-algebras of proper foliations

We study the -algebras of proper foliations. In case of finite depth they are described by a tower of exact sequences. We conclude with remarks about foliations almost without holonomy and AF-embeddability.

     

Commutativity criteria for Banach -algebras

Let be a Banach -algebra with an identity. Necessary and sufficient conditions are given for to be commutative modulo its -radical and for to be commutative if has a faithful -representation as operators on a Hilbert space.

     

Continuous-trace groupoid -algebras. III

Suppose that is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid -algebra has continuous trace if and only if there is a Haar system for the isotropy groupoid and the action of the quotient groupoid is proper on the unit space of .

     

On the definition of real -algebras

We prove that, if is a real -algebra having a predual , then is the unique predual of and the product of is -continuous.

     

Type I -algebras of real rank zero

We show that a separable -algebra of type I has real rank zero if and only if where is a modified dimension. We also show that a separable -algebra of type I has real rank zero if and only if it is an AF-algebra.

     

Actions of compact quantum groups on -algebras

In this paper we investigate a structure of the fixed point algebra under an action of compact matrix quantum group on a -algebra . We also show that the categories of -comodules in and inner endomorphisms restricted to the fixed point algebra coincide when the relative commutant of the fixed point algebra is trivial. Next we show a version of the Tannaka duality theorem for twisted unitary groups.

     

Normal operators in -algebras without nice approximants

The second author constructed a separable direct limit -algebra with real rank zero containing a normal element whose spectrum is the closed unit disk that is not the limit of normal elements in the limiting algebras, and is not a limit of normals in the algebra having finite spectrum. We use Fredholm index theory to modify and simplify this construction to obtain such examples that are not limits of any ``nice' types of elements.

     

Compact operators and the geometric structure of -algebras

Given a -algebra and an element , we give necessary and sufficient geometric conditions equivalent to the existence of a representation of so that is a compact or a finite-rank operator. The implications of these criteria on the geometric structure of -algebras are also discussed.

     

On -algebras associated with locally compact groups

Let be a locally compact group, and let denote the same group with the discrete topology. There are various associated to and We are concerned with the question of when these are isomorphic. This is intimately related to amenability. The results can be reformulated in terms of Fourier and Fourier-Stieltjes algebras and of weak containment properties of unitary representations.

     

On factor states of -algebras and their extensions

We obtain some results on the unique extension of (factor) states of -algebras which complement various existing results. Our results also lead to a class of -algebras whose states are -convex sums of factor states.

     

WKB-type approximations for second-order differential equations in -algebras

Rigorous asymptotic approximations of the WKB (or Liouville-Green) type are obtained for a basis of solutions to in the framework of -algebras. Both cases and are included, thus generalizing the classical theory for scalar equations developed by F.W.J. Olver to matrix as well as to infinite-dimensional equations.

     

The -algebras of infinite graphs

We associate -algebras to infinite directed graphs that are not necessarily locally finite. By realizing these algebras as Cuntz-Krieger algebras in the sense of Exel and Laca, we are able to give criteria for their uniqueness and simplicity, generalizing results of Kumjian, Pask, Raeburn, and Renault for locally finite directed graphs.

     

Simply connected nilpotent Lie groups with quasi-standard -algebras

The problem of when the group -algebra of a locally compact group is quasi-standard is investigated for certain simply connected nilpotent Lie groups. The characterization is in terms of the coadjoint orbit structure in the dual of the Lie algebra.

     

Homogeneous ideals in Wick -algebras

The necessary and sufficient condition for the family of homogeneous elements to determine a Wick ideal is presented. The structure of homogeneous Wick ideals with degree higher than 2 is discussed. For the braided operator a formula to calculate the largest cubic ideal when the quadratic one is known is obtained. Irreducible -representations of the -CAR algebra are classified.

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

The Furuta inequality in Banach -algebras

Let be real numbers with and Furuta (1987) proved that if bounded linear operators on a Hilbert space satisfy , then . This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach -algebra with continuous involution.