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.

     

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 .

     

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.

     

Irreducible restriction and zeros of characters

Let be a finite group, let be normal in and suppose that is an irreducible complex character of . Then is not irreducible if and only if vanishes on some coset of in .

     

Diagonal type conditions on group C-algebras

Let be a locally compact group with and its enveloping and reduced C-algebras respectively. We show that if is residually finite dimensional, then is maximally almost periodic, and is residually finite dimensional if and only if is both amenable and maximally almost periodic. Letting be the left regular representation of , we show that a certain quasidiagonality condition on implies that is amenable.

     

Line bundles for which a projectivized jet bundle is a product

We characterize the triples , consisting of line bundles and on a complex projective manifold , such that for some positive integer , the -th holomorphic jet bundle of , , is isomorphic to a direct sum .

     

Splitting sets in integral domains

Let be an integral domain. A saturated multiplicatively closed subset of is a splitting set if each nonzero may be written as where and for all . We show that if is a splitting set in , then is a splitting set in , a multiplicatively closed subset of , and that is a splitting set in is an lcm splitting set of , i.e., is a splitting set of with the further property that is principal for all and . Several new characterizations and applications of splitting sets are given.

     

Minimal index of a -crossed product by a finite group

Let be the -crossed product of a simple unital -algebra by a finite group . In this paper we show that the canonical conditional expectation from to has the minimal index if is simple. It is also proved that if is an outer action, then the canonical one is the unique conditional expectation of index-finite type from to , while there are infinitely many conditional expectations when a nontrivial subgroup of acts innerly on .

     

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 .

     

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, .

     

Serial subalgebras of finitary Lie algebras

A Lie subalgebra of is said to be finitary if it consists of elements of finite rank. We show that, if acts irreducibly on , and if is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of acts irreducibly on too. When , it follows that the locally solvable radical of such is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic are hyperabelian.

     

Normal bases for Hopf-Galois algebras

Let be a Hopf algebra over a commutative ring such that is a finitely generated, projective module over , let be a right -comodule algebra, and let be the subalgebra of -coinvariant elements of . If is a Galois extension of and is a local subalgebra of the center of , then is a cleft right -comodule algebra or, equivalently, there is a normal basis for over .

     

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.

     

Compact range property and operators on -algebras

We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.

     

Unique ergodicity on compact homogeneous spaces

Extending results of a number of authors, we prove that if is the unipotent radical of an -split solvable epimorphic subgroup of a real algebraic group which is generated by unipotents, then the action of on is uniquely ergodic for every cocompact lattice in . This gives examples of uniquely ergodic and minimal two-dimensional flows on homogeneous spaces of arbitrarily high dimension. Our main tools are the Ratner classification of ergodic invariant measures for the action of a unipotent subgroup on a homogeneous space, and a simple lemma (the `Cone Lemma') about representations of epimorphic subgroups.

     

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 .

     

Projections of polytopes on the plane and the generalized Baues problem

Given an affine projection of a -polytope onto a polygon , it is proved that the poset of proper polytopal subdivisions of which are induced by has the homotopy type of a sphere of dimension if maps all vertices of into the boundary of . This result, originally conjectured by Reiner, is an analogue of a result of Billera, Kapranov and Sturmfels on cellular strings on polytopes and explains the significance of the interior point of present in the counterexample to their generalized Baues conjecture, constructed by Rambau and Ziegler.

     

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.

     

On contractible

We present a very short proof of a well-known result, that for each there exists a contractible -dimensional compactum, non-embeddable into .

     

A numerical range characterization of uniformly smooth Banach spaces

We prove that a Banach space is uniformly smooth if and only if, for every -valued bounded function on the unit sphere of , the intrinsic numerical range of is equal to the closed convex hull of the spatial numerical range of .