Free products of operator spaces and free Markov processes

Certain (reduced) free product is introduced in the framework of operator spaces. Under the construction, the free product of preduals of von Neumann algebras agrees with the predual of the free product of von Neumann algebras. This answers a question asked by Effros affirmatively. An example is presented to show that the C*-algebra reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of the operator space reduced free product of the two C*-algebras. Free Markov processes are also investigated in Voiculescu\u27s free probability theory. This highly non-commutative notion generalizes that of free Brownian motion and free Levy processes. Some free Markov processes are realized as solutions to free stochastic differential equations driven by free Levy processes. A special and rather interesting kind of free Markov processes, free Ornstein-Uhlenbeck processes, is studied in some details. It is shown that a probability measure on R is free self-decomposable if and only if it is the stationary distribution of a stationary free Ornstein-Uhlenbeck process (driven by a free Levy process). Finally, the notion of free fractional Brownian motion is introduced. Examples of fractional free Brownian motion are given, which are based on creation and annihilation operators on full Fock spaces. It is proved that the Langevin equation driven by fractional free Brownian motion has a unique solution. We call the solution a fractional free Ornstein-Uhlenbeck process

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids $M$ and $N$ is maximal with respect to the weak order among matroids having $M$ as a submatroid, with complementary contraction equal to $N$. Any minor of the free product of $M$ and $N$ is a free product of a repeated truncation of the corresponding minor of $M$ with a repeated Higgs lift of the corresponding minor of $N$. We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra $\cal C$ of a family of matroids $\cal M$ that is closed under formation of minors and free products: namely, $\cal C$ is cofree, cogenerated by the set of irreducible matroids belonging to $\cal M$.Comment: Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for publication in the Journal of Combinatorial Theory (A). See arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this subjec

Automatic continuity for homomorphisms into free products

A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.Comment: 15 pages, 1 table. Final version. To appear in the Journal of Symbolic Logi