A remark onC*-algebras

It is shown that any complex Banach algebra with hermitean involution and the weakC*-property |x|²=|x ²| for allx=x* is aC*-algebra.The research in this paper was partially supported by the U. S. Army Research Office, Durham.     

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C ^*-Hopf algebra H and its dual Ĥ we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory. is the observable algebra of a generalized quantum spin chain with H-order and Ĥ-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If is a group algebra then becomes an ordinary G-spin model. We classify all DHR-sectors of – relative to some Haag dual vacuum representation – and prove that their symmetry is described by the Drinfeld double . To achieve this we construct localized coactions and use a certain compressibility property to prove that they are universal amplimorphisms on . In this way the double can be recovered from the observable algebra as a universal cosymmetry. Received: 4 September 1995\,/\,Accepted: 3 December 1996     

CQG algebras: A direct algebraic approach to compact quantum groups

The purely algebraic notion of CQG algebra (algebra of functions on a compact quantum group) is defined. In a straightforward algebraic manner, the Peter-Weyl theorem for CQG algebras and the existence of a unique positive definite Haar functional on any CQG algebra are established. It is shown that a CQG algebra can be naturally completed to aC ^*-algebra. The relations between our approach and several other approaches to compact quantum groups are discussed.     

A Locally Compact Quantum Group Analogue of the Normalizer of SU(1,1) in SL(2, ℂ)

 S.L. Woronowicz proved in 1991 that quantum SU(1,1) does not exist as a locally compact quantum group. Results by L.I. Korogodsky in 1994 and more recently by Woronowicz gave strong indications that the normalizer of SU(1,1) in SL(2,ℂ) is a much better quantization candidate than SU(1,1) itself. In this paper we show that this is indeed the case by constructing , a new example of a unimodular locally compact quantum group (depending on a parameter 0<q<1) that is a deformation of . After defining the underlying von Neumann algebra of we use a certain class of q-hypergeometric functions and their orthogonality relations to construct the comultiplication. The coassociativity of this comultiplication is the hardest result to establish. We define the Haar weight and obtain simple formulas for the antipode and its polar decomposition. As a final result we produce the underlying C ^* -algebra of . The proofs of all these results depend on various properties of q-hypergeometric ₁ϕ₁ functions. Received: 28 June 2001 / Accepted: 25 July 2002 Published online: 10 December 2002 RID="*" ID="*" Post-doctoral researcher of the Fund for Scientific Research – Flanders (Belgium) (F.W.O.) Communicated by L. Takhtajan     

The higher order Hamiltonian structures for the modified classical Yang-Baxter equation

We consider constructing the higher order Hamiltonian structures on the dual of the Lie algebra from the first Hamiltonian structure of the coadjoint orbit method. For this purpose we show that the structure of the Lie algebrag is inherited to the algebra of vector fields ong ^* through the solution of the Modified Classical Yang-Baxter equation (Classicalr matrix). We study the algebra that generates the compatible Poisson brackets.This work was supported by Grant Aid for Scientific Research, the Ministry of Education.     

Unbounded elements affiliated withC*-algebras and non-compact quantum groups

The affiliation relation that allows to include unbounded elements (operators) into theC ^*-algebra framework is introduced, investigated and applied to the quantum group theory. The quantum deformation of (the two-fold covering of) the group of motions of Euclidean plane is constructed. A remarkable radius quantization is discovered. It is also shown that the quantumSU(1, 1) group does not exist on theC ^*-algebra level for real value of the deformation parameter.Supported by Japan Society for the Promotion of Science     

C*-algebras and Mackey's axioms

A non-commutative version of probability theory is outlined, based on the concept of a*-algebra of operators (sequentially weakly closedC*-algebra of operators). Using the theory of*-algebras, we relate theC*-algebra approach to quantum mechanics as developed byKadison with the probabilistic approach to quantum mechanics as axiomatized byMackey. The*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by theW*-algebra approach. By considering the*-algebra, rather than the von Neumann algebra, generated by the givenC*-algebraA in its reduced atomic representation, we show that a difficulty encountered byGuenin concerning the domain of a state can be resolved.     

