactness Theorem of the Space of Free Boundary <Emphasis Type="Italic">f</Emphasis>-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

Let \((M^3,g,e^{-f}d\mu _M)\) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry–Émery Ricci curvature and the boundary \(\partial M\) is strictly f-mean convex. We prove that there exists a properly embedded smooth f-minimal surface \(\Sigma \) in M with free boundary \(\partial \Sigma \) on \(\partial M\). If we further assume that the boundary \(\partial M\) is strictly convex, then we prove that \(M^3\) is diffeomorphic to the 3-ball \(B^3\), and a compactness theorem for the space of properly embedded f-minimal surfaces with free boundary in such \((M^3,g,e^{-f}d\mu _M)\), when the topology of these f-minimal surfaces is fixed.     

On countably {\mu}-paracompact spaces

A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {F_i} of non-empty closed sets with \({\bigcap_{i=1}^{\infty} F_{i} = \emptyset}\) there exists a sequence {G_i} of open sets such that \({\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}\) and \({F_{i} \subset G_{i}}\) for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a \({\mu}\)-normal generalized topological space satisfying the analogue of A which is not even countably \({\mu}\)-metacompact. Then we study the relationships between countably \({\mu}\)-paracompactness, countably \({\mu}\)-metacompactness and the condition corresponding to condition A in generalized topological spaces.     

On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ?²ⁿ⁺¹ which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ?²ⁿ⁺¹.     

Submanifolds with parallel normalized mean curvature vector in a unit sphere

Let M ⁿ be an n-dimensional closed submanifold of a sphere with parallel normalized mean curvature vector. Denote by S and H the squared norm of the second fundamental form and the mean curvature of M ⁿ , respectively. Assume that the fundamental group \({\pi_{1}(M^{n})}\) of M ⁿ is infinite and \({S\, \leqslant\, S(H)=n+\frac{n^{3}H^{2}}{2(n-1)}-\frac{n(n-2)H}{2(n-1)}\sqrt{n^{2}H^{2}+4(n-1)}}\), then S is constant, S = S(H), and M ⁿ is isometric to a Clifford torus \({S^{1}(\sqrt{1-r^{2}})\times S^{n-1}(r)}\) with \({r^{2}\leqslant \frac{n-1}{n}}\).     

Complete spacelike submanifolds with parallel mean curvature vector in a semi-Euclidean space

Our aim in this article is to study the geometry of n-dimensional complete spacelike submanifolds immersed in a semi-Euclidean space \({\mathbb{R}^{n+p}_{q}}\) of index q, with \({1\leq q\leq p}\). Under suitable constraints on the Ricci curvature and on the second fundamental form, we establish sufficient conditions to a complete maximal spacelike submanifold of \({\mathbb{R}^{n+p}_{q}}\) be totally geodesic. Furthermore, we obtain a nonexistence result concerning complete spacelike submanifolds with nonzero parallel mean curvature vector in \({\mathbb{R}^{n+p}_{p}}\) and, as a consequence, we get a rigidity result for complete constant mean curvature spacelike hypersurfaces immersed in the Lorentz–Minkowski space \({\mathbb{R}^{n+1}_{1}}\).     

Generalized fractional integrals in <Emphasis Type="Italic">p</Emphasis>-adic morrey and Herz spaces

For Riesz potential I ^β (f) on p-adic linear space Q _p ⁿ and its modification \(\widetilde{I^\beta }(f)\) we give sufficient conditions of their boundedness from radialMorrey space to anotherMorrey or Campanato space. Also we study the boundedness of modified Riesz potential \(\widetilde{I^\beta }(f)\) from Herz space to special Campanato spaces.     

