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1.
The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: Chu and Lau replaced the algebra
L
∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L
∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L
∞(G) by to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action
of σ on VN(G) can be extended to . We study the corresponding space of “σ-harmonic operators”, i.e., fixed points in under the action of σ. We show, under mild conditions on either σ or G, that is in fact a von Neumann subalgebra of . Our investigation of relies, in particular, on a notion of support for an arbitrary operator in that extends Eymard’s definition for elements of VN(G). Finally, we present an approach to via ideals in , where denotes the trace class operators on L
2(G), but equipped with a product different from composition, as it was pioneered for harmonic functions by Willis.
M. Neufang was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach.
V. Runde was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach. 相似文献
2.
Let G be an additive subgroup of a normed space X. We say that a point is weakly separated (resp. -separated) from G if it can be separated from G by a continuous character (resp. by a continuous positive definite function). Let T : X → Y be a continuous linear operator. Consider the following conditions:
(ws) if , then x is weakly separated from G;
(ps) if , then x is -separated from G;
(wp) if Tx is -separated from T(G), then x is weakly separated from G.
By (resp. , ) we denote the class of operators T : X → Y which satisfy (ws) (resp. (ps), (wp)) for all and all subgroups G of X. The paper is an attempt to describe the above classes of operators for various Banach spaces X, Y. It is proved that if X, Y are Hilbert spaces, then is the class of Hilbert-Schmidt operators. It is also shown that if T is a Hilbert-to-Banach space operator with finite ℓ-norm, then .
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3.
Javier Pérez Alvarez 《Mathematische Zeitschrift》2009,262(1):17-26
We shall call quantum states of a principal bundle π : P → M with structure group a semi-simple Lie group G, the elements of certain space of sections of the adjoint bundle , associated to the G-bundle of connections . An inner product of sections of is defined for which is a Hilbert space such that the Gauge group gau(P) of the given bundle represents in a family of self-adjoint operators. This work crystallizes some heuristic considerations,
on the unitary representations of Gauge algebras, of Garcia in the already a classical article (J. Differ. Geom. 12, 209–227, 1977). 相似文献
4.
Let be a scroll over a smooth curve C and let denote the hyperplane bundle. The special geometry of X implies that certain sheaves related to the principal part bundles of are locally free. The inflectional loci of X can be expressed in terms of these sheaves, leading to explicit formulas for the cohomology classes of the loci. The formulas
imply that the only uninflected scrolls are the balanced rational normal scrolls.
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5.
Julia Weber 《Mathematische Zeitschrift》2007,256(1):57-74
We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented
cobordism ring of a topological space X. This homology theory Eh
* has coefficients in every nonnegative dimension. There exists a natural transformation that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds,
which are singular objects defined below. An isomorphism of graded -modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh
*, generalizing the equivariant Euler characteristic. 相似文献
6.
7.
The theory of principal G-bundles over a Lie groupoid is an important one unifying various types of principal G-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal G-bundles. In this paper, we study differential geometry of these objects, including connections and holonomy maps. We also
introduce a Chern–Weil map for these principal bundles and prove that the characteristic classes obtained coincide with the
universal characteristic classes. As an application, we recover the equivariant Chern–Weil map of Bott–Tu. We also obtain
an explicit chain map between the Weil model and the simplicial model of equivariant cohomology which reduces to the Bott–Shulman
map when the manifold is a point.
P. Xu Research partially supported by NSF grant DMS-03-06665. 相似文献
8.
Ali Ben Amor 《Mathematische Zeitschrift》2007,255(3):627-647
Let
be a regular Dirichlet form on L
2(X,m), μ a positive Radon measure charging no sets of zero capacity and Φ an N-function. We prove that the Sobolev-Orlicz inequality(SOI)
for every
is equivalent to a capacitary-type inequality. Further we show that if
is continuously embedded into L
2(X,μ), the latter one implies some integrability condition, which is nothing else but the classical uniform integrability condition
if μ is finite. We also prove that a SOI for
yields a Nash-type inequality and if further μ = m and Φ is admissible, it yields the ultracontractivity of the corresponding semigroup. After, in the spirit of SOIs, we derive
criteria for
to be compactly embedded into L
2(μ), provided μ is finite. As an illustration of the theory, we shall relate the compactness of the latter embedding to the
discreteness of the spectrum of the time changed Dirichlet form and shall derive lower bounds for its eigenvalues in term
of Φ.
This work has been supported by the Deutsche Forschungsgemeinschaft. 相似文献
9.
James Gillespie 《Mathematische Zeitschrift》2007,257(4):811-843
We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact
and semi-separated scheme X. The approach generalizes and simplifies the method used by the author in (Trans Am Math Soc 356(8) 3369–3390, 2004) and
(Trans Am Math Soc 358(7), 2855–2874, 2006) to build monoidal model structures on the category of chain complexes of modules
over a ring and chain complexes of sheaves over a ringed space. Indeed, much of the paper is dedicated to showing that in
any Grothendieck category , any nice enough class of objects induces a model structure on the category Ch() of chain complexes. The main technical requirement on is the existence of a regular cardinal κ such that every object satisfies the following property: Each κ-generated subobject of F is contained in another κ-generated subobject S for which . Such a class is called a Kaplansky class. Kaplansky classes first appeared in Enochs and López-Ramos (Rend Sem Mat Univ Padova 107, 67–79,
2002) in the context of modules over a ring R. We study in detail the connection between Kaplansky classes and model categories. We also find simple conditions to put
on which will guarantee that our model structure is monoidal. We will see that in several categories the class of flat objects
form such Kaplansky classes, and hence induce monoidal model structures on the associated chain complex categories. We will
also see that in any Grothendieck category , the class of all objects is a Kaplansky class which induces the usual (non-monoidal) injective model structure on Ch(). 相似文献
10.
