Properties of subgenerators of regularized semigroups

We introduce two operations , in the set of subgenerators of a given - regularized semigroup and prove that is a complete partially ordered lattice with respect to , and the operator inclusion . Also presented are some other properties and examples for

     

A pointwise spectrum and representation of operators

For a self-adjoint operator commuting with an increasing family of projections we study the multifunction an open set of the topology containing , where is the spectrum of on . Let be the measure of maximal spectral type. We study the condition that is essentially a singleton, is not a singleton. We show that if is the density topology and if satisfies the density theorem, in particular if it is absolutely continuous with respect to the Lebesgue measure, then this condition is equivalent to the fact that is a Borel function of . If is the usual topology then the condition is equivalent to the fact that is approched in norm by step functions , where the set of intervals covers the set where is a singleton.

     

A twistor correspondence and Penrose transform for odd-dimensional hyperbolic space

For odd-dimensional hyperbolic space , we construct transforms between the cohomology of certain line bundles on (a twistor space for ) and eigenspaces of the Laplacian and of the Dirac operator on . The transforms are isomorphisms. As a corollary we obtain that every eigenfunction of or on extends as a holomorphic eigenfunction of the corresponding holomorphic operator on a certain region of the complexification of . We also obtain vanishing theorems for the cohomology of a class of line bundles on .

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

Automorphic-differential identities and actions of pointed coalgebras on rings

In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let be a ring, its left Martindale quotient ring and a right ideal of having no nonzero left annihilator. (1) Let be a pointed coalgebra which measures such that the group-like elements of act as automorphisms of . If is prime and for , then . Furthermore, if the action of extends to and if such that , then . (2) Let be an endomorphism of given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If is semiprime and , then .

     

On uniqueness of invariant means

The following results on uniqueness of invariant means are shown:

(i) Let be a connected almost simple algebraic group defined over . Assume that , the group of the real points in , is not compact. Let be a prime, and let be the compact -adic Lie group of the -points in . Then the normalized Haar measure on is the unique invariant mean on .

(ii) Let be a semisimple Lie group with finite centre and without compact factors, and let be a lattice in . Then integration against the -invariant probability measure on the homogeneous space is the unique -invariant mean on .

     

Subnormal Subgroups of Group Ring Units

Let be an arbitrary group. If satisfies , , then the units , generate a nonabelian free subgroup of units. As an application we show that if is contained in an almost subnormal subgroup of units in then either contains a nonabelian free subgroup or all finite subgroups of are normal. This was known before to be true for finite groups only.

     

The property and the reals

The property may be generalized by using filters on in a very natural way. We analyze the necessary requirements for a space to have property for a filter . We construct special filters for which has the property, in particular a P-point and a Q-point.

     

A bound for the nilpotency of a group of self homotopy equivalences

Let be the group of homotopy classes of self-homotopy equivalences of such that . We prove that is a nilpotent group and that .

     

Every complete doubling metric space carries a doubling measure

We prove that a complete metric space carries a doubling measure if and only if is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on and of the homogeneity exponents of are equal. We also show that a closed subset of carries a measure of homogeneity exponent . These results are based on the case of compact due to Volberg and Konyagin.

     

Elementary abelian 2-group actions on flag manifolds and applications

Let denote the unoriented cobordism ring. Let and let denote the equivariant cobordism ring of smooth manifolds with smooth -actions having finite stationary points. In this paper we show that the unoriented cobordism class of the (real) flag manifold is in the subalgebra generated by , where , and . We obtain sufficient conditions for indecomposability of an element in . We also obtain a sufficient condition for algebraic independence of any set of elements in . Using our criteria, we construct many indecomposable elements in the kernel of the forgetful map in dimensions , for , and show that they generate a polynomial subalgebra of .

     

Normalizers of nest algebras

For a nest with associated nest algebra , we define , the normalizer of . We develop a characterization of elements of based on certain order homomorphisms of into itself. This characterization enables us to prove several structure theorems.

     

A remark on Gelfand-Kirillov dimension

Let be a finitely generated non-PI Ore domain and the quotient division algebra of . If is the center of , then .

     

Stability of weakly almost conformal mappings

We prove a stability of weakly almost conformal mappings in for not too far below the dimension by studying the -quasiconvex hull of the set of conformal matrices. The study is based on coercivity estimates from the nonlinear Hodge decompositions and reverse Hölder inequalities from the Ekeland variational principle.

     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Inradius and integral means for green's functions and conformal mappings

Let be a convex planar domain of finite inradius . Fix the point and suppose the disk centered at and radius is contained in . Under these assumptions we prove that the symmetric decreasing rearrangement in of the Green's function , for fixed , is dominated by the corresponding quantity for the strip of width . From this, sharp integral mean inequalities for the Green's function and the conformal map from the disk to the domain follow. The proof is geometric, relying on comparison estimates for the hyperbolic metric of with that of the strip and a careful analysis of geodesics.

     

Duality and local group cohomology

Let be a group, let be a field, and let be a local system - an upwardly directed collection of subgroups whose union is . In this paper we give a short, elementary proof of the following result: If either is a --bimodule, or else is finite dimensional over its center, then . From this we deduce as easy corollaries some recent results of Meierfrankenfeld and Wehrfritz on the cohomology of a finitary module.

     

Derivations mapping into the socle, II

Let be a unital Banach algebra. Several characterizations of continuous derivations of having only elements with finite spectrum in their ranges are obtained. In particular, if is semisimple, then such derivations map into its socle.