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.
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 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 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.
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.
We prove the existence of invariant projections from the Banach space of -pseudomeasures onto with for closed neutral subgroup of a locally compact group . As a main application we obtain that every closed neutral subgroup is a set of -synthesis in and in fact locally -Ditkin in . We also obtain an extension theorem concerning the Fourier algebra.
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.
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.
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.
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, .
In 1992, Blanc and Brylinski showed the following property for a -adic group , called the ``abstract Selberg principle': the orbital integrals on conjugacy classes of non-compact elements of the Hattori rank of a finitely generated projective smooth representation of vanish. The proof is by explicit computations of ``low' level ( and cyclic and Hochschild cohomologies. Here we intend to show that this property is actually a direct consequence of two facts: Clozel's integration formula (which leads us to assume the defining characteristic to be zero) and the triviality of the action of unramified characters on the of (which is also proven here, using a standard -theoretic argument due to Grothendieck).