Measurable Kac cohomology for bicrossed products

We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a measurable version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.

     

Classifying bicrossed products of two Taft algebras

We classify all Hopf algebras which factor through two Taft algebras ${\mathbb{T}}_{n^2}(\stackrel{￣}{q})$ and respectively $T_{m^2}(q)$. To start with, all possible matched pairs between the two Taft algebras are described: if $\stackrel{￣}{q}\ne q^{n?1}$ then the matched pairs are in bijection with the group of d-th roots of unity in k, where $d=(m,n)$ while if $\stackrel{￣}{q}=q^{n?1}$ then besides the matched pairs above we obtain an additional family of matched pairs indexed by $k^{?}$. The corresponding bicrossed products (double cross product in Majid's terminology) are explicitly described by generators and relations and classified. As a consequence of our approach, we are able to compute the number of isomorphism types of these bicrossed products as well as to describe their automorphism groups.     

Ergodic theorems for tensor products

An ergodic theorem is proved for tensor products of Banach spaces. As a special case, an ergodic theorem is proved for vector-valued L^p-spaces. This theorem generalizes results of Aribaud, J. Funct. Anal.5 (1970), 395–411, and Dinculeanu, J. Funct. Anal.12 (1973), 229–235.     

Bernstein type theorems for higher codimension

We show a Bernstein theorem for minimal graphs of arbitrary dimension and codimension under a bound on the slope that improves previous results and is independent of the dimension and codimension. The proof depends on the regularity theory for the harmonic Gauss map and the geometry of Grassmann manifolds. Received: September 25, 1998 / Accepted: October 23, 1998     

Uniqueness theorems for Korenblum type spaces

For a scale of spaces X of functions analytic in the unit disc, including the Korenblum space, and for some natural families ɛ of uniqueness subsets for X, we describe minorants for (X, ɛ), that is, non-decreasing functions M: (0, 1) → (0, ∞) such that f ∈ X, E ∈ ɛ, and log |f(z)| ≤ −M(|z|) on E imply f = 0. We give an application of this result to approximation by simple fractions with restrictions on the coefficients. The first author was partially supported by the ANR project DYNOP. The second author was partially supported by the Research Council of Norway, grant 160192/V30.     

Schreier loops

We study systematically the natural generalization of Schreier’s extension theory to obtain proper loops and show that this construction gives a rich family of examples of loops in all traditional common, important loop classes.     

Comparison theorems for hyperbolic type metrics

The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.     

Beurling-Pollard type theorems

We establish a version of the Beurling–Pollard theoremfor operator synthesis and apply it to derive some results onlinear operator equations and to prove a Beurling–Pollardtype theorem for Varopoulos tensor algebras. In addition, weestablish a Beurling–Pollard theorem for weighted Fourieralgebras and use it to obtain ascent estimates for operatorsthat are functions of generalized scalar operators.     

Some Schroeder-Bernstein type theorems for Banach spaces

We first introduce the notion of (p,q,r)-complemented subspaces in Banach spaces, where p,q,r∈N. Then, given a couple of triples {(p,q,r),(s,t,u)} in N and putting Λ=(q+r−p)(t+u−s)−ru, we prove partially the following conjecture: For every pair of Banach spaces X and Y such that X is (p,q,r)-complemented in Y and Y is (s,t,u)-complemented in X, we have that X is isomorphic Y if and only if one of the following conditions holds:

(a)

Λ≠0, Λ divides p−q and s−t, p=1 or q=1 or s=1 or t=1.

(b)

p=q=s=t=1 and gcd(r,u)=1.

The case {(2,1,1),(2,1,1)} is the well-known Pe?czyński's decomposition method. Our result leads naturally to some generalizations of the Schroeder-Bernstein problem for Banach spaces solved by W.T. Gowers in 1996.     

Otto Schreier

Ohne Zusammenfassung     

Multiplier theorems for Herz type Hardy spaces

In this paper, the authors establish a multiplier theorem for Herz type Hardy spaces.

     

Poincaré type theorems for non-autonomous systems

In this paper we establish analytic equivalence theorems of Poincaré and Poincaré-Dulac type for analytic non-autonomous differential systems based on the dichotomy spectrum of their linear part. As applications of the theorem, normal forms linearize for two illustrative examples.