A geometric approach to data representation incorporating informationtheoretic ideas is presented. The task of finding a faithful representation, where the input distribution is evenly partitioned into regions of equal mass, is addressed. For input consisting of mixtures of statistically independent sources, we treat independent component analysis (ICA) as a computational geometry problem. Firs...

We prove that a free group F2 admits a faithful discrete representation into Diff +[0, 1]. We also prove that F2 admits a faithful discrete representation of bi-Lipschitz class into Homeo+[0, 1]. Some properties of these representations have been studied. In the last section of the paper we raise several questions.

The braid groups Bn were originally introduced by Emil Artin in 1926. They have many interpretations, for example, as the group of geometric braids in R, as the Artin group of type An, as the fundamental group of a certain hyperplane arrangement, as a subgroup of the automorphism group of a free group, and so on. In this paper we will use the interpretation of Bn as the mapping class group of a...

It is proved that the free product state, in the reduced free product of C–algebras, is faithful if the states one started with are faithful. The reduced free product of C–algebras was introduced by Voiculescu as the appropriate construction for C–algebras in the setting of his theory of freeness [1]. (See also the book [2]). Given a set I and for each ι ∈ I a unital C–algebra Aι with a state φ...

Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.

Let Bn denote the n-strand braid group. We recall that this admits a representation fin : Bn ---+ GLnl (Z[ t , t l ] ) the (reduced) Burau representation [1]. Although open for a long time, it is now known that this representation is not faithful for n > 6, (See [8, 5]) and it is an old result that it is faithful for n = 3. (See [7]) Despite these counterexamples, there is no understanding of t...

In this paper we analyse the matrix differential system X ′ = [N,X], where N is skew-symmetric and X(0) is symmetric. We prove that it is isospectral and that it is endowed with a Poisson structure, and discuss its invariants and Casimirs. Formulation of the Poisson problem in a Lie–Poisson setting, as a flow on a dual of a Lie algebra, requires a computation of its faithful representation. Alt...

We show that every (continuous) faithful product system admits a (continuous) faithful nondegenerate representation. For Hilbert spaces this is equivalent to Arveson’s result that every Arveson system comes from an E0–semigroup. We point out that for Hilbert modules this is not so. As applications we show a C–algebra version of a result for von Neumann algebras due to Arveson and Kishimoto, and...