Let denote the spectral radius of an operator . We construct operators and on such that is discontinuous almost everywhere on the unit disk.
Let , where and is a Banach space. Let be an extension of to all of (i.e., ) such that has minimal (operator) norm. In this paper we show in particular that, in the case and the field is R, there exists a rank- such that for all if and only if the unit ball of is either not smooth or not strictly convex. In this case we show, furthermore, that, for some , there exists a choice of basis such that ; i.e., each is a Hahn-Banach extension of .
The following dichotomy is established for any pair , of hereditary families of finite subsets of : Given , an infinite subset of , there exists an infinite subset of so that either , or , where denotes the set of all finite subsets of .
A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .
Let be a positive matrix-valued measure on a locally compact abelian group such that is the identity matrix. We give a necessary and sufficient condition on for the absence of a bounded non-constant matrix-valued function on satisfying the convolution equation . This extends Choquet and Deny's theorem for real-valued functions on .
A variety is a class of Banach algebras , for which there exists a family of laws such that is precisely the class of all Banach algebras which satisfies all of the laws (i.e. for all , . We say that is an -variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras , for which there exists a family of homogeneous laws such that is precisely the class of all Banach algebras , for which there exists 0$"> such that for all homogeneous polynomials , , where . However, there is no variety between the variety of all -algebras and the variety of all -algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.
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.
Let be an odd prime number. We prove that if acts freely on a product of equidimensional lens spaces, then . This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian -groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A -group can act freely on a product of lens spaces with a trivial action on homology if and only if and has the -extension property. The main technique is to study group extensions associated to free actions.
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 .
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 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.