On Definition of Skew Frames

Skew frames represent a common generalization of frames and orthomodular lattices. They could serve as Lindenbaum algebras of quantum intuitionistic logic as well as invariants of noncommutative C*-algebras. It is shown that lattices of open projections with skew (partial) operations are complete invariants of C*-algebras and that these operations are preserved by morphims of C*-algebras.     

Comparison Theory and Smooth Minimal C*-Dynamics

We prove that the C*-algebra of a minimal diffeomorphism satisfies Blackadar’s Fundamental Comparability Property for positive elements. This leads to the classification, in terms of K-theory and traces, of the isomorphism classes of countably generated Hilbert modules over such algebras, and to a similar classification for the closures of unitary orbits of self-adjoint elements. We also obtain a structure theorem for the Cuntz semigroup in this setting, and prove a conjecture of Blackadar and Handelman: the lower semicontinuous dimension functions are weakly dense in the space of all dimension functions. These results continue to hold in the broader setting of unital simple ASH algebras with slow dimension growth and stable rank one. Our main tool is a sharp bound on the radius of comparison of a recursive subhomogeneous C*-algebra. This is also used to construct uncountably many non-Morita-equivalent simple separable amenable C*-algebras with the same K-theory and tracial state space, providing a C*-algebraic analogue of McDuff’s uncountable family of II₁ factors. We prove in passing that the range of the radius of comparison is exhausted by simple C*-algebras. This research was supported in part by an NSERC Discovery Grant.     

States on Partial Rings

Recently, the structure theory of JB*-tripleshas received considerable attention. The reason is thatJB*-triples and those JB*-triples which are dual spaces,the JBW*-triples, not only form natural generalizations of Jordan C*-algebras and C*-algebras, andJordan W*-algebras and W*-algebras, but also provide acontext for the study of infinite-dimensional holomorphyand infinite-dimensional Lie algebras. In a JBW*-triple the tripotents play the role of the projectionsin a W*-algebra. In analogy to the projection lattice ofa W*-algebra, we investigate the partial ring oftripotents of JBW*-triple. Unlike on W*-algebras, states, i.e., positive normalized homomorphismsfrom the partial ring of tripotents of a JBW*-tripleinto the partial ring of real numbers, have not yet beendiscussed in the literature. We show that the partial ring of tripotents of a JBW*-tripleadmits a unital set of Jauch-Piron states.     

A New Light on Nets of C*-Algebras and Their Representations

The present paper deals with the question of representability of nets of C*-algebras whose underlying poset, indexing the net, is not upward directed. A particular class of nets, called C*-net bundles, is classified in terms of C*-dynamical systems having as group the fundamental group of the poset. Any net of C*-algebras has a canonical morphism into a C*-net bundle, the enveloping net bundle, which generalizes the notion of universal C*-algebra given by Fredenhagen to nonsimply connected posets. This allows a classification of nets; in particular, we call injective those nets such that the canonical morphism is faithful. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity to a generalized Čech cocycle of the net, and this allows us to give examples of nets exhausting the above classification.     

The Picard Groupoid in Deformation Quantization

In this Letter we give an overview on recent developments in representation theory of star product algebras. In particular, we relate the *-representation theory of *-algebras over rings C = R(i) with an ordered ring R and i²=?1 to the *-representation theory of *-algebras over and point out some properties of the Picard groupoid corresponding to the notion of strong Morita equivalence. Some Morita invariants are interpreted as arising from actions of this groupoid     

The Picard Groupoid in Deformation Quantization

In this Letter we give an overview on recent developments in representation theory of star product algebras. In particular, we relate the *-representation theory of *-algebras over rings C = R(i) with an ordered ring R and i²=–1 to the *-representation theory of *-algebras over and point out some properties of the Picard groupoid corresponding to the notion of strong Morita equivalence. Some Morita invariants are interpreted as arising from actions of this groupoid     

Ergodic Coactions with Large Multiplicity and Monoidal Equivalence of Quantum Groups

We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.     

The Muhly–Renault–Williams Theorem for Lie Groupoids and its Classical Counterpart

A theorem of Muhly–Renault–Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.     

Extremal decomposition of Wightman functions and of states on nuclear *-algebras by Choquet theory

We give a short proof for the decomposability of states on nuclear *-algebras into extremal states by using the integral decompositions of Choquet and the nuclear spectral theorem, recovering a recent result by Borchers and Yngvason. The decomposition of Wightman fields into irreducible fields is a special case of this. We also indicate a quick solution of the moment problem on nuclear spaces.     

