Vol 53, No 2 (2019)

Let Ω be a metric space. By A^t we denote the metric neighborhood of radius t of a set A ⊂ Ω and by \(\mathfrak{D}\), the lattice of open sets in Ω with partial order ⊆ and order convergence. The lattice of \(\mathfrak{D}\)-valued functions of t ∈ (0, ∞) with pointwise partial order and convergence contains the family I\(\mathfrak{D}\) = {A(·)| A(t) = A^t, A ∈ \(\mathfrak{D}\)}. Let ̃Ω be the set of atoms of the order closure \(\overline{I\mathfrak{D}}\). We describe a class of spaces for which the set ̃Ω equipped with an appropriate metric is isometric to the original space Ω.

The space ̃Ω is the key element of the construction of the wave spectrum of a lower bounded symmetric operator, which was introduced in a work of one of the authors. In that work, a program for constructing a functional model of operators of the aforementioned class was laid down. The present paper is a step in the realization of this program.

Several necessary conditions for the topological flatness of Banach modules are given. The main result is as follows: a Banach module over a relatively amenable Banach algebra which is topologically flat as a Banach space is topologically flat as a Banach module. Finally examples of topologically flat modules among classical modules of analysis are given.

Realizations of five-dimensional Lie algebras as algebras of holomorphic vector fields on homogeneous real hypersurfaces of a three-dimensional complex space are studied. In view of already known results, in the problem of describing such varieties only Levy nondegenerate hypersurfaces with exactly five-dimensional Lie algebras are of interest. It is shown that only two of the nine existing distinct nilpotent Lie algebras admit realizations associated with such varieties, and the varieties corresponding to these exceptional algebras are standard quadrics.

We show that on the sphere S^m, m ≠ 1, 3, 7, there exists an n_m-valued multiplication with unit for some n_m ∈ {2, 4, 8}. We also explicitly construct a 2^k−1-fold branched covering of \(S^{m_1}\;\times\cdots\times\;S^{m_k}\) the product S^m1× ··· × S^mk of k spheres over the sphere S^m, m = m₁ + ··· + m_k.

The present note deals with operator Hilbert systems, which are quantizations of unital cones in Hilbert spaces. One central result of the note is that the Pisier operator Hilbert space is an operator system whose quantum cone of positive elements is described in terms of the quantum ball of the relevant conjugate Hilbert space. Finally, we obtain a solution to the problem of Paulsen, Todorov and Tomforde on separable morphisms between operator systems and characterize minmax- completely positive maps between Archimedean order unit spaces.

Questions related to “very weak” solutions of physical models of acoustic and shallow water wave propagation with singular dissipation are studied. The existence of a new type of solutions is proved. An existence theorem for a very weak solution of the problem is obtained. Finally it is shown that very weak solutions are consistent with classical ones in a certain sense, provided that the latter exist.