Noncommutative lattices as finite approximations

Lattice discretizations of continuous manifolds are common tools used in a variety of physical contexts. Conventional discrete approximations, however, cannot capture all aspects of the original manifold, notably its topology. In this paper we discuss an approximation scheme due to Sorkin (1991) which correctly reproduces important topological aspects of continuum physics. The approximating topological spaces are partially ordered sets (posets), the partial order encoding the topology. Now, the topology of a manifold M can be reconstructed from the commutativè C^*algebra C(M) of continuous functions defined on it. In turn, this algebra is generated by continuous probability densities in ordinary quantum physics on M. The latter also serves to specify the domains of observables like the Hamiltonian. For a poset, the role of this algebra is assumed by a noncommutative C^*-algebra A. This fact makes any poset a genuine ‘noncommutative’ (‘quantum’) space, in the sense that the algebra of its ‘continuous functions’ is a noncommutative C^*-algebra. We therefore also have a remarkable connection between finite approximations to quantum physics and noncommutative geometries. We use this connection to develop various approximation methods for doing quantum physics using A.     

On the Borel structure ofC*-algebras

We provide a method of embedding aC*-algebra in aC*-algebra called its -envelope. is contained in the enveloping algebra of but is generally much smaller, and if is commutative with identity then can be identified with the algebra of bounded Baire functions on the spectrum of. The main result is to completely determine the structure of for all separable G. C. R. algebras. This provides a good basis for a non-commutative theory of probability.We should like to thankJ. T. Lewis, G. W. Mackey andR. J. Plymen, who have given us considerable encouragement and insight into the quantum mechanical relevance of the ideas developed here.     

Symplectic structure of the moduli space of flat connection on a Riemann surface

We consider the canonical symplectic structure on the moduli space of flatg-connections on a Riemann surface of genusg withn marked points. Forg being a semisimple Lie algebra we obtain an explicit efficient formula for this symplectic form and prove that it may be represented as a sum ofn copies of Kirillov symplectic form on the orbit of dressing transformations in the Poisson-Lie groupG ^* andg copies of the symplectic structure on the Heisenberg double of the Poisson-Lie groupG (the pair (G, G ^*) corresponds to the Lie algebrag).Supported by Swedish Natural Science Research Council (NFR) under the contract F-FU 06821-304Supported in part by a Soros Foundation Grant awarded by the American Physical Society     

Generalized Diagonal Crossed Products and Smash Products for Quasi-Hopf Algebras. Applications

In this paper we introduce generalizations of diagonal crossed products, two-sided crossed products and two-sided smash products, for a quasi-Hopf algebra H. The results we obtain may then be applied to H ^*-Hopf bimodules and generalized Yetter-Drinfeld modules. The generality of our situation entails that the “generating matrix” formalism cannot be used, forcing us to use a different approach. This pays off because as an application we obtain an easy conceptual proof of an important but very technical result of Hausser and Nill concerning iterated two-sided crossed products.Research partially supported by the EC programme LIEGRITS, RTN 2003, 505078, and by the bilateral projects “Hopf Algebras in Algebra, Topology, Geometry and Physics” and “New techniques in Hopf algebras and graded ring theory” of the Flemish and Romanian Ministries of Research. The first two authors have been also partially supported by the programme CERES of the Romanian Ministry of Education and Research, contract no. 4-147/2004.     

Quantum Spheres and Projective Spaces as Graph Algebras

 The C ^* -algebras of continuous functions on quantum spheres, quantum real projective spaces, and quantum complex projective spaces are realized as Cuntz-Krieger algebras corresponding to suitable directed graphs. Structural results about these quantum spaces, especially about their ideals and K-theory, are then derived from the general theory of graph algebras. It is shown that the quantum even and odd dimensional spheres are produced by repeated application of a quantum double suspension to two points and the circle, respectively. Received: 31 January 2001 / Accepted: 29 July 2002 Published online: 7 November 2002 RID="*" ID="*" Supported by grant No. R04–2001–000–00117–0 from the Korea Science & Engineering Foundation. RID="**" ID="**" Partially supported by the Research Management Committee of the University of Newcastle.     

