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 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.
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 show that a family of functions meromorphic in some domain is normal, if for all the derivative omits the value and if the values that can take at the zeros of satisfy certain restrictions. As an application we obtain a new proof of a theorem of Langley which classifies the functions meromorphic in the plane such that and have no zeros.
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 .
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 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.
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).
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.
Let , a prime (resp. , act freely on a finitistic space with (resp. rational) cohomology ring isomorphic to that of . In this paper we determine the possible cohomology algebra of the orbit space .
We show that there exists a finite polyhedron homotopy dominating infinitely many finite polyhedra of different homotopy types such that there isn't any homotopy type between and . This answers negatively the question raised by K. Borsuk in 1975: Does every FANR have only finitely many left neighbors?
Some consequences of these extension properties are also studied.