Characterizations of Categories of Commutative C*-Subalgebras

We aim to characterize the category of injective *-homomorphisms between commutative C*-subalgebras of a given C*-algebra A. We reduce this problem to finding a weakly terminal commutative subalgebra of A, and solve the latter for various C*-algebras, including all commutative ones and all type I von Neumann algebras. This addresses a natural generalization of the Mackey–Piron programme: which lattices are those of closed subspaces of Hilbert space? We also discuss the way this categorified generalization differs from the original question.     

States of physical systems

States of physical systems may be represented by states onB*-algebras, satisfying certain requirements of physical origin. We discuss such requirements as are associated with the presence of unbounded observables or invariance under a group. It is possible in certain cases to obtain a unique decomposition of states invariant under a group into extremal invariant states. Our main results is such a decomposition theorem when the group is the translation group in dimensions and theB*-algebra satisfies a certain locality condition. An application of this theorem is made to representations of the canonical anticommutation relations.     

Some Examples of Graded C*-Algebras

We apply the theory of C*-algebras graded by a semilattice to crossed products of C*-algebras. We establish a correspondence between the spectrum of commutative graded C*-algebras and the spectrum of their components. This will allow us to compute the spectrum of some commutative examples of graded C*-algebras.     

The smallestC*-algebra for canonical commutations relations

We consider theC*-algebras which contain the Weyl operators when the symplectic form which defines the C.C.R. is possibly degenerate. We prove that the C.C.R. are all obtained as a quotient of a universalC*-algebra by some of its ideals, and we characterize all these ideals.     

Self-adjoint algebras of unbounded operators

Unbounded *-representations of *-algebras are studied. Representations called self-adjoint representations are defined in analogy to the definition of a self-adjoint operator. It is shown that for self-adjoint representations certain pathologies associated with commutant and reducing subspaces are avoided. A class of well behaved self-adjoint representations, called standard representations, are defined for commutative *-algebras. It is shown that a strongly cyclic self-adjoint representation of a commutative *-algebra is standard if and only if the representation is strongly positive, i.e., the representations preserves a certain order relation. Similar results are obtained for *-representations of the canonical commutation relations for a finite number of degrees of freedom.Work supported in part by U.S. Atomic Energy Commission under Contract AT(30-1)-2171 and by the National Science Foundation.Alfred P. Sloan Foundation Fellow.     

Representation of completely positive maps between partial *-algebras

A characterization of the invariant completely positive conjugate-bilinear maps from an arbitrary partial *-algebra to a semiassociative, locally convex partial *-algebra is given. The result generalizes Stinespring's characterization of completely positive maps onC*-algebras, as well as its recent extensions by a number of authors.     

Large groups of automorphisms of C*-algebras

Groups of *-automorphisms ofC*-algebras and their invariant states are studied. We assume the groups satisfy a certain largeness condition and then obtain results which contain many of those known for asymptotically abelianC*-algebras and for inner automorphisms and traces ofC*-algebras. Our key result is the construction in certain finite cases, where the automorphisms are spatial, of an invariant linear map of theC*-algebra onto the fixed point algebra carrying with it most of the relevant information.     

Coherent States on Lie Algebras: A Constructive Approach

We generalize the notion of coherent states toarbitrary Lie algebras by making an analogy with the GNSconstruction in C*-algebras. The method is illustratedwith examples of semisimple and nonsemisimple finite-dimensional Lie algebras as well as loopand Kac–Moody algebras. A deformed addition on theparameter space is also introduced simplifying someexpressions and some applications to conformal field theory are pointed out, e.g., differentialoperator and free field realizations found.     

Noncommutative mean ergodic theorem for partialW *-dynamical semigroups

A noncommutative mean ergodic theorem for dynamical semigroups of maps on partial W^*-algebras of linear operators from a pre-Hilbert space into its completion is proved. This generalizes a similar result of Watanabe for dynamical semigroups of maps onW ^*-algebras of operators.     

Non-Commutative Methods for the K-Theory of C*-Algebras of Aperiodic Patterns from Cut-and-Project Systems

We investigate the C*-algebras associated to aperiodic structures called model sets obtained by the cut-and-project method. These C*-algebras are Morita equivalent to crossed product C*-algebras obtained from dynamics on a disconnected version of the internal space. This construction may be made from more general data, which we call a hyperplane system. From a hyperplane system, others may be constructed by a process of reduction and we show how the C*-algebras involved are related to each other. In particular, there are natural elements in the Kasparov KK-groups for the C*-algebra of a hyperplane system and that of its reduction. The induced map on K-theory fits in a six-term exact sequence. This provides a new method of the computation of the K-theory of such C*-algebras which is done completely in the setting of non-commutative geometry.