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 .

     

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 -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.

     

Refinement monoids with weak comparability and applications to regular rings and -algebras

We prove a cancellation theorem for simple refinement monoids satisfying the weak comparability condition, first introduced by K.C. O'Meara in the context of von Neumann regular rings. This result is then applied to von Neumann regular rings and -algebras of real rank zero via the monoid of isomorphism classes of finitely generated projective modules.

     

A note on invariance of spectrum for symmetric banach -algebras

Let be a symmetric Banach -algebra, let be a Banach algebra, and assume that . A result is proved giving conditions which imply that every element of has the same spectrum in both and .

     

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.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

Products of images

Let be the \u{C}ech-Stone remainder . We show that there exists a large class of images of such that whenever is a subset of of cardinality at most the continuum, then is again an image of . The class contains all separable compact spaces, all compact spaces of weight at most and all perfectly normal compact spaces.

     

Standard systems for semifinite O-algebras

We shall continue the study of standard systems which make it possible to develop the Tomita-Takesaki theory in O-algebras. The main purpose of this paper is to give the necessary and sufficient conditions for which a standard system of an O-algebra , a generalized vector and the commutant is unitarily equivalent to a standard system constructed by a standard tracial generalized vector for an O-algebra and a non-singular positive self-adjoint operator affiliated with the commutant of .

     

-algebras generated by a subnormal operator

Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under cannot be orthogonal to each other. Then we show that the -algebra generated by and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that -algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy space and the Bergman space of the unit ball.

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

Finite and -resolvability

A topological space is -resolvable if has disjoint dense subsets. In this paper, we prove that if is -resolvable for each positive integer , then is -resolvable.

     

Morita equivalence for crossed products by Hilbert -bimodules

We introduce the notion of the crossed product of a -algebra by a Hilbert -bimodule . It is shown that given a -algebra which carries a semi-saturated action of the circle group (in the sense that is generated by the spectral subspaces and ), then is isomorphic to the crossed product . We then present our main result, in which we show that the crossed products and are strongly Morita equivalent to each other, provided that and are strongly Morita equivalent under an imprimitivity bimodule satisfying as Hilbert -bimodules. We also present a six-term exact sequence for -groups of crossed products by Hilbert -bimodules.

     

Semiconstant measures on hyperbolic logics

We characterize the set of all semiconstant measures on the hyperbolic logics of projections in indefinite metric spaces and describe the set of all probability measures on these logics.

     

The relationship between fragmentable spaces and class spaces

In this note, we show that each fragmentable space introduced by Jayne and Rogers in 1985 is of class which was introduced by Kenderov in 1984. Our example shows that a space which is of class may not be a fragmentable space.

     

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.

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .