Algebra Of Compact Operators Research Articles

Hyper-instability of Banach algebras

<abstract><p>In this paper, we introduce and study the concept of hyper-instability as a strong version of multiplicative instability. This concept provides a powerful tool to study the multiplicative instability of Banach algebras. It replaces the condition of the iterated limits in the definition of multiplicative instability with conditions that are easier to examine. In particular, special conditions are suggested for Banach algebras that admit bounded approximate identities. Moreover, these conditions are preserved under isomorphisms. This enlarges the class of studied Banach algebras. We prove that many interesting Banach algebras are hyper-unstable, such as $ C^* $-algebras, Fourier algebras, and the algebra of compact operators on Banach spaces, each under certain conditions.</p></abstract>

Non-linear traces on the algebra of compact operators and majorization

We study non-linear traces of the Choquet type and the Sugeno type on the algebra of compact operators. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and the Sugeno type respectively. There exists a close relation between non-linear traces of the Choquet type and majorization theory. We study trace class operators for non-linear traces of the Choquet type. More generally we discuss Schatten–von Neumann p-class operators for non-linear traces of the Choquet type. We determine when they form Banach spaces. This is an attempt at non-commutative integration theory for non-linear traces of the Choquet type on the algebra of compact operators. We also consider the triangle inequality for non-linear traces of the Sugeno type.

Approximate Identities in Algebras of Compact Operators on Frechet Spaces

Bounded approximate identity(bai) is a key concept in the theory of amenability of algebras. In this paper, we show that algebra of compact operators on Frechet space X has both the right and left locally bounded approximate identities. Sufficient conditions for the existence of these identities are established based on the geometry properties of the Frechet space X and its dual space X' respectively.

Weighted Cuntz algebras

We study the C∗-algebra T/K where T is the C∗-algebra generated by d weighted shifts on the Fock space of Cd, F(Cd), (where the weights are given by a sequence {Zk} of matrices Zk∈Mdk(C)) and K is the algebra of compact operators on the Fock space. If Zk=I for every k, T/K is the Cuntz algebra Od. We show that T/K is isomorphic to a Cuntz--Pimsner algebra and use it to find conditions for the algebra to be simple. We present examples of simple and of nonsimple algebras of this type. We also describe the C∗-representations of T/K.

Minimal hermitian compact operators related to a C*-subalgebra of K(H)

Let K(H) be the C*-algebra of compact operators on a separable Hilbert space H. This paper studies the properties of hermitian compact operators Y such that‖Y‖≤‖Y+W‖for all W∈W(H), where W(H) is a C*-subalgebra of K(H). Such a Y is called minimal related to W(H). The necessary and sufficient conditions that are required for Y to be minimal related to W(H) are characterized. Moreover, a particular C*-subalgebra W(H) such that there is a quasi-conditional expectation E from K(H) onto it is considered, and several examples are provided.

Hypercyclic Translation Operators on the Algebra of Compact Operators

In this paper we characterize hypercyclic translation operators on the space of all compact linear operators on a Hilbert space H. Also, we give some sufficient condition for a related cosine operator function to be chaotic or topologically transitive

Aspects of p-adic operator algebras

In this article, we propose a $p$-adic analogue of complex Hilbert space and consider generalizations of some well-known theorems from functional analysis and the basic study of operators on Hilbert spaces. We compute the $K$-theory of the analogue of the algebra of compact operators and the algebra of all bounded operators. This article contains a survey on results from the thesis of the first author.

The Operator-Valued Parallelism and Norm-Parallelism in Matrices

Let ℌ be a Hilbert space, and let K(ℌ) be the C*-algebra of compact operators on ℌ. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert K(ℌ)-module by employing the Birkhoff-James orthogonality. Among other things, we present a characterization of transitive relation of the norm-parallelism for elements in a certain Hilbert K(ℌ)-module. We also give some characterizations of the Schatten p-norms and the operator norm-parallelism for matrices.

Frame-Less Hilbert C$$^*$$-modules II

We show that if A is a non-unital $$C^*$$-algebra of compact operators, which is $$ *$$-isomorphic to $$\oplus _{i \in I} K(H_{i})$$, where I is an arbitrary index set and for every $$ i \in I $$, $$H_{i} $$ is a separable Hilbert space, then there exists a Hilbert $$A_1$$-module admitting no frames, where $$A_1$$ is the unitization of A.

The Covariant Stone–von Neumann Theorem for Actions of Abelian Groups on $$ C^{*} $$-Algebras of Compact Operators

In this paper, we formulate and prove a version of the Stone-von Neumann Theorem for every $ C^{\ast} $-dynamical system of the form $ (G,\mathbb{K}(\mathcal{H}),\alpha) $, where $ G $ is a locally compact Hausdorff abelian group and $ \mathcal{H} $ is a Hilbert space. The novelty of our work stems from our representation of the Weyl Commutation Relation on Hilbert $ \mathbb{K}(\mathcal{H}) $-modules instead of just Hilbert spaces, and our introduction of two additional commutation relations, which are necessary to obtain a uniqueness theorem. Along the way, we apply one of our basic results on Hilbert $ C^{\ast} $-modules to significantly shorten the length of Iain Raeburn's well-known proof of Takai-Takesaki Duality.