11.
Damiano Testa 《Mathematische Zeitschrift》2008,259(1):61-64
Let be a smooth complex complete intersection such that . Let f : S → X be a generically finite morphism from a smooth projective variety to X. Under some positivity assumption on the anticanonical divisor of S, if 2 ≤ dim S ≤ dim X − 2 we prove that the deformations of f are contained in a subvariety of codimension at least 2. 相似文献
12.
Amílcar Pacheco 《Mathematische Zeitschrift》2009,261(4):787-804
Let k be a field of characteristic q, a smooth geometrically connected curve defined over k with function field . Let A/K be a non-constant abelian variety defined over K of dimension d. We assume that q = 0 or > 2d + 1. Let p ≠ q be a prime number and a finite geometrically Galois and étale cover defined over k with function field . Let (τ′, B′) be the K′/k-trace of A/K. We give an upper bound for the -corank of the Selmer group Sel
p
(A ×
K
K′), defined in terms of the p-descent map. As a consequence, we get an upper bound for the -rank of the Lang–Néron group A(K′)/τ′B′(k). In the case of a geometric tower of curves whose Galois group is isomorphic to , we give sufficient conditions for the Lang–Néron group of A to be uniformly bounded along the tower.
This work was partially supported by CNPq research grant 305731/2006-8. 相似文献
13.
J. F. Jardine 《Mathematische Zeitschrift》2006,254(4):811-836
The usual notion of the site associated to a stack is expanded to a definition to a site
fibred over a presheaf of categories A on a site
. If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalant to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site
is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on
. In particular, stack cohomology can be calculated on the fibred site for any representing presheaf of groupoids within a fixed homotopy type. 相似文献
14.
J. Ruppenthal 《Mathematische Zeitschrift》2009,263(2):447-472
Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L
p
-sense on Y for 1 ≤ p ≤ ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.
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15.
Robert J. Taggart 《Mathematische Zeitschrift》2009,261(4):933-949
Suppose that {T
t
: t ≥ 0} is a symmetric diffusion semigroup on L
2(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem
and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence. 相似文献
16.
Jaehong Kim 《Mathematische Zeitschrift》2009,263(1):89-102
This paper is motivated by Grothendieck’s splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles.
We consider a compact complex manifold X and a holomorphic Banach bundle E → X that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that E splits into the sum of a finite rank bundle and a trivial bundle, provided . 相似文献
17.
For a smooth curve C it is known that a very ample line bundle on C is normally generated if Cliff() < Cliff(C) and there exist extremal line bundles (:non-normally generated very ample line bundle with Cliff() = Cliff(C)) with . However it has been unknown whether there exists an extremal line bundle with . In this paper, we prove that for any positive integers (g, c) with g = 2c + 5 and (mod 2) there exists a smooth curve of genus g and Clifford index c carrying an extremal line bundle with . In fact, a smooth quadric hypersurface section C of a general projective K3 surface always has an extremal line bundle with . More generally, if C has a line bundle computing the Clifford index c of C with , then C has such an extremal line bundle .
For all authors, this work was supported by Korea Research Foundation Grant funded by Korea Government (MOEHRD, Basic Reasearch
Promotion Fund)(KRF-2005-070-C00005). 相似文献
18.
Michele Bolognesi 《Mathematische Zeitschrift》2009,261(1):149-168
Let C be a genus 2 curve and the moduli space of semi-stable rank 2 vector bundles on C with trivial determinant. In Bolognesi (Adv Geom 7(1):113–144, 2007) we described the parameter space of non stable extension
classes of the canonical sheaf ω of C by ω−1. In this paper, we study the classifying rational map that sends an extension class to the corresponding rank two vector bundle. Moreover, we prove that, if we blow up along a certain cubic surface S and at the point p corresponding to the bundle , then the induced morphism defines a conic bundle that degenerates on the blow up (at p) of the Kummer surface naturally contained in . Furthermore we construct the -bundle that contains the conic bundle and we discuss the stability and deformations of one of its components. 相似文献
19.
Let X be a set and
the full transformation semigroup on X. Let ρ be an equivalence relation on X and
Then T(X,ρ) is a subsemigroup of
. In this note, we describe the equivalence relations ρ on X for which
in the semigroup T(X,ρ). 相似文献
20.
The subexponentiality of products revisited 总被引:1,自引:0,他引:1
Qihe Tang 《Extremes》2006,9(3-4):231-241
Following the work of Cline and Samorodnitsky (Stoch. Process. Their Appl. 49(1):75–98, 1994), we reexamine the subexponentiality of the product of two random variables, X and Y, which are independent and have distributions F and G, respectively. The main result is the following: If F belongs to the class [that is to say, F is subexponential and holds for some v>1] and G, with G(0–)=0 and G(0)<0, satisfies for each u>0, then the distribution of XY also belongs to the class .
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