-ideals of compact operators are separably determined

We prove that the space of compact operators on a Banach space is an -ideal in the space of bounded operators if and only if has the metric compact approximation property (MCAP), and is an -ideal in for all separable subspaces of having the MCAP. It follows that the Kalton-Werner theorem characterizing -ideals of compact operators on separable Banach spaces is also valid for non-separable spaces: for a Banach space is an -ideal in if and only if has the MCAP, contains no subspace isomorphic to and has property It also follows that is an -ideal in for all Banach spaces if and only if has the MCAP, and is an -ideal in .

     

groups and quasi-equivalence

A torsion-free abelian group is if every map from a pure subgroup of into lifts to an endomorphism of The class of groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous groups, those homogeneous groups such that, for pure in every has a lifting to a quasi-endomorphism of An irreducible group is if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A group is if and only if every endomorphism of is an integral multiple of an automorphism. A group has minimal test for quasi-equivalence ( if whenever and are quasi-isomorphic pure subgroups of then and are equivalent via a quasi-automorphism of For homogeneous groups, we show that in almost all cases the and properties coincide.

     

Layered circlepackings and the type problem

We study the geometric type of a surface packed with circles. For circles packed in concentric layers of uniform degree, the circlepacking is specified by this sequence of degrees. We write an infinite sum whose convergence discerns the geometric type: if layers of degree follow the th layer of degree , and the th layer of degree has circles, then converges/diverges as the circlepacking is hyperbolic/Euclidean. We illustrate a hyperbolic circlepacking with surprisingly few layers of degree .

     

Character degrees and local subgroups of -separable groups

Let be a finite -solvable group for different primes and . Let and be such that . We prove that every of -degree has -degree if and only if and .

     

A unified extension of two results of Ky Fan on the sum of matrices

Let be an Hermitian matrix with where are the ordered eigenvalues of . A result of Ky Fan (1949) asserts that if and are Hermitian matrices, then is majorized by . We extend the result in the framework of real semisimple Lie algebras in the following way. Let be a noncompact real semisimple Lie algebra with Cartan decomposition . We show that for any given , , where is the unique element corresponding to , in a fixed closed positive Weyl chamber of a maximal abelian subalgebra of in . Here the ordering is induced by the dual cone of . Fan's result corresponds to the Lie algebra . The compact case is also discussed. As applications, two unexpected singular values inequalities concerning the sum of two real matrices and the sum of two real skew symmetric matrices are obtained.

     

Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

Applications of pseudo-monotone operators with some kind of upper semicontinuity in generalized quasi-variational inequalities on non-compact sets

Let be a topological vector space and be a non-empty subset of . Let and be two maps. Then the generalized quasi-variational inequality (GQVI) problem is to find a point and a point such that for all . We shall use Chowdhury and Tan's 1996 generalized version of Ky Fan's minimax inequality as a tool to obtain some general theorems on solutions of the GQVI on a paracompact set in a Hausdorff locally convex space where the set-valued operator is either strongly pseudo-monotone or pseudo-monotone and is upper semicontinuous from to the weak-topology on for each non-empty finite subset of .

     

Singular extensions of the trace and the relative Dixmier property in the type factors

If is an inclusion of type factors with we study the connection between the existence of singular states on which extend the trace on and the Dixmier approximation property in with unitaries in We also prove the existence of singular conditional expectations from certain free product factors onto irreducible hyperfinite subfactors.

     

First order differential operators in white noise analysis

Let be the space of test white noise functionals. We first introduce a family of products on including Wiener and Wick products, and then show that with each product , we can associate a first order differential operator, called a first order -differential operator. We next show that a first order -differential operator is indeed a continuous derivation under the product . We finally characterize by means of rotation-invariance and continuous derivation under the product . Here and are the Gross Laplacian and the number operator on , respectively.

     

Hardy spaces of spaces of homogeneous type

Let be a space of homogeneous type, and be the generator of a semigroup with Gaussian kernel bounds on . We define the Hardy spaces of for a range of , by means of area integral function associated with the Poisson semigroup of , which is proved to coincide with the usual atomic Hardy spaces on spaces of homogeneous type.

     

Cycle rank of Lyapunov graphs and the genera of manifolds

We show that the cycle-rank of a Lyapunov graph on a manifold satisfies: , where is the genus of . This generalizes a theorem of Franks. We also show that given any integer with , for some Lyapunov graph on .

     

The intersection of three spheres in a sphere and a new application of the Sato-Levine invariant

Take transverse immersions such that (1) is an embedding, (2) and is connected, and (3) . Then we obtain three surface-links = (, ) in , where =(1,2,3), (2,3,1), (3,1,2). We prove that, we have the equality where is the Sato-Levine invariant of , if all are semi-boundary links.

     

On complementary subspaces of Hilbert space

Every pair of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form on a Hilbert space . Here is possibly , is a positive injective contraction and denotes the graph of . For such a pair the following are equivalent: (i) is similar to a pair in generic position; (ii) and have a common algebraic complement; (iii) is similar to for some operators on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.

     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.

     

On the relative commutants of subfactors

Let be factors generated by a periodic tower of finite dimensional -algebras. We prove that for sufficiently large , is -isomorphic to a subalgebra of .

     

On uniqueness of -adic meromorphic functions

Let be a complete ultrametric algebraically closed field of characteristic zero, and let be the field of meromorphic functions in . For all set in and for all we denote by the subset of : zero of order After studying unique range sets for entire functions in in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every , there exist sets of elements in such that, if have the same poles (counting multiplicities), and satisfy , then . We show how to construct such sets.

     

The Fuglede-Putnam theorem and a generalization of Barría's lemma

Let and be bounded linear operators, and let be a partial isometry on a Hilbert space. Suppose that (1) , (2) , (3) and (4) . Then we have .

     

Eigenvalue pinching theorems on compact symmetric spaces

We prove two first eigenvalue pinching theorems for Riemannian symmetric spaces (Theorems 1 and 2). As their application, we answer negatively a question raised by Elworthy and Rosenberg, who proposed to show that for every compact simple Lie group with a bi-invariant Riemannian metric on with respect to , being the Killing form of the Lie algebra , the first eigenvalue would satisfy

for all orthonormal bases of tangent spaces of (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that for a Lie group .

     

When do connected spaces have nice connected preimages?

We prove that every connected Tychonoff space is an open monotone continuous image of a connected strictly -discrete left-separated Tychonoff space. For wide classes of connected spaces it is established that they have a finer Hausdorff strictly -discrete connected topology. Another result is that a finer Tychonoff connected strictly -discrete topology exists for any Tychonoff topology with a countable network. We show that there are Tychonoff connected spaces with countable network which are not continuous images of connected second countable spaces. It is established also that every connected Tychonoff space is an open retract of a connected homogeneous Tychonoff space, while it is not always possible to find a finer connected homogeneous topology on .

     

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .