Algebra Of Compact Operators Research Articles

Projection lifting from the corona algebra using UCT

In this paper we consider the projection lifting from the corona algebra of C(X)⊗B where X is a compact, Hausdorff, finite dimensional space and B is either the algebra of compact operators on a separable infinite dimensional Hilbert space or a stable Kirchberg algebra whose multiplier algebra has real rank zero. Assuming a geometric condition on the projection, we suggest a new approach to the projection lifting using major classification results of continuous fields of C⁎-algebras.

Inclusion systems of Hilbert modules over the C^∗-algebra of compact operators

Inclusion systems of Hilbert modules over the C^∗-algebra of compact operators

Elliptic complexes over C∗-algebras of compact operators

For a $C^*$-algebra $A$ of compact operators and a compact manifold $M,$ we prove that the Hodge theory holds for $A$-elliptic complexes of pseudodifferential operators acting on smooth sections of finitely generated projective $A$-Hilbert bundles over $M.$ For these $C^*$-algebras, we get also a topological isomorphism between the cohomology groups of an $A$-elliptic complex and the space of harmonic elements. Consequently, the cohomology groups appear to be Banach spaces. We prove as well, that if the Hodge theory holds for a complex in the category of Hilbert $A$-modules and continuous adjointable Hilbert $A$-module homomorphisms, the complex is self-adjoint parametrix possessing.

The corona algebra of the stabilized Jiang–Su algebra

Let Z be the Jiang–Su algebra and K the C ⁎ -algebra of compact operators on an infinite dimensional separable Hilbert space. We prove that the corona algebra M ( Z ⊗ K ) / Z ⊗ K has real rank zero. We actually prove a more general result.

Norm-parallelism in the geometry of Hilbert [formula omitted]-modules

Utilizing the Birkhoff–James orthogonality, we present some characterizations of the norm-parallelism for elements of B(ℋ) defined on a finite dimensional Hilbert space, elements of a Hilbert C∗-module over the C∗-algebra of compact operators and elements of an arbitrary C∗-algebra. We also consider the characterization of norm parallelism problem for operators on a finite dimensional Hilbert space when the operator norm is replaced by the Schatten p-norm. Some applications and generalizations are discussed for certain elements of a Hilbert C∗-module.

On the ideals of a C*-algebra generated by a family of partial isometries and multipliers

The C*-subalgebra of the algebra of all bounded operators on the Hilbert space l2 generated by the multiplier algebra and a family of partial isometries satisfying certain relations is considered. The ideals of the algebra under examination and of its quotient by the algebra of compact operators are studied. It is shown that the quotient algebra can be represented as a direct sum of two principal ideals and has nontrivial center.

Which multiplier algebras are $W^*$-algebras?

We consider the question of when the multiplier algebra $M(\mathcal{A})$ of a $C^*$-algebra $\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebra $\mathcal{A}$ which is Morita equivalent to a $ W^*$-algebra must be a $ W^*$-algebra.

Kinds of derivations on Hilbert $C^*$-modules and their operator algebras

Let M be a Hilbert C -module. A linear mapping dW M! M is called a deriva- tion if d. ·/D ·C ·C d· for all x;y;·2 M. We give some results for derivations and automatic continuity of them on M. Also, we will characterize generalized derivations and strong higher derivations on the algebra of compact operators and adjointable operators of HilbertC -modules, respectively. 2010 Mathematics Subject Classification: 46L08; 16W25

EP modular operators and their products

We study first EP modular operators on Hilbert C⁎-modules and then we provide necessary and sufficient conditions for the product of two EP modular operators to be EP. These enable us to extend some results of Koliha (2000) [13] for an arbitrary C⁎-algebra and the C⁎-algebras of compact operators.