On a Class of Weakly Coupled Systems of Elliptic Operators with Unbounded Coefficients

We consider a class of weakly coupled systems of elliptic operators \({\mathcal{A}}\) with unbounded coefficients defined in \({\mathbb{R}^N}\). We prove that a semigroup (T(t))t ≥ 0 of bounded linear operators can be associated with \({\mathcal{A}}\), in a natural way, in the space of all bounded and continuous functions. We prove a compactness property of the semigroup as well as some uniform estimates on the derivatives of the function T(t)f, when f belongs to some spaces of Hölder continuous functions, which are the key tools to prove some optimal Schauder estimates for the solution to some nonhomogeneous elliptic equations and Cauchy problems associated with the operator \({\mathcal{A}}\). Under suitable additional conditions, we then prove that the restriction of the semigroup to the subspace of smooth and compactly supported functions extends by a strongly continuous semigroup to L ^p -spaces over \({\mathbb{R}^N}\), related to the Lebesgue measure, when \({p \in [1,\infty)}\). We also provide sufficient conditions for this semigroup to be analytic when \({p \in [1,\infty)}\). Finally, we prove some L ^p ?L ^q -estimates.     

Conjugations on 6-manifolds

Conjugation spaces are spaces with an involution such that the fixed point set of the involution has \({\mathbb{Z} _2}\)-cohomology ring isomorphic to the \({\mathbb{Z} _2}\)-cohomology of the space itself, with the difference that all degrees are divided by two (e.g. \({\mathbb{C} {\rm P}^n}\) with the complex conjugation has \({\mathbb{R} {\rm P}^n}\) as fixed point set). One also requires that a certain conjugation equation is fulfilled. We give a new characterisation of conjugation spaces and apply it to the following realization problem: given M, a closed orientable 3-manifold, does there exist a simply connected 6-manifold X and a conjugation on X with fixed point set M? We give an affirmative answer.     

Hyperplanes of $DW(5,{\mathbb{K}})$ with ${\mathbb{K}}$ a perfect field of characteristic 2

Let \({\mathbb{K}}\) be a perfect field of characteristic 2. In this paper, we classify all hyperplanes of the symplectic dual polar space \(DW(5,{\mathbb{K}})\) that arise from its Grassmann embedding. We show that the number of isomorphism classes of such hyperplanes is equal to 5+N, where N is the number of equivalence classes of the following equivalence relation R on the set \(\{\lambda\in {\mathbb{K}}\,|\,X^{2}+\lambda X+1\mbox{ isirreducible}\) \(\mbox{in }{\mathbb{K}}[X]\}\): (λ ₁,λ ₂)∈R whenever there exists an automorphism σ of \({\mathbb{K}}\) and an \(a\in {\mathbb{K}}\) such that (λ ₂ ^σ )^?1=λ ₁ ^?1 +a ²+a.     

The Vectorial Ribaucour Transformation for Submanifolds of Constant Sectional Curvature

We obtain a reduction of the vectorial Ribaucour transformation that preserves the class of submanifolds of constant sectional curvature of space forms, which we call the L-transformation. It allows to construct a family of such submanifolds starting with a given one and a vector-valued solution of a system of linear partial differential equations. We prove a decomposition theorem for the L-transformation, which is a far-reaching generalization of the classical permutability formula for the Ribaucour transformation of surfaces of constant curvature in Euclidean three space. As a consequence, we derive a Bianchi-cube theorem, which allows to produce, from k initial scalar L-transforms of a given submanifold of constant curvature, a whole k-dimensional cube all of whose remaining \(2^k-(k+1)\) vertices are submanifolds with the same constant sectional curvature given by explicit algebraic formulae. We also obtain further reductions, as well as corresponding decomposition and Bianchi-cube theorems, for the classes of n-dimensional flat Lagrangian submanifolds of \({\mathbb {C}}^n\) and n-dimensional Lagrangian submanifolds with constant curvature c of the complex projective space \({\mathbb {C}}{\mathbb {P}}^n(4c)\) or the complex hyperbolic space \({\mathbb {C}}{\mathbb {H}}^n(4c)\) of complex dimension n and constant holomorphic curvature 4c.