A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.
Let be a locally compact Hausdorff space. We define a quasi-measure in , a quasi-integral on , and a quasi-integral on . We show that all quasi-integrals on are bounded, continuity properties of the quasi-integral on , representation of quasi-integrals on in terms of quasi-measures, and unique extension of quasi-integrals on to .
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 .
Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .
Let be a locally compact group, the Fourier algebra of and the von Neumann algebra generated by the left regular representation of . We introduce the notion of -spectral set and -Ditkin set when is an -invariant linear subspace of , thus providing a unified approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of -spectral sets and -Ditkin sets, and an injection theorem for -spectral sets.
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.
B.Y. Wang conjectured that if and are subspaces of an -dimensional complex inner product space , and their dimensions are and , respectively, where , then there exists a -dimensional subspace having two orthonormal bases and with and for all .
We prove this conjecture and its real counterpart. The proof is in essence an application of a real version of the Bézout Theorem for the product of several projective spaces.
Under the assumption that there exists in the unit interval an uncountable set with the property that every continuous mapping from a Baire metric space into is constant on some non-empty open subset of , we construct a Banach space such that belongs to Stegall's class but is not fragmentable.
Let be a separable inner product space over the field of real numbers. Let (resp., denote the orthomodular poset of all splitting subspaces (resp., complete-cocomplete subspaces) of . We ask whether (resp., can be a lattice without being complete (i.e. without being Hilbert). This question is relevant to the recent study of the algebraic properties of splitting subspaces and to the search for ``nonstandard' orthomodular spaces as motivated by quantum theories. We first exhibit such a space that is not a lattice and is a (modular) lattice. We then go on showing that the orthomodular poset may not be a lattice even if . Finally, we construct a noncomplete space such that with being a (modular) lattice. (Thus, the lattice properties of (resp. do not seem to have an explicit relation to the completeness of though the Ammemia-Araki theorem may suggest the opposite.) As a by-product of our construction we find that there is a noncomplete such that all states on are restrictions of the states on for being the completion of (this provides a solution to a recently formulated problem).
For an algebraically closed field, let denote the quotient field of the power series ring over . The ``Newton-Puiseux theorem' states that if has characteristic 0, the algebraic closure of is the union of the fields over . We answer a question of Abhyankar by constructing an algebraic closure of for any field of positive characteristic explicitly in terms of certain generalized power series.
Let be a compact manifold, and let be a transitive homologically full Anosov flow on . Let be a -cover for , and let be the lift of to . Babillot and Ledrappier exhibited a family of measures on , which are invariant and ergodic with respect to the strong stable foliation of . We provide a new short proof of ergodicity.
We consider real spaces only.
Definition. An operator between Banach spaces and is called a Hahn-Banach operator if for every isometric embedding of the space into a Banach space there exists a norm-preserving extension of to .
A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces such that there exists a Hahn-Banach operator of rank . The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman.
Let denote the unit ball in This paper characterizes the subsets of with the property that for all harmonic functions on which have finite Dirichlet integral. It also examines, in the spirit of a celebrated paper of Brelot and Doob, the associated question of the connection between non-tangential and fine cluster sets of functions on at points of the boundary.
Let be the algebra of all bounded operators on a complex Hilbert space and let be an invertible self-adjoint (or skew-symmetric) operator of . Corach-Porta-Recht proved that
The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht Inequality; (ii) a necessary condition (resp. necessary and sufficient condition, when for the invertible positive operators to satisfy the operator-norm inequality for all in ; (iii) a necessary and sufficient condition for the invertible operator in to satisfy
In this paper we deal with the interpolation from Lebesgue spaces and , into an Orlicz space , where and for some concave function , with special attention to the interpolation constant . For a bounded linear operator in and , we prove modular inequalities, which allow us to get the estimate for both the Orlicz norm and the Luxemburg norm,
where the interpolation constant depends only on and . We give estimates for , which imply . Moreover, if either or , then . If , then , and, in particular, for the case this gives the classical Orlicz interpolation theorem with the constant .
Let be a covariant system and let be a covariant representation of on a Hilbert space . In this note, we investigate the representation of the covariance algebra and the -weakly closed subalgebra generated by and in the case of or when there exists a pure, full, -invariant subspace 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 .