On a geometric realization of C*-algebras

Further to the functional representations of C$^*$-algebras proposed by R. Cirelli, A. Mania and L. Pizzocchero, we consider in this article the uniform K\"ahler bundle (in short, UKB) description of some C$^*$-algebraic subjects. In particular, we obtain an one-to-one correspondence between closed ideals of a C$^*$-algebra $\mathcal{A}$ and full uniform K\"ahler sub bundles over open subsets of the base space of the UKB associated with $\mathcal{A}$. In addition, we will present a geometric description of the pure state space of hereditary C$^*$-subalgebras and show that that if $\mathcal{B}$ is a hereditary C$^*$-subalgebra of $\mathcal{A}$, the UKB of $\mathcal{B}$ is a kind of K\"ahler subbundle of the UKB of $\mathcal{A}$. As a simple example, we consider hereditary C$^*$-subalgebras of the C$^*$-algebra of compact operators on a Hilbert space. Finally, we remark that hereditary C$^*$-subalgebras also naturally can be characterized as uniform holomorphic Hilbert subbundles.

Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and \S, where \cpt denotes the C^*-algebra of compact operators and \S denotes the algebra of Schatten class operators. We introduce assembly maps with finite coefficients and under an additional hypothesis, we prove that such a group also satisfies the algebraic K-theoretic Novikov conjecture over \bar{\mathbb{Q}} and \mathbb{C} with finite coefficients. For all torsion free Gromov hyperbolic groups G, we demonstrate that the canonical algebra homomorphism \cpt[G]\map C^*_r(G)\hat{\otimes}\cpt induces an isomorphism between their algebraic K-theory groups.

Models of p-adic mechanics

We segregate the class of ultrametric (p-adic) systems within the standard models of classical and quantum mechanics. We show that ultrametric models can be described in the language of standard models but also have several distinguishing properties. In particular, we show that a stronger Poincare recurrence theorem holds for classical ultrametric dynamical systems. As an example of a quantum p-adic system, we consider the algebra of commutation relations of the one-dimensional quantum mechanics. We show that this algebra, as in the real case, is isomorphic to the algebra of compact operators.

Homotopy Classification of Projections in the Corona Algebra of a Non-simple C*-algebra

AbstractWe study projections in the corona algebra of C(X) ⊗ K, where K is the C*-algebra of compact operators on a separable infinite dimensional Hilbert space and X = [0, 1], [0,∞), (−∞,∞), or [0, 1]/﹛0, 1﹜. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in K0, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

Finite-dimensional Hilbert C*-modules

In this paper we obtain a characterization of finite-dimensional Hilbert C*-modules. It is known that those are the modules for which both underlying C*-algebras are finite-dimensional. We show that such modules can be described by a certain property of bounded sequences of their elements. It turns out that similar property leads to another characterization of Hilbert C*-modules over C*-algebras of compact operators.

Solvable Jordan algebras of compact operators

It is proved that a Jordan algebra of compact operators which is closed is either an Engel Jordan algebra, or contains a nonzero finite rank operator; Moreover, it is showed that any solvable Jordan algebra of compact operators on an infinite dimensional Banach space is triangularizable.

EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

Compactness properties of operator multipliers

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, vollständig beschränkte Abbildungen mit Werten in einer nuklearen C ∗ -Algebra, Diplomarbeit, Universität des Saarlandes, Saarbrücken, 1982].

Certain invariants of extension algebras of $$C({\mathbb{T}}^{2})$$

In this paper, we compute certain invariants of extension algebras of the torus algebra \(C({\mathbb{T}}^{2})\) by \({\mathcal{K}}\), where \({\mathcal{K}}\) is the C*-algebra of compact operators on an infinite dimensional separable Hilbert space H. These extension algebras are also constructed up to isomorphism.

Another characterization of Hilbert [formula omitted]-modules over compact operators

In this paper we discuss a certain class of sets of generators for a full Hilbert C ∗ -module which characterize Hilbert C ∗ -modules over the C ∗ -algebra of compact operators. An example of such sets of generators are orthonormal bases.

Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules

AbstractC*-algebrasAof compact operators are characterized as those C*-algebras of coefficients of Hilbert C*-modules for which (i) every boundedA-linear operator between two HilbertA-modules possesses an adjoint operator, (ii) the kernels of all bounded A-linear operators between HilbertA-modules are orthogonal summands, (iii) the images of all boundedA-linear operators with closed range between HilbertA-modules are orthogonal summands, and (iv) for every HilbertA-module every HilbertA-submodule is a topological summand. Thus, the theory of Hilbert C*-modules over C*-algebras of compact operators has similarities with the theory of Hilbert spaces. In passing, we obtain a general closed graph theorem for bounded module operators on arbitrary Hilbert C*-modules.