Shift invariant preduals of ℓ 1(ℤ)

The Banach space ? ₁(?) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ? ₁(?) weak*-continuous. This is equivalent to making the natural convolution multiplication on ? ₁(?) separately weak*-continuous and so turning ? ₁(?) into a dual Banach algebra. We call such preduals shift-invariant. It is known that the only shift-invariant predual arising from the standard duality between C ₀(K) (for countable locally compact K) and ? ₁(?) is c ₀(?). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak*-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c ₀. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ?. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c ₀.     

Uniqueness of preduals of certain Banach spaces

In [1], the authors have shown the existence of non-quasireflexive Banach spaces having unique isomorphic preduals. In fact, certain James-Lindenstrauss’ spaces have this property. In this paper it is shown that there are many such separable spaces. More precisely, there exist infinitely many different isomorphic types of James-Lindenstrauss’ spaces which are non-quasireflexive and have unique isomorphic preduals. The research of both authors was partially supported by N.S.F. Grant No. MPS75-07115     

Embedding of Semigroups of Lipschitz Maps into Positive Linear Semigroups on Ordered Banach Spaces Generated by Measures

Interpretation, derivation and application of a variation of constants formula for measure-valued functions motivate our investigation of properties of particular Banach spaces of Lipschitz functions on a metric space and semigroups defined on their (pre)duals. Spaces of measures densely embed into these preduals. The metric space embeds continuously in these preduals, even isometrically in a specific case. Under mild conditions, a semigroup of Lipschitz transformations on the metric space then embeds into a strongly continuous semigroups of positive linear operators on these Banach spaces generated by measures.      

A Nadel vanishing theorem via injectivity theorems

The purpose of this paper is to establish Nadel type vanishing theorems with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu’s metrics). For this purpose, we generalize Kollár’s injectivity theorem to an injectivity theorem for line bundles equipped with singular metrics, by making use of the theory of harmonic integrals. Moreover we give asymptotic cohomology vanishing theorems for high tensor powers of line bundles.     

The Hopf Structure of Some Dual Operator Algebras

We study the Hopf structure of a class of dual operator algebras corresponding to certain semigroups. This class of algebras arises in dilation theory, and includes the noncommutative analytic Toeplitz algebra and the multiplier algebra of the Drury–Arveson space, which correspond to the free semigroup and the free commutative semigroup respectively. The preduals of the algebras in this class naturally form Hopf (convolution) algebras. The original algebras and their preduals form (non-self-adjoint) dual Hopf algebras in the sense of Effros and Ruan. We study these algebras from this perspective, and obtain a number of results about their structure.     

Vanishing moment conditions for wavelet atoms in higher dimensions

We provide explicit criteria for wavelets to give rise to frames and atomic decompositions in L²(?^d), but also in more general Banach function spaces. We consider wavelet systems that arise by translating and dilating the mother wavelet, with the dilations taken from a suitable subgroup of GL(?^d), the so-called dilation group.The paper provides a unified approach that is applicable to a wide range of dilation groups, thus giving rise to new atomic decompositions for homogeneous Besov spaces in arbitrary dimensions, but also for other function spaces such as shearlet coorbit spaces. The atomic decomposition results are obtained by applying the coorbit theory developed by Feichtinger and Gröchenig, and they can be informally described as follows: Given a function ψ ∈ L²(?^d) satisfying fairly mild decay, smoothness and vanishing moment conditions, any sufficiently fine sampling of the translations and dilations will give rise to a wavelet frame. Furthermore, the containment of the analyzed signal in certain smoothness spaces (generalizing the homogeneous Besov spaces) can be decided by looking at the frame coefficients, and convergence of the frame expansion holds in the norms of these spaces. We motivate these results by discussing nonlinear approximation.     

Generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on domains

In this paper, we consider a non‐smooth atomic decomposition by using a smooth atomic decomposition. Applying the non‐smooth atomic decomposition, a local means characterization and a quarkonical decomposition, we obtain a pointwise multiplier and a trace operator for generalized Besov–Morrey spaces and generalized Triebel–Lizorkin–Morrey spaces on the whole space. We also develop the theory of those spaces on domains. We consider an extension operator and a trace operator on the upper half space and on compact oriented Riemannian manifolds.     

