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We describe the Shilov boundary for a q -analog of the algebra of holomorphic functions on the unit ball in the space of symmetric 2×2 matrices.

The notion of reduced synthesis in the context of harmonic analysis on general locally compact groups is introduced; in the classical situation of commutative groups, this notion means that a function f in the Fourier algebra is annihilated by any pseudofunction supported on f (-1)(0). A relationship between reduced synthesis and compact synthesis (i.e., the possibility of approximating compact operators by pseudointegral ones without increasing the support) is determined, which makes it possible to obtain new results both in operator theory and in harmonic analysis. Applications to the theory of linear operator equations are also given.

We study bounded and unbounded *-representations of Twisted Generalized Weyl Algebras and algebras similar to them for different choices of involutions.

All *-representations of generalized Weyl algebras by bounded operators are classified with respect to the * preserving principal grading.

We introduce a twisted higher rank generalization of generalized Weyl algebras and study simple weight modules over such algebras.

We continue the study of multidimensional operator multipliers initiated in [K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc., in press]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C ∗ -algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C ∗ -algebra of compact operators in terms of tensor products, generalising results of Saar [H. Saar, Kompakte, vollstandig beschrankte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universitat des Saarlandes, Saarbrucken, 1982].