On tensor products of nuclear operators in Banach spaces

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

New examples of Bernoulli algebraic actions

We give an example of a principal algebraic action of the non-commutative free group ${\mathbb {F}}$ of rank two by automorphisms of a connected compact abelian group for which there is an explicit measurable isomorphism with the full Bernoulli 3-shift action of ${\mathbb {F}}$ . The isomorphism is defined using homoclinic points, a method that has been used to construct symbolic covers of algebraic actions. To our knowledge, this is the first example of a Bernoulli algebraic action of ${\mathbb {F}}$ without an obvious independent generator. Our methods can be generalized to a large class of acting groups.

Banach space valued weak second order stochastic processes

Banach space valued stochastic processes of weak second order on a locally compact abelian group G G is considered. These processes are recognized as operator valued processes on G G . More fully, letting U \mathfrak {U} be a Banach space and H \mathfrak {H} a Hilbert space, we study B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes. Since B ( U , H ) B(\mathfrak {U},\mathfrak {H}) has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued gramian, every B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued process has a B ( U , U ∗ ) B(\mathfrak {U},\mathfrak {U}^*) -valued covariance function. Using this property we can define operator stationarity, operator harmonizability and operator V V -boundedness for B ( U , H ) B(\mathfrak {U},\mathfrak {H}) -valued processes, in addition to scalar ones. Interrelations among these processes are obtained together with the operator stationary dilation.

Popular differences for corners in Abelian groups

For a compact abelian group G, a corner in G × G is a triple of points (x, y), (x, y+d), (x+d, y). The classical corners theorem of Ajtai and Szemerédi implies that for every α > 0, there is some δ > 0 such that every subset A ⊂ G × G of density α contains a δ fraction of all corners in G × G, as x, y, d range over G. Recently, Mandache proved a “popular differences” version of this result in the finite field case $G = {\mathbb{F}}_p^n$ , showing that for any subset A ⊂ G × G of density α, one can fix d ≠ 0 such that A contains a large fraction, now known to be approximately α4, of all corners with difference d, as x, y vary over G. We generalise Mandache’s result to all compact abelian groups G, as well as the case of corners in $\mathbb{Z}^2$ .

On a group analogue of the Heyde theorem

Heyde proved that a Gaussian distribution on a real line is characterized by the symmetry of the conditional distribution of one linear form given another. The present article is devoted to an analogue of the Heyde theorem in the case when random variables take values in a locally compact Abelian group and the coefficients of the linear forms are integers.

A bridge theorem for the entropy of semigroup actions

The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

The Pedersen Rigidity Problem

If is an action of a locally compact abelian group G on a C*-algebra A, Takesaki-Takai duality recovers (A, α) up to Morita equivalence from the dual action of Ĝ on the crossed product A × α G. Given a bit more information, Landstad duality recovers (A, α) up to isomorphism. In between these, by modifying a theorem of Pedersen, (A, α) is recovered up to outer conjugacy from the dual action and the position of A in M(A ×α G). Our search (still unsuccessful, somehow irritating) for examples showing the necessity of this latter condition has led us to formulate the "Pedersen Rigidity problem". We present numerous situations where the condition is redundant, including G discrete or A stable or commutative. The most interesting of these "no-go theorems" is for locally unitary actions on continuous-trace algebras.