Utilizing the linking algebra of a Hilbert $C^*$-module $\big(\mathscr{V}, {\|\!\cdot\!\|}_{_{\mathscr{V}}}\big)$, we introduce $\Omega(x)$ as definition numerical radius for an element $x\in\mathscr{V}$ and then show that $\Omega(\cdot)$ is norm on $\mathscr{V}$ such $\frac{1}{2}{\|x\|}_{_{\mathscr{V}}} \leq \Omega(x) {\|x\|}_{_{\mathscr{V}}}$. In addition, obtain equivalent condition $\Omega(...

It’s well known that the functional Hilbert space H2 over the unit ball Bd ⊂ C , with kernel function K(z, ω) = 1 1−z1ω1−···−zdωd , admits a natural A(Bd)-module structure. We show the rank of a nonzero submodule M ⊂ H2 is infinity if and only if M is of infinite codimension. Together with Arveson’s dilation theory, our result shows that Hilbert modules stand in stark contrast with Hilbert basi...

The goal of this talk is to prove an analog of the Beilinson-Bernstein localization theorem for Cherednik algebras. Strictly speaking we will only do this for categories O, in fact, the localization theorem for all modules follows from here. Let us recall the notation and some definitions. We consider the reflection representation h of the symmetric group Sn. By X we denote the “normalized” Hil...

In this paper we introduce the notion of linking algebra of a Hilbert module over a locally C -algebra and we extend in the context of locally C -algebras a result of Brown, Green and Rie¤el [Paci…c J.,1977] which states that two C -algebras are strongly Morita equivalent if and only if they are isomorphic with two complementary full corners of a C -algebra. Full text

We review some of our results from the theory of product systems of Hilbert modules [BS00, BBLS00, Ske00a, Ske01, Ske02, Ske03]. We explain that the product systems obtained from a CP-semigroup in [BS00] and in [MS02] are commutants of each other. Then we use this new commutant technique to construct product systems from E0–semigroups on Ba(E) where E is a strongly full von Neumann module. (Thi...

Let X and Y be Hilbert C *-modules over a C *-algebra, and ϕ : X × Y → [0, ∞) be a function. A (not necessarily linear) mapping f : X → Y is called a ϕ-perturbed adjointable mapping if there exists a (not necessarily linear) mapping g : Y → X such that In this paper, we investigate the stability of adjointable mappings on Hilbert C *-modules and discuss the case where the modules are self-dual....

‎let $pa$ be a commutative banach algebra and $ex$ be a left banach $pa$-module‎. ‎we study the set‎ ‎$dec_{pa}(ex)$ of all elements in $pa$ which induce a decomposable multiplication operator on $ex$‎. ‎in the case $ex=pa$‎, ‎$dec_{pa}(pa)$ is the well-known apostol algebra of $pa$‎. ‎we show that $dec_{pa}(ex)$ is intimately related with the largest spectrally separable subalgebra of $pa$ and...

In this paper we give characterizations of essential left ideals of a C*-algebra A in terms of their properties as operator A-modules. Conversely, we seek C*-algebraic characterizations of those ideals J in A such that A is an essential extension of J in various categories of operator modules. In the case of two-sided ideals, we prove that all the above concepts coincide. We obtain results, ana...