Additive Units of Inclusion Systems of Hilbert Modules over the $$C^*$$-Algebra of Compact Operators

We introduce the notion of additive units (addits) of a pointed inclusion system of Hilbert modules over the \(C^*\)-algebra of all compact operators acting on a Hilbert space G. By a pointed inclusion system, we mean an inclusion system with a fixed normalised reference unit. We prove that if G is a Hilbert space of finite dimension, then there is a bijection between the set of addits of a pointed inclusion system and the set of addits of the generated product system. We also consider addits of spatial product systems of Hilbert modules and, as an example, we find all continuous addits in the product system from Barreto et al. (J Funct Anal 212:121–181, 2004, Example 4.2.4).

Extensions of the Lax–Milgram theorem to Hilbert $$C^*$$-modules

We present three versions of the Lax-Milgram theorem in the framework of Hilbert C*-modules, two for those over W*-algebras and one for those over C*-algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert C*-modules over C*-algebras of compact operators, our Lax-Milgram theorem turns out to be valid for all of them. We also give several examples to illustrate our results, in particular, we show that the main theorem is not true for Hilbert modules over arbitrary C*-algebras.

Quantum Calculus and Ideals in the Algebra of Compact Operators

One of the goals of noncommutative geometry is to translate the basic notions of analysis into the language of Banach algebras. This translation is based on the quantization procedure. The arising operator calculus is called, following Connes, the quantum calculus. In this paper we give several assertions from this calculus concerning the interpretation of Schatten ideals of compact operators in a Hilbert space in terms of function theory. The main focus is on the case of Hilbert-Schmidt operators.

Orthogonal complementing in Hilbert $C^{*}$ -modules

We characterize orthogonally complemented submodules in Hilbert C∗-modules by their orthogonal closures. Applying Magajna’s characterization of Hilbert C∗-modules over C∗-algebras of compact operators by the complementing property of submodules, we give an elementary proof of Schweizer’s characterization of Hilbert C∗-modules over C∗-algebras of compact operators. Also, we prove analogous characterization theorems for C∗-algebras of compact operators related to topological properties of submodules of strict completions of Hilbert modules over a nonunital C∗-algebra.

Parallelism in Hilbert $K(\mathcal{H})$-modules

Let $\mathcal{H}$ be a Hilbert space and $K(\mathcal{H})$ be the $C^*$-algebra of compact operators on $\mathcal{H}$. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert $K(\mathcal{H})$-module $\mathcal{E}$ by employing the minimal projections on $\mathcal{H}$. In addition, we give some equivalence assertions about the norm-parallelism of compact operators on a Hilbert $C^*$-module.

Chain conditions for C*-algebras coming from Hilbert C*-modules

In this paper we define and study chain conditions for Hilbert C*-modules through their C*-algebras of compact operators and discuss their perseverance under Morita equivalence and tensor products. We show that these chain conditions are passed from the C*-algebra to its Hilbert module under certain conditions. We also study chain conditions for Hilbert modules coming from inclusion of C*-algebra with a faithful conditional expectation.

Naimark’s problem for graph $C^{*}$-algebras

Naimark’s problem asks whether a $C^{*}$-algebra that has only one irreducible $*$-representation up to unitary equivalence is isomorphic to the $C^{*}$-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable $C^{*}$-algebras and Type I $C^{*}$-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a $C^{*}$-algebra with $\aleph_{1}$ generators that is a counterexample to Naimark’s Problem. More precisely, they showed that the statement “There exists a counterexample to Naimark’s Problem that is generated by $\aleph_{1}$ elements.” is independent of the axioms of ZFC. Whether Naimark’s problem itself is independent of ZFC remains unknown. In this paper, we examine Naimark’s problem in the setting of graph $C^{*}$-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph $C^{*}$-algebras as well as for $C^{*}$-algebras of graphs in which each vertex emits a countable number of edges.

Self-adjoint approximations of the degenerate Schrödinger operator

The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have selfadjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on the abelian algebra can be presented by the Kraus decomposition with two terms. Both of these terms are corresponded to the unitary and shift components of Wold’s decomposition of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.

The Inner Corona Algebra of a C0(X)-Algebra

AbstractLet A = C(X) ⊗ K(H), where X is a compact Hausdorff space and K(H) is the algebra of compact operators on a separable infinite-dimensional Hilbert space. Let As be the algebra of strong*-continuous functions from X to K(H). Then As/A is the inner corona algebra of A. We show that if X has no isolated points, then As/A is an essential ideal of the corona algebra of A, and Prim(As/A), the primitive ideal space of As/A, is not weakly Lindelof. If X is also first countable, then there is a natural injection from the power set of X to the lattice of closed ideals of As/A. If X = βℕ\ℕ and the continuum hypothesis (CH) is assumed, then the corona algebra of A is a proper subalgebra of the multiplier algebra of As/A. Several of the results are obtained in the more general setting of C0(X)-algebras.

FRAMES IN HILBERT C*-MODULES AND MORITA EQUIVALENT C*-ALGEBRAS

AbstractWe show that the property of a C*-algebra that all its Hilbert modules have a frame, in the case of σ-unital C*-algebras, is preserved under Rieffel–Morita equivalence. In particular, we show that a σ-unital continuous-trace C*-algebra with trivial Dixmier–Douady class, all of whose Hilbert modules admit a frame, has discrete spectrum. We also show this for the tensor product of any commutative C*-algebra with the C*-algebra of compact operators on any Hilbert space.