We determine the dimension of the space of Hodge cycles for the generic fibers of the Kuga fiber varieties associated to certain quaternion algebras.

We investigate the invariant subspace structure of subalgebras of groupoid C*-algebras that are determined by automorphism groups implemented by cocycles on the groupoids. The invariant subspace structure is intimately tied to the asymptotic behavior of the cocycle.

We consider the Riesz product with a constant coefficient and odometer action over infinite product spaces. By studying the ratio set we can conclude the type of the above dynamical systems is III1.

It is known that every frame is isomorphic to the generalized Gleason algebra of an essentially unique bi-Stonian space (X, σ, τ) in which σ is T0. Let (X, σ, τ) be as above. The specialization order ≤σ, of (X, σ) is τ × τ-closed. By Nachbin's Theorem there is exactly one quasi-uniformity U on X such that ∩U = ≤σ and J(U*) = τ. This quasi-uniformity is compatible with σ and is coarser than the Pervin quasi-uniformity U of (X, σ). Consequently, τ is coarser than the Skula topology of σ and coincides with the Skula topology if and only if U = P.

Given two topological algebra sheaves, we seek that their tensor product be an (algebra) sheaf of the same type. We further study the latter sheaf in connection with the set of morphisms which are defined on it. As an application, we finally consider fundamental notions and results related to algebras of holomorphic functions in the framework of topological algebra sheaves.

Let X be a (real or complex) rearrangement-invariant function space on Ω (where Ω = [0, 1] or Ω ⊆ N) whose norm is not proportional to the L2-norm. Let H be a separable Hilbert space. We characterize surjective isometries of X (H). We prove that if T is such an isometry then there exist Borel maps a: Ω → + K and σ: Ω → Ω and a strongly measurable operator map S of Ω into B (H) so that for almost all ω, S(ω) is a surjective isometry of H, and for any f ∈ X(H), T f(ω) = a(ω)S(ω)(f(σ(ω))) a.e. As a consequence we obtain a new proof of the characterization of surjective isometries in complex rearrangement-invariant function spaces.

We characterize the Hardy spaces Hp(G) of a compact Lie groupG by means of S-functions in analogy with the theorem of Fefferman-Stein for Rn. We also characterize Hp(G) by means of the -functions.

It is shown that, over any ring R, the direct sum M = ⊕i∈IMi of uniform right R-modules Mi with local endomorphism rings is a CS-module if and only if every uniform submodule of M is essential in a direct summand of M and there does not exist an infinite sequence of non-isomorphic monomorphisms , with distinct in ∈ I. As a consequence, any CS-module which is a direct sum of submodules with local endomorphism rings has the exchange property.

In this paper, the concept of essential equilibria for production economies is first given. We then prove that in ‘most’ production economies (in the sense of Baire category) all equilibria are essential.

We use the second derivative of intertwining operators to realize a unitary structure for the irreducible subrepresentations in the reducible spherical principal series of U(1, n). These representations can also be realized as the kernels of certain invariant first-order differential operators acting on sections of homogeneous bundles over the hyperboloid (U(1) × U(n))/U(1, n).

Kallenberg [2] introduced the concept of F-exchangeable sequences of random variables and produced some characterizations of F-exchangeability in terms of stopping times. In this paper ways of extending the concept of F-exchangeability to doubly indexed arrays of random variables are explored and some characterizations obtained for row and column exchangebale arrays, weakly exchangeable arrays and separately exchangeable continuous processes.

The Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

The purpose of this paper is to complete results concerning the class ℋ of expansive homeomorphisms having the pseudo orbits tracing property on a compact metric space. We show that hyperbolic homeomorphisms introduced by Mañé in [8] are exactly those in the class ℋ then by the result of [12, 20] they form a class equal to the Smale space introduced by Ruelle in [18]. Next, assuming that the phase space is a smooth manifold, we show that a diffeomorphism is Anosov if and only if it is in the class ℋ and is a lower semi-continuity point of the map which assigns to any diffeomorphism the supremum of its expansive constants (possibly zero). Then we discuss the behavior of the dynamical systems generated by homeomorphisms from ℋ near their basic sets.

We describe measurable Hilbert sheaves as Hilbert space objects in a sheaf category constructed from a measure space. These are quite useful for the interpretation of the direct integral of Hilbert spaces as an indexed functor. We set up a framework to put this and similar constructions of operator theory on an indexed categorical footing.

In this paper, we study the set of rational μ in (−2, 2) for which the group Gu generated by is not free by using quadratic Diophantine equations of the form ax2 −by2 = ±1. We give a new set of accumulation points for rational values of μ in (−2, 2) for which Gμ is not free, thereby extending the results of Beardon where he showed that are accumulation points, where N is an integer which is not a perfect square. In particular, we exhibit an infinite set of accumulation points for μ between 1 and 2 including the point 1.

A strong, natural duality is established for the variety by a dihedral gruop of order 2m with m odd. This is the first natural duality for a non-abelian variety of groups.

If G is a finite solvable group, we show that Isaacs' theory on partial characters on Hall π-subgroups can be developed for the nilpotent injectors of G. Therefore, the irreducible characters of G are partitioned into blocks associated to some nilpotent subgroups of G.

A weighted generalization of a p-Sidon set, called an (a, p)-Sidon set, is introduced and studied for infinite, non-abelian, connected, compact groups G. The entire dual object Ĝ is shown never to be central (p − 1, p)-Sidon for 1 ≦ p < 2, nor central (1 + ε, 2)-Sidon for ε > 0. Local (p, p)-Sidon sets are shown to be identical to local Sidon sets. Examples are constructed of infinite non-Sidon sets which are central (2p − 1, p)-Sidon, or (p − 1, p)-Sidon, for 1 < p < 2. Full m-fold FTR sets are proved not to be central (a, 2m/(m + 1))-Sidon for any a > 1.

We show that p-groups of order p5 are determined by their group algebras over the field of p elements. Many cases have been dealt with in earlier work of ourselves and others. The only case whose details remain to be given here is that of groups of nilpotency class 3 for p odd.

We present sufficient conditions on an approximate mapping F: X → Y of approximate inverse systems in order that the limit f: X → Y of F is a universal map in the sense of Holsztyński. A similar theorem holds for a more restrictive concept of a proximately universal map introduced recently by the second author. We get as corollaries some sufficient conditions on an approximate inverse system implying that the its limit has the (proximate) fixed point property. In particular, every chainable compact Hausdorif space has the proximate fixed point property.