The Shilov boundary for a qq-analog of the holomorphic functions on the unit ball of 2×22 \times 2 symmetric matrices

We describe the Shilov boundary for a $q$-analog of the algebra of holomorphic functions on the unit ball in the space of symmetric $2 \times 2$ matrices.Comment: 14 page

Beurling-Fourier Algebras and Complexification

In this paper, we develop a new approach that allows to identify the Gelfand spectrum of weighted Fourier algebras as a subset of an abstract complexification of the corresponding group for a wide class of groups and weights. This generalizes some recent results of Ghandehari-Lee-Ludwig-Spronk-Turowska on the spectrum of Beurling-Fourier algebras on some Lie groups. In the case of discrete groups we consider a more general concept of weights and classify them in terms of finite subgroups.Comment: 40 page

On the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups

We extend the results by Froelich and Spronk and Turowska on the connection between operator synthesis and spectral synthesis for A(G) to second countable locally compact groups G. This gives us another proof that one-point subset of G is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis. Furthermore we show that ``non-triangular'' sets are strong operator Ditkin sets and we establish a connection between operator Ditkin sets and Ditkin sets. These results are applied to prove that any closed subgroup of $G$ is a local Ditkin set.Comment: 21 page

Operator synthesis II. Individual synthesis and linear operator equations

The second part of our work on operator synthesis deals with individual operator synthesis of elements in some tensor products, in particular in Varopoulos algebras, and its connection with linear operator equations. Using a developed technique of ``approximate inverse intertwining'' we obtain some generalizations of the Fuglede and the Fuglede-Weiss theorems. Additionally, we give some applications to spectral synthesis in Varopoulos algebras and to partial differential equations.Comment: 42 page

Shilov boundary for "holomorphic functions" on a quantum matrix ball

We describe the Shilov boundary ideal for a q-analog of algebra of holomorphic functions on the unit ball in the space of $2\times 2$ matrices.Comment: 14 page

On bounded and unbounded idempotents whose sum is a multiple of the identity

We study bounded and unbounded representations of the $*$-algebra $Q_{n,\lambda}(*)$ generated by $n$ idempotents whose sum equals $\lambda e$ ($\lambda\in{\mathbb C}$, $e$ is the identity)

Operator synthesis. I. Synthetic sets, bilattices and tensor algebras

The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by projection-valued measures. We propose a ''coordinate'' approach which nevertheless does not use the technique of pseudo-integral operators, as well as a coordinate free one which allows to extend to non-separable spaces some important results and constructions of [W.Arveson, Operator Alegebras and Invariant subspaces, Ann. of Math. (2) 100 (1974)] and solve some problems posed there.Comment: 32 pages. to appear in Journal of Functional Analysi

Beurling-Fourier algebras on compact groups: spectral theory

For a compact group $G$ we define the Beurling-Fourier algebra $A_\omega(G)$ on $G$ for weights $\omega$ defined on the dual \what G and taking positive values. The classical Fourier algebra corresponds to the case $\omega$ is the constant weight 1. We study the Gelfand spectrum of the algebra realizing it as a subset of the complexification $G_{\mathbb C}$ defined by McKennon and Cartwright and McMullen. In many cases, such as for polynomial weights, the spectrum is simply $G$. We discuss the questions when the algebra $A_\omega(G)$ is symmetric and regular. We also obtain various results concerning spectral synthesis for $A_\omega(G)$.Comment: 37 page