加权的Hardy空间和面积积分的加权模不等式

本文建立了具有高阶消失矩的加权的原子Ｈａｒｄｙ空间，证明了０＜Ｐ＜＝１时面积积分的双向加权模不等式。     

On the Choquet integrals associated to Bessel capacities

Czechoslovak Mathematical Journal - We characterize the Choquet integrals associated to Bessel capacities in terms of the preduals of the Sobolev multiplier spaces. We make use of the boundedness...     

On the Boundedness of Singular Integrals in Morrey Spaces and its Preduals

We reduce the boundedness of operators in Morrey spaces \(L_p^r\left( {\mathbb R}^n\right) \), its preduals, \(H^{\varrho }L_p ({\mathbb R}^n)\), and their preduals \(\overset{\circ }{L}{}^r_{p}\left( \mathbb {R}^n\right) \) to the boundedness of the appropriate operators in Lebesgue spaces, \(L_p({\mathbb R}^n)\). Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications in harmonic analysis.     

Derived categories of Keum's fake projective planes

We conjecture that derived categories of coherent sheaves on fake projective n  -spaces have a semi-orthogonal decomposition into a collection of n+1$n+1$ exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum's surface). Then by passing to equivariant categories we construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing.     

Asymptotic behavior of Markov semigroups on preduals of von Neumann algebras

We develop a new approach for investigation of asymptotic behavior of Markov semigroup on preduals of von Neumann algebras. With using of our technique we establish several results about mean ergodicity, statistical stability, and constrictiviness of Markov semigroups.     

A Class of Bilinear forms on Dirichlet type Spaces

This paper characterizes the boundedness of certain bilinearforms on Dirichlet type spaces. The main tool is the Littlewood-Paleytheory. Applications to the preduals of the related functionspaces including BMO and Bloch spaces, and the boundedness ofcertain commutators are also indicated.     

Fourier Algebras on Topological Foundation *-Semigroups.

We introduce the notion of the Fourier and Fourier-Stieltjes algebras of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.     

Near isometries in the class ofL 1-preduals

We construct preduals ofl ₁ which are nearly isometric without being isometric. We also show that ifX is nearly isometric to aC(K) space withK first countable, then they are in fact isometric.     

Atomic decompositions for tensor products and polynomial spaces

We study the existence of atomic decompositions for tensor products of Banach spaces and spaces of homogeneous polynomials. If a Banach space X admits an atomic decomposition of a certain kind, we show that the symmetrized tensor product of the elements of the atomic decomposition provides an atomic decomposition for the symmetric tensor product , for any symmetric tensor norm μ. In addition, the reciprocal statement is investigated and analogous consequences for the full tensor product are obtained. Finally we apply the previous results to establish the existence of monomial atomic decompositions for certain ideals of polynomials on X.     

Lipschitz continuity of the absolute value in preduals of semifinite factors

We prove a weak-type estimate for the absolute value mapping in the preduals of semifinite factors which extends an earlier result of Kosaki for the trace class.Research supported by the Australian Research Council (ARC) and the Netherlands Organization for Scientific Research (NWO)     

Some non-quasireflexive spaces having unique isomorphic preduals

It is shown that the dual spaces of certain James-Lindenstrauss spaces are spaces which are non-quasireflexive but have unique isomorphic preduals. The research of both authors was partially supported by N.S.F. Grant No. 20150.     

Singular and fractional integral operators on preduals of Campanato spaces with variable growth condition

We investigate the boundedness of singular and fractional integral operators on generalized Hardy spaces defined on spaces of homogeneous type, which are preduals of Campanato spaces with variable growth condition. To do this we introduce molecules with variable growth condition. Our results are new even for R~n case.     

区域上Besov空间的原子分解和限制定理

本文在区域Ω（Ω 　 Ｒｎ，ｎ≥１）上定义了某类在边界上消失的Ｂｅｓｏｖ空 间Ｂ~（ｓ，ｑ）_（ｐ，ｏ）（Ω）（ｓ∈Ｒ，０＜ｐ，ｑ≤∞），并给出了它的原子分解，然后证明了当区域Ω∈ Ｄε（ｎ）（０＜ε≤１，ｐｓ＜ε）时，得到了限制定理Ｂ~（ｓ，ｑ）_（ｐ，ｏ）（Ω）＝Ｂ~（ｓ，ｑ）_（ｐ）（Ω）．