TheC*-algebra of bosonic strings

We give a rigorous definition of Witten'sC ^*-string-algebra. To this end we present a new construction ofC ^*-algebras associated to special geometric situations (Kähler foliations) and generalize this later construction to the string case. Through this we get a natural geometrical interpretation of the string of semi-infinite forms as well as the fermionic algebra structure. Using the (non-commutative) geometric concepts for investigating the string algebra we get a natural Fredholm module representation of dimension 26+.Work partially supported by the DFG (under contract MU 75712.3)     

Clustering for algebraicK-systems

We prove that for a von Neumann algebra that is an algebraicK system with respect to some automorphism, the invariant state isK-clustering andr-clustering. Further, we study by using examples how far the von Neumann algebra inherits theK property from the underlyingC ^* algebra.     

Local Index Formula on the Equatorial Podleś Sphere

We discuss spectral properties of the equatorial Podleś sphere S _q ². As a preparation we also study the ‘degenerate’ (i.e. q=0) case (related to the quantum disk). Over S _q ² we consider two different spectral triples:one related to the Fock representation of the Toeplitz algebra and the isopectral one given in [7]. After the identification of the smooth pre-C ^*-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the C ^*-algebra (the nontrivial generator of the K ₀-group) as well as the pairing with the q-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.     

Algebraic Rieffel induction, formal Morita equivalence, and applications to deformation quantization

In this paper, we consider algebras with involution over a ring C which is given by the quadratic extension by i of an ordered ring R. We discuss the ^*-representation theory of such ^*-algebras on pre-Hilbert spaces over C and develop the notions of Rieffel induction and formal Morita equivalence for this category analogously to the situation for C^*-algebras. Throughout this paper, the notion of positive functionals and positive algebra elements will be crucial for all constructions. As in the case of C^*-algebras, we show that the GNS construction of ^*-representations can be understood as Rieffel induction and, moreover, that formal Morita equivalence of two ^*-algebras, which is defined by the existence of a bimodule with certain additional structures, implies the equivalence of the categories of strongly non-degenerate ^*-representations of the two ^*-algebras. We discuss various examples like finite rank operators on pre-Hilbert spaces and matrix algebras over ^*-algebras. Formal Morita equivalence is shown to imply Morita equivalence in the ring-theoretic framework. Finally, we apply our considerations to deformation theory and in particular to deformation quantization and discuss the classical limit and the deformation of equivalence bimodules.     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Positive linear maps on operator algebras

Given a family of completely positive maps, indexed by a group, from aC*-algebra into itself, we are concerned with its dilation to a group of *-automorphisms on a larger algebra. A Schwarz-type inequality forn-positive *-linear mappings from an involutive algebra into the bounded linear operators on a hilbert space is obtained. Strongly continuous one-parameter semigroups and groups onC*-algebras, which have certain positivity properties, are studied.     

Isotropic solutions of the Einstein-Boltzmann equations

We prove that theP()₂ quantum field theory satisfies the spectral condition. The space time translationa=(x, t) is implemented by the unitary groupU(a)=exp(itH–ixP), and the joint spectrum of the energy operatorH and the momentum operatorP is contained in the forward cone. We also obtain bounds on certain vacuum expectation values of products of field operators. Our proofs involve an analysis of the limitV for approximate theories in a periodic box of volumeV. Assuming the existence of a uniform mass gap, we are able to establish all the Wightman axioms with the exception of the Lorentz invariance of the vacuum.Supported in part by the Air Force Office of Scientific Research, Contract AF 49(638)-1719.Alfred P. Sloan Foundation Fellow. Supported in part by the Air Force Office of Scientific Research, Contract F 44620-70-C-0030.