AN EIGENSPACE OF LARGE DIMENSION FOR A HECKE ALGEBRA ON AN BUILDING

Let Δ be an affine building of type $\tilde A_2$ and let 𝔸 be its fundamental apartment. We consider the set 𝕌0 of vertices of type 0 of 𝔸 and prove that the Hecke algebra of all W0-invariant difference operators with constant coefficients acting on 𝕌0 has three generators. This property leads us to define three Laplace operators on vertices of type 0 of Δ. We prove that there exists a joint eigenspace of these operators having dimension greater than ∣W0 ∣. This implies that there exist joint eigenfunctions of the Laplacians that cannot be expressed, via the Poisson transform, in terms of a finitely additive measure on the maximal boundary Ω of Δ.

Invariant measures on groups satisfying various chain conditions

For any group satisfying a suitable chain condition, we construct a finitely additive measure on it that is invariant under certain actions.

Smoothness conditions on measures using Wallman spaces

In this paper,Xdenotes an arbitrary nonempty set,ℒa lattice of subsets ofXwith∅,X∈ℒ,A(ℒ)is the algebra generated byℒandM(ℒ)is the set of nontrivial, finite, and finitely additive measures onA(ℒ), andMR(ℒ)is the set of elements ofM(ℒ)which areℒ-regular. It is well known that anyμ∈M(ℒ)induces a finitely additive measureμ¯on an associated Wallman space. Wheneverμ∈MR(ℒ),μ¯is countably additive.We consider the general problem of givenμ∈MR(ℒ), how do properties ofμ¯imply smoothness properties ofμ? For instance, what conditions onμ¯are necessary and sufficient forμto beσ-smooth onℒ, or stronglyσ-smooth onℒ, or countably additive? We consider in discussing these questions either of two associated Wallman spaces.

Some remarks concerning finitely Subadditive outer measures with applications

The present paper is intended as a first step toward the establishment of a general theory of finitely subadditive outer measures. First, a general method for constructing a finitely subadditive outer measure and an associated finitely additive measure on any space is presented. This is followed by a discussion of the theory of inner measures, their construction, and the relationship of their properties to those of an associated finitely subadditive outer measure. In particular, the interconnections between the measurable sets determined by both the outer measure and its associated inner measure are examined. Finally, several applications of the general theory are given, with special attention being paid to various lattice related set functions.

Weakly Purely Finitely Additive Measures

Let L be an orthomodular poset. A positive measure ξ on L is said to be weakly purely finitely additive if the zero measure is the only completely additive measure majorized by ξ. It was shown in [15] that, in an arbitrary orthomodular poset L, every positive measures μ is the sum v + ξ of a positive completely additive measure v and a weakly purely finitely additive measure ξ. We give sufficient conditions for this Yosida-Hewitt-type decomposition to be unique.A positive measure λ on L is said to be filtering if every non-zero element p in L majorizes a non-zero element q on which λ vanishes. A filtering measure is weakly purely finitely additive. Filtering measures play a mediator role throughout these investigations since some of the aforementioned conditions are given in terms of these.The results obtained here are then viewed in the context of Boolean lattices and applied to lattices of idempotents of non-associative JBW-algebras.

Measures Below Outer Measures

We give a simple proof that, for any ॉ > 0, there is an outer measure μ* on a finite set X such that, for any measure Thus there is a non-zero outer (finitely subadditive) measure v* on the clopen subsets of the Cantor set such that, if v ≤ v* is a finitely additive measure on the clopen subsets of the Cantor set, then v ≡ 0.

Measures on coallocation and normal lattices

Letℒ1andℒ2be lattices of subsets of a nonempty setX. Supposeℒ2coallocatesℒ1andℒ1is a subset ofℒ2. We show that anyℒ1-regular finitely additive measure on the algebra generated byℒ1can be uniquely extended to anℒ2-regular measure on the algebra generated byℒ2. The case whenℒ1is not necessary contained inℒ2, as well as the measure enlargement problem are considered. Furthermore, some discussions on normal lattices and separation of lattices are also given.

On maximal measures with respect to a lattice

Outer measures are used to obtain measures that are maximal with respect to a normal lattice. Alternate proofs are then given extending the measure theoretic characterizations of a normal lattice to an arbitrary, non-negative finitely additive measure on the algebra generated by the lattice. Finally these general results are used to considerσ-smooth measures with respect to the lattice when further conditions on the lattice hold.