В работе построено разрешенное относительно неизвестных уравнение в свободной группе ранга 2, левая часть которого принадлежит коммутанту (имеет нулевую сумму показателей по каждой переменной), а правая - простейший коммутатор, не имеющее решения этой группе, но решение любом конечном гомоморфном образе группы. We construct an equation (resolved with respect to unknowns) such that (i) it has no...

We construct a weak dilation of a not necessarily unital CP-semigroup to an E–semigroup acting on the adjointable operators of a Hilbert module with a unit vector. We construct the dilation in such a way that the dilating E–semigroup has a pre-assigned product system. Then, making use of the commutant of von Neumann correspondences, we apply the dilation theorem to proof that covariant represen...

A new class of operator algebras, Kadison-Singer (KS-) algebras, is introduced. These highly noncommutative, non self-adjoint algebras generalize triangular matrix algebras. They are determined by certain minimally generating lattices of projections in the von Neumann algebras corresponding to the commutant of the diagonals of the KS-algebras. It is shown that these lattices and their reduced f...

An algorithm is given for the problem of finding the finest simultaneous block-diagonalization of a given set of square matrices. This problem has been studied independently in the area of semidefinite programming and independent component analysis. The proposed algorithm considers the commutant algebra of the matrix ∗-algebra generated by the given matrices. It is simpler than other existing m...

Motivated by the multivariate wavelet theory, and by the spectral theory of transfer operators, we construct an abstract affine structure and a multiresolution associated to a matrix-valued weight. We describe the one-to-one correspondence between the commutant of this structure and the fixed points of the transfer operator. We show how the covariant representation can be realized on Rn if the ...

We study the following question: suppose that A and B are two algebras of complex n×n matrices such that the ring commutator [A, B] = AB−BA is “small” for each A ∈ A and B ∈ B. Does this imply that A and B have a common non-trivial invariant subspace? This question is motivated by a series of papers studying the structure of “almost-commutative” algebras and, more generally, semigroups. A simpl...