Exchange Relation Planar Algebras

An inclusion of II ₁ factors NM of finite index has as an invariant, a double sequence of finite-dimensional algebras known as the standard invariant. Planar algebras were introduced by V. Jones as a geometric tool for computing standard invariants of existing subfactors as well as generating standard invariants for new subfactors. In this paper we define a class of planar algebras, termed exchange relation planar algebras, that provides a general framework for understanding several classes of known subfactor inclusions: the Fuss–Catalan algebras (i.e. those coming from the presence of intermediate subfactors) and all depth 2 subfactors. In addition, we present a new class of planar algebras (and thus a new class of subfactors) coming from automorphism subgroups of finite groups.     

Commutator inequalities associated with the polar decomposition

Let be a polar decomposition of an complex matrix . Then for every unitarily invariant norm , it is shown that

where denotes the operator norm. This is a quantitative version of the well-known result that is normal if and only if . Related inequalities involving self-commutators are also obtained.

     

On translations of subsets of the real line

In this paper we discuss various questions connected with translations of subsets of the real line. Most of these questions originate from W. Sierpinski. We discuss the number of translations a single subset of the reals may have. Later we discuss almost invariant subsets of Abelian groups.

     

Finite SAGBI bases for polynomial invariants of conjugates of alternating groups

It is well-known, that the ring of polynomial invariants of the alternating group has no finite SAGBI basis with respect to the lexicographical order for any number of variables . This note proves the existence of a nonsingular matrix such that the ring of polynomial invariants , where denotes the conjugate of with respect to , has a finite SAGBI basis for any .     

On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group of homeomorphisms of the circle possesses a free subgroup with two generators or there is an invariant probabilistic measure on S ¹ . In the present paper, we prove the following strengthening of Margulis's statement: an invariant probabilistic measure for a group exists if and only if the quotient group does not contain a free subgroup with two generators (here is some specific subgroup of G defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups of homeomorphisms of the line.     

The Eta Invariant and the Connective K-Theory of the Classifying Space for Cyclic 2 Groups

We use the eta invariant to study the connective K-theory groups ko _m (B ) of the classifying space for the cyclic group where - 2 2.     

Integral representation formula for generalized normal derivations

For generalized normal derivations, acting on the space of all bounded Hilbert space operators, the following integral representation formulas hold:

 

and

 

whenever is a Hilbert-Schmidt class operator and is a Lipschitz class function on Applying this formula, we extend the Fuglede-Putnam theorem concerning commutativity modulo Hilbert-Schmidt class, as well as trace inequalities for covariance matrices of Muir and Wong. Some new monotone matrix functions and norm inequalities are also derived.

     

逆极限的不变测度和一致正熵性质

In this paper, the interconnection of some ergodic properties between a continuous selfmap and its inverse limit is studied. It has been proved that (1) their invariant Borel probability measures are identical up to homeomorphism and (2) they preserve uniform positive entropy property simuitaneously. As applications, it is also proved that the upper semi-continu-ous properties of their entropy maps are restricted each other, and the entropy map of the asymptotically h-expansive continuous map is upper semi-contlnuous, at the same time a continuous map having u, p.e. is topological weakmixing.     

A mean value theorem on bounded symmetric domains

Let be a Cartan domain of rank and genus and , , the Berezin transform on ; the number can be interpreted as a certain invariant-mean-value of a function around . We show that a Lebesgue integrable function satisfying , , must be -harmonic. In a sense, this result is reminiscent of Delsarte's two-radius mean-value theorem for ordinary harmonic functions on the complex -space , but with the role of radius played by the quantity .

     

Invariant Differential Operators on Symmetric Cones and Hermitian Symmetric Spaces

We give a brief survey on the study of constructions of invariant differential operators on Riemannian symmetric spaces and of combinatorial and analytical properties of their eigenvalues, and pose some open questions.