Diagonalization of compact operators in Hilbert modules over finiteW*-algebras

It is known that a continuous family of compact self-adjoint operators can be diagonalized pointwise. One can consider this fact as a possibility of diagonalization of the compact operators on Hilbert modules over a commutativeW*-algebra. The aim of the present paper is to generalize this fact to a finiteW*-algebraA not necessarily commutative. We prove that for a compact operatorK acting on the right HilbertA-moduleH* _A dual toH _A under slight restrictions one can find a set of eigenvectorsx _i H* _A and a non-increasing sequence of eigenvalues _i A such thatK x _i=x _{i i} and the selfdual HilbertA-module generated by these eigenvectors is the wholeH* _A. As an application we consider the Schrödinger operator in a magnetic field with irrational magnetic flow as an operator acting on a Hilbert module over the irrational rotation algebraA and discuss the possibility of its diagonalization.     

Compact operators on Hilbert modules

We prove that an adjointable contraction acting on a countably generated Hilbert module over a separable unital C*-algebra is compact if and only if the set of its second contractive perturbations is separable.

     

On Hilbert modules over locally <Emphasis Type="Italic">C</Emphasis>*-algebras II

Summary In this paper we study the unitary equivalence between Hilbert modules over a locally <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>C^{*}$-algebra. Also, we prove a stabilization theorem for countably generated modules over an arbitrary locally $C^{*}$-algebra and show that a Hilbert module over a Fr\'{e}chet locally $C^{*}$-algebra is countably generated if and only if the locally $C^{*}$-algebra of all ``compact' operators has an approximate unit.     

Wigner's theorem in Hilbert -algebras of compact operators

Let be a Hilbert -module over the -algebra of all compact operators on a Hilbert space. It is proved that any function which preserves the absolute value of the -valued inner product is of the form , where is a phase function and is an -linear isometry. The result generalizes Molnár's extension of Wigner's classical unitary-antiunitary theorem.

     

On the classification of certain real rank zero C*-algebras

In this paper,we give a classification of real rank zero C^*-algebras that can be expressed as inductive limits of a sequence of a subclass of Elliott-Thomsen algebras C.     

Hilbert modules over a class of semicrossed products

Given the disk algebra and an automorphism , there is associated a non-self-adjoint operator algebra called the semicrossed product of with . Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules over and pairs of contractions and on satisfying . In this paper, we show that the orthogonally projective and Shilov Hilbert modules over correspond to pairs of isometries on satisfying . The problem of commutant lifting for is left open, but some related results are presented.     

The structure of compact disjointness preserving operators on continuous functions

Let T be a compact disjointness preserving linear operator from C₀(X) into C₀(Y), where X and Y are locally compact Hausdorff spaces. We show that T can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, T = Σ_n δ ?h_n for a (possibly finite) sequence {x_n }_n of distinct points in X and a norm null sequence {h_n }_n of mutually disjoint functions in C₀(Y). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Isomorphism of Hilbert modules over stably finite C-algebras

It is shown that if A is a stably finite C^∗-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C^∗-algebra that are not isomorphic.     

Essentially normal Hilbert modules and K-homology IV: Quasi-homogenous quotient modules of Hardy module on the polydisks

In this paper,we give a complete characterization for the essential normality of quasi-homogenous quotient modules of the Hardy modules H2 (D2).Also,we show that if d 3,then all the principle homogenous quotient modules of H 2 (Dd) are not essentially normal.     

Vector lattices of weakly compact operators on Banach lattices

A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.

     

Unconditional ideals of finite rank operators

Let X be a Banach space. We give characterizations of when is a u-ideal in for every Banach space Y in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when is a u-ideal in for every Banach space Y, when is a u-ideal in for every Banach space Y, and when is a u-ideal in for every Banach space Y.     

A note on supercyclic vectors of Hilbert space operators

Let H be an infinite dimensional complex Hilbert space and T be a bounded linear operator on H. We show that if there exists $x\in H$ such that the closure of $\{\alpha T^{n}x:\alpha \in \mathbb{C},n?0\}$ is H, then there is a subsequence ${(n_k)}_{k=1}^{\infty }$ such that the closed linear span of $\{T^{n_k}x:k?1\}$ is not the whole space H.     

Topological characterisation of weakly compact operators

Let X be a Banach space. Then there is a locally convex topology for X, the “Right topology,” such that a linear map T, from X into a Banach space Y, is weakly compact, precisely when T is a continuous map from X, equipped with the “Right” topology, into Y equipped with the norm topology. When T is only sequentially continuous with respect to the Right topology, it is said to be pseudo weakly compact. This notion is related to Pelczynski's Property (V).     

On algebras of polynomially compact operators

In this paper, we find sufficient and necessary conditions for a triangularizable closed algebra of polynomially compact operators to be commutative modulo the radical. We also prove that an algebraic algebra of operators of a bounded degree on a Banach space is triangularizable under some mild additional conditions. As a special case we obtain a result stating that every algebraic algebra of operators of bounded degree is triangularizable whenever its commutators are nilpotent operators.     

Equivalence of quotient Hilbert modules

LetM be a Hilbert module of holomorphic functions over a natural function algebraA(Ω), where Ω ⊆ ℂ ^m is a bounded domain. LetM ₀ ⊆M be the submodule of functions vanishing to orderk on a hypersurfaceZ ⊆ Ω. We describe a method, which in principle may be used, to construct a set of complete unitary invariants for quotient modulesQ =M ⊖M ₀ The invariants are given explicitly in the particular case ofk = 2.     

On equivariant embedding of Hilbert <Emphasis Type="Italic">C</Emphasis>* modules

We prove that an arbitrary (not necessarily countably generated) Hilbert G - module on a G - C ^* algebra admits an equivariant embedding into a trivial G - module, provided G is a compact Lie group and its action on is ergodic.     

Weakly compact composition operators on vector-valued BMOA

Weak compactness of the analytic composition operator f?f○φ is studied on BMOA(X), the space of X-valued analytic functions of bounded mean oscillation, and its subspace VMOA(X), where X is a complex Banach space. It is shown that the composition operator is weakly compact on BMOA(X) if X is reflexive and the corresponding composition operator is compact on the scalar-valued BMOA. A concrete example is given which shows that BMOA(X) differs from the weak vector-valued BMOA for infinite dimensional Banach spaces X.     

Some remarks on the real rank of non-unital C*-algebras

For a non-unital C-algebra , let be the C-algebra obtained from by adjoining an identity. In this paper we show that

where is a locally compact Hausdorff space with .

     

Equivalence of quotient Hilbert modules-II

For any open, connected and bounded set , let be a natural function algebra consisting of functions holomorphic on . Let be a Hilbert module over the algebra and let be the submodule of functions vanishing to order on a hypersurface . Recently the authors have obtained an explicit complete set of unitary invariants for the quotient module in the case of . In this paper, we relate these invariants to familiar notions from complex geometry. We also find a complete set of unitary invariants for the general case. We discuss many concrete examples in this setting. As an application of our equivalence results, we characterise certain homogeneous Hilbert modules over the bi-disc algebra.