Types in reductive -adic groups: The Hecke algebra of a cover

In this paper, is a non-Archimedean local field and is the group of -points of a connected reductive algebraic group defined over . Also, is an irreducible representation of a compact open subgroup of , the pair being a type in . The pair is assumed to be a cover of a type in a Levi subgroup of . We give conditions, generalizing those of earlier work, under which the Hecke algebra is the tensor product of a canonical image of and a sub-algebra , for a compact open subgroup of containing .

     

Lifting wreath product extensions

Let and be finite groups and let be a hilbertian field. We show that if has a generic extension over and satisfies the arithmetic lifting property over , then the wreath product of and also satisfies the arithmetic lifting property over . Moreover, if the orders of and are relatively prime and is abelian, then any extension of by (which is necessarily a semidirect product) has the arithmetic lifting property.

     

A note on the existence of a largest topological factor with zero entropy

Given a topological system on a -compact Hausdorff space and its factor we show the existence of a largest topological factor containing such that for each -invariant measure , . When a relative variational principle holds, .

     

On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .

We prove that the following are equivalent:

1. is Lipschitz homeomorphic to .

2. is Lipschitz homeomorphic to .

3. is Lipschitz homeomorphic to .

This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.

     

Dense subsets of maximally almost periodic groups

A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a)  \vert G\vert$">, (b) each is dense in , and (c) distinct satisfy ; a totally bounded topological group with such a family is a strongly extraresolvable topological group.

We give two theorems, the second generalizing the first.

Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup.

Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable.

Theorem 2. Let be maximally almost periodic. Then there are a subgroup of and a family such that

(i) is dense in every totally bounded group topology on ;

(ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and

(iii) admits a totally bounded group topology as in (ii).

Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice is impossible.

     

Splitting for subalgebras of tensor products

We prove splitting results for subalgebras of tensor products of operator algebras. In particular, any -algebra s.t. is a tensor product provided is simple and nuclear.

     

Convergence of cascade algorithms associated with nonhomogeneous refinement equations

This paper is devoted to a study of multivariate nonhomogeneous refinement equations of the form where is the unknown, is a given vector of functions on , is an dilation matrix, and is a finitely supported refinement mask such that each is an (complex) matrix. Let be an initial vector in . The corresponding cascade algorithm is given by In this paper we give a complete characterization for the -convergence of the cascade algorithm in terms of the refinement mask , the nonhomogeneous term , and the initial vector of functions .

     

Quasitilted extensions of algebras I

Let be a connected finite dimensional -algebra, and let be a nonzero decomposable -module such that the one-point extension is quasitilted. We show here that every nonzero indecomposable direct summand of is directing and is a tilted algebra.

     

Between the Lindelöf property and countable tightness

We consider a class of compact spaces for which the space of probability Radon measures on has countable tightness in the topology. We show that that class contains those compact zero-dimensional spaces for which is weakly Lindelöf, and, under MA + CH, all compact spaces with having property (C) of Corson.

     

A note on branching theorems

Let be a complex, simply connected semisimple analytic group with a closed connected reductive subgroup. Suppose is an irreducible holomorphic -module and an irreducible holomorphic -module. We prove that Hom possesses the structure of an irreducible -module whenever is . Moreover, for all and if and only if is commutative.

     

Normal essential eigenvalues in the boundary of the numerical range

A purely geometric property of a point in the boundary of the numerical range of an operator on Hilbert space is examined which implies that such a point is the value at of a multiplicative linear functional of the -algebra, , generated by and the identity operator. Roughly speaking, such a property means that the boundary of the numerical range (of ) has infinite curvature at that point. Furthermore, it is shown that if such a point is not a sharp linear corner of the numerical range of , then the multiplicative linear functional vanishes on the compact operators in .

     

The Riesz decomposition property for the space of regular operators

If and are Banach lattices such that is separable and has the countable interpolation property, then the space of all continuous regular operators has the Riesz decomposition property. This result is a positive answer to a conjecture posed by A. W. Wickstead.

     

On classification of polarized varieties with non-integral nef values

Let be an -dimensional normal projective variety with only Gorenstein, terminal, -factorial singularities. Let be an ample line bundle on . Let denote the nef value of . The classification of via the nef value morphism is given for the situations when satisfies or .

     

Approximating discrete valuation rings by regular local rings

Let be a field of characteristic zero and let be a discrete rank-one valuation domain containing with . Assume that the fraction field of has finite transcendence degree over . For every positive integer , we prove that can be realized as a directed union of regular local -subalgebras of of dimension .

     

The gap between probability and prevalence: Loneliness in vector spaces

The best available definition of a subset of an infinite dimensional, complete, metric vector space being ``small' is Christensen's Haar zero sets, equivalently, Hunt, Sauer, and Yorke's shy sets. The complement of a shy set is a prevalent set. There is a gap between prevalence and likelihood. For any probability on , there is a shy set with . Further, when is locally convex, any i.i.d. sequence with law repeatedly visits neighborhoods of only a shy set of points if the neighborhoods shrink to at any rate.

     

Le degré de Lindelöf est -invariant

Two Tychonoff spaces and are said to be -equivalent if and are linearly homeomorphic. It is shown that if and are -equivalent, then the Lindelöf numbers of and are the same. The proof given is a strengthening of the one given by N.V. Velichko to show that the Lindelöf property is -invariant.

     

The Brauer group of Sweedler's Hopf algebra

We calculate the Brauer group of the four dimensional Hopf algebra introduced by M. E. Sweedler. This Brauer group is defined with respect to a (quasi-) triangular structure on , given by an element . In this paper is a field . The additive group of is embedded in the Brauer group and it fits in the exact and split sequence of groups: where is the well-known Brauer-Wall group of . The techniques involved are close to the Clifford algebra theory for quaternion or generalized quaternion algebras.

     

A projection theorem and tangential boundary behavior of potentials

Let be the Weinstein operator on the half space, . Suppose there is a sequence of Borel sets such that a certain tangential projection of onto forms a pairwise disjoint subset of the boundary. Let be a finite test measure on the boundary for a specific non-isotropic Hausdorff measure. The measure is carried back to a measure on a subset of by the projection. We give an upper bound for the Weinstein potential corresponding to the measure in terms of a universal constant and a Weinstein subharmonic function. We use this upper bound to deduce a result concerning tangential behavior of Weinstein potentials at the boundary with the exception of sets on the boundary of vanishing non-isotropic Hausdorff measure.

     

A characterization of algebras with polynomial growth of the codimensions

Let be an associative algebras over a field of characteristic zero. We prove that the codimensions of are polynomially bounded if and only if any finite dimensional algebra with has an explicit decomposition into suitable subalgebras; we also give a decomposition of the -th cocharacter of into suitable -characters.

We give similar characterizations of finite dimensional algebras with involution whose -codimension sequence is polynomially bounded. In this case we exploit the representation theory of the hyperoctahedral group.

     

Linear maps preserving ideals of -algebras

We show that unital self-adjoint linear bijections of matrix algebras, type factors and abelian -algebras preserving maximal left ideals are isomorphisms and we show that a unital continuous linear map of a -algebra that maps the minimal left ideal into itself is the identity map.