Twisted Lie Group C*-Algebras as Strict Quantization

A nonzero 2-cocycle Z2(g, R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie–Poisson structure on the dual Lie algebra g*, leading to a Poisson algebra C (g*()). Similarly, a multiplier c Z2(G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C-algebra C*(G,c). Further to some superficial yet enlightening analogies between C (g*()) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Plancks constant assumes values in (Z\{0})-1. This means that there exists a continuous field of C*-algebras, indexed by 0 (Z\{0})-1, for which A0= C0(g*) and A=C*(G,c) for 0, along with a cross-section of the field satisfying Diracs condition asymptotically relating the commutator in A to the Poisson bracket on C(g*()). Note that the quantization of does not occur for =0.     

Nonassociative Strict Deformation Quantization of C*-Algebras and Nonassociative Torus Bundles

In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with H-flux. In particular, the Octonions fit nicely into our theory.     

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     

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.     

Lp-Spaces for C*-Algebras with a State

We present here a construction of noncommutative L ^p -spaces for a C ^*-algebrawith respect to a state on the algebra. Their properties are deduced fromwell-established properties of corresponding Haagerup and Kosaki spaces. Twoexamples are considered.     

Re-explanation of Weyl Quantization Scheme via Weyl Ordering Approach

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl ordered product of operators, i.e., the Weyl correspondence rule can be reconstructed by classical functions＇ Fourier transformation followed by an inverse Fourier transformation within Weyl ordering of operators. As an application of this reconstruction, we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.     

Crossed Module Actions on Continuous Trace C*-Algebras

We lift an action of a torus \({\mathbb{T}^n}\) on the spectrum of a continuous trace algebra to an action of a certain crossed module of Lie groups that is an extension of \({\mathbb{R}^n}\). We compute equivariant Brauer and Picard groups for this crossed module and describe the obstruction to the existence of an action of \({\mathbb{R}^n}\) in our framework.     

Re-explanation of Weyl Quantization Scheme via Weyl Ordering Approach

We re-explain the Weyl quantization scheme by virtue of the technique of integration within Weyl ordered product of operators, i.e., the Weyl correspondence rule can be reconstructed by classical functions' Fourier transformation followed by an inverse Fourier transformation within Weyl ordering of operators. As an application of this reconstruction, we derive the quantum operator coresponding to the angular spectrum amplitude of a spherical wave.     

Representations of Nets of C*-Algebras over S 1

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C^*-algebras over S ¹ admits faithful representations, and when the net is covariant under Diff(S ¹), it admits representations covariant under any amenable subgroup of Diff(S ¹).     

Integration of Lie 2-Algebras and Their Morphisms

Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module. On the other hand, the integration procedure of Getzler and Henriques will also produce a 2-group. In this paper, we show that these two integration results are Morita equivalent. As an application, we integrate a non-strict morphism between Lie algebra crossed modules to a generalized morphism between their corresponding Lie group crossed modules.     

C*-Algebras associated with one dimensional almost periodic tilings

For each irrational number, 0<α<1, we consider the space of one dimensional almost periodic tilings obtained by the projection method using a line of slope α. On this space we put the relation generated by translation and the identification of the “singular pairs”. We represent this as a topological spaceX _α with an equivalence relationR _α. OnR _α there is a natural locally Hausdorff topology from which we obtain a topological groupoid with a Haar system. We then construct the C^*-algebra of this groupoid and show that it is the irrational rotation C^*-algebra,A _α. Research supported by the Natural Sciences and Engineering Research Council of Canada and the Fields Institute for Research in Mathematical Sciences.     

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.     

Crossed Products of C*-Algebras and Spectral Analysis of Quantum Hamiltonians

We study spectral properties of a hamiltonian by analyzing the structure of certain C ^*-algebras to which it is affiliated. The main tool we use for the construction of these algebras is the crossed product of abelian C ^*-algebras (generated by the classical potentials) by actions of groups. We show how to compute the quotient of such a crossed product with respect to the ideal of compact operators and how to use the resulting information in order to get spectral properties of the hamiltonians. This scheme provides a unified approach to the study of hamiltonians of anisotropic and many-body systems (including quantum fields). Received: 5 November 2001 / Accepted: 10 March 2002     

C*-Algebras Associated with Endomorphisms and Polymorphisms of Compact Abelian Groups

A surjective endomorphism or, more generally, a polymorphism in the sense of Schmidt and Vershik [Erg Th Dyn Sys 28(2):633–642, 2008], of a compact abelian group H induces a transformation of L ²(H). We study the C*-algebra generated by this operator together with the algebra of continuous functions C(H) which acts as multiplication operators on L ²(H). Under a natural condition on the endo- or polymorphism, this algebra is simple and can be described by generators and relations. In the case of an endomorphism it is always purely infinite, while for a polymorphism in the class we consider, it is either purely infinite or has a unique trace. We prove a formula allowing to determine the K-theory of these algebras and use it to compute the K-groups in a number of interesting examples.     

Effect Algebras with the Riesz Decomposition Property and AF C*-Algebras

Relations between effect algebras with Riesz decomposition properties and AF C^*-algebras are studied. The well-known one-one correspondence between countable MV-algebras and unital AF C^*-algebras whose Murray-von Neumann order is a lattice is extended to any unital AF C^* algebras and some more general effect algebras having the Riesz decomposition property. One-one correspondence between tracial states on AF C^*-algebras and states on the corresponding effect algebras is proved. In particular, pure (faithful) tracial states correspond to extremal (faithful) states on corresponding effect algebras.     

Quantization of triangular Lie bialgebras

We derive a universal twisting element for an arbitrary triangularγ-matrix using a simple analogue of the Fedosov quantization method. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002. The work of SLL is partially supported by RFBR grant 00-02-17-956 and the grant INTAS 00-262. The work of AASh is supported by RFBR grant 02-02-06879 and Russian Ministry of Education under the grant E-00-33-184. The work of VAD is partially supported by the grant INTAS 00-561 and by the Grant for Support of Scientific Schools 00-15-96557. The work of API is partially supported by the RFBR grant 00-01-00299.