Borel and Volume Classes for Dense Representations of Discrete Groups

We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

On the Borel Classes of Set-Valued Maps of Two Variables

Using the Borel classification of set-valued maps, we present here some new results on set-valued maps which are similar to some of the well known theorems on functions due to Lebesgue and Kuratowski. We consider set-valued maps of two variables in perfectly normal topological spaces. It was proved in [11] that a set-valued map lower semicontinuous (i.e. of lower Borel class 0) in the first and upper semicontinuous (i.e. of upper Borel class 0) in the second variable is of upper Borel class 1 and also (with stronger assumptions) of lower Borel class 1. This result cannot be generalized into higher Borel classes. In this paper we show that a set-valued map of the upper (resp. lower) Borel class α in the first and lower semicontinuous and upper quasicontinuous (upper semicontinuous and lower quasicontinuous) in the second variable is of the lower (resp. upper) Borel class α + 1. Also other cases are considered.

Measurable functions similar to the itô integral and the Paley-Wiener-Zygmund integral over continuous paths

Let C[0,T] denote an analogue of generalizedWiener space, the space of continuous real-valued functions on the interval [0,T]. On the space C[0,T], we introduce a finite measure w?,?,? and investigate its properties, where ? is an arbitrary finite measure on the Borel class of R. Using the measure w?,?,?, we also introduce two measurable functions on C[0,T], one of them is similar to the It? integral and the other is similar to the Paley-Wiener-Zygmund integral. We will prove that if ?(R) = 1, then w?,?,? is a probability measure with the mean function ? and the variance function ?, and the two measurable functions are reduced to the Paley-Wiener-Zygmund integral on the analogue ofWiener space C[0,T]. As an application of the integrals, we derive a generalized Paley-Wiener-Zygmund theorem which is useful to calculate generalized Wiener integrals on C[0,T]. Throughout this paper, we will recognize that the generalized It? integral is more general than the generalized Paley-Wiener-Zygmund integral.

A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

Using a simple formula for conditional expectations over continuous paths, we will evaluate conditional expectations which are types of analytic conditional Fourier-Feynman transforms and conditional convolution products of generalized cylinder functions and the functions in a Banach algebra which is the space of generalized Fourier transforms of the measures on the Borel class of L2[0,T]. We will then investigate their relationships. Particularly, we prove that the conditional transform of the conditional convolution product can be expressed by the product of the conditional transforms of each function. Finally we will establish change of scale formulas for the conditional transforms and the conditional convolution products. In these evaluation formulas and change of scale formulas, we use multivariate normal distributions so that the conditioning function does not contain present positions of the paths.

Computing Borel's regulator

We present an infinite series formula based on the Karoubi–Hamida integral, for the universal Borel class evaluated on

Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces

Let X be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of X.