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     

Mixing properties of quantum systems

We generalize the classical notion of topological mixing for automorphisms ofC ^*-algebras in two ways. We show that for Galilean-invariant Fermi systems the weaker form of mixing is satisfied. With some additional requirement on the range of the interaction we can also demonstrate the stronger mixing property.     

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)     

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.     

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.     

Inaccurate experiments and environment-induced superselection rules

It is argued that approximate superselection rules induced by the environment cannot account for the emergence of definite measurement results in single experiments. The reason for this is that the inaccuracy necessary for an experiment failing to distinguish between exact and approximate mixtures requires the pointer observable to be strictly classical. This is shown in the case that the observables of physical systems generateW ^*-algebras.     

Logic of infinite quantum systems

Limits of sequences of finite-dimensional (AF)C ^*-algebras, such as the CAR algebra for the ideal Fermi gas, are a standard mathematical tool to describe quantum statistical systems arising as thermodynamic limits of finite spin systems. Only in the infinite-volume limit one can, for instance, describe phase transitions as singularities in the thermodynamic potentials, and handle the proliferation of physically inequivalent Hilbert space representations of a system with infinitely many degrees of freedom. As is well known, commutative AFC ^*-algebras correspond to countable Boolean algebras, i.e., algebras of propositions in the classical two-valued calculus. We investigate thenoncommutative logic properties of general AFC ^*-algebras, and their corresponding systems. We stress the interplay between Gödel incompleteness and quotient structures in the light of the nature does not have ideals program, stating that there are no quotient structures in physics. We interpret AFC ^*-algebras as algebras of the infinite-valued calculus of Lukasiewicz, i.e., algebras of propositions in Ulam's twenty questions game with lies.     

Topological entropy for endomorphisms of localC *-algebras

A notion of topological entropy for endomorphisms of localC ^*-algebras is introduced as a generalisation of the topological entropy of classical dynamical systems. The basic properties are derived and a series of calculations are presented.     

Generators of KMS Symmetric Markov Semigroups on {\mathcal{B}({\rm h})} Symmetry and Quantum Detailed Balance

We find the structure of generators of norm-continuous quantum Markov semigroups on B(h){\mathcal{B}({\rm h})} that are symmetric with respect to the scalar product tr (ρ ^1/2 x*ρ ^1/2 y) induced by a faithful normal invariant state ρ and satisfy two quantum generalisations of the classical detailed balance condition related with this non-commutative notion of symmetry: the so-called standard detailed balance condition and the standard detailed balance condition with an antiunitary time reversal.     

A generalization of the classical moment problem on *-algebras with applications to relativistic quantum theory. I.

A (non-commutative) generalization of the classical moment problem is formulated on arbitrary *-algebras with units. This is used to produce aC*-algebra associated with the space of test functions for quantum fields. ThisC*-algebra plays a role in theories of bounded localized observables in Hilbert space which is similar to that of the space of test functions in quantum field theories (namely it is represented in Hilbert space). The case of local quantum fields which satisfy a slight generalization of the growth condition is investigated. Laboratorie associé au Centre National de la Recherche Scientifique.     

Dynamical Entropy of Generalized Quantum Markov Chains on Gauge-invariant C *-algebras

We prove that the mean entropy and the dynamical entropy are equal for generalized quantum Markov chains on gauge-invariant C ^*-algebras.     

The Atiyah-Singer index theorem as passage to the classical limit in quantum mechanics

A new approach to the Atiyah-Singer index theorem is described, using the technique of continuous fields ofC ^*-algebras. The proof is given in the case of elliptic pseudodifferential operators on ℝ ⁿ .     

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.     

Deformation Quantization of the n-tuple Point

Contrary to the classical methods of quantum mechanics, the deformation quantization can be carried out on phase spaces which are not even topological manifolds. In particular, the Moyal star product gives rise to a canonical functor F from the category of affine analytic spaces to the category of associative (in general, non-commutative) ℂ-algebras. Curiously, if X is the n-tuple point, x ⁿ =0, then F(X) is the algebra of n×n matrices. Received: 4 November 1998 / Accepted: 3 March 1999     

On <Emphasis Type="Italic">C</Emphasis>*-algebras related to constrained representations of a free group

We consider representations of the free group F ₂ on two generators for which the norm of the sum of the generators and their inverses is bounded by some number μ ∈ [0, 4]. These μ-constrained representations determine a C*-algebra A _μ for each μ ∈ [0, 4]. If μ = 4, this gives the full group C*-algebra of F ₂. We prove that these C*-algebras form a continuous bundle of C*-algebras over [0, 4] and evaluate their K-groups.     

Bell's inequality forC *-algebras

Bell's inequality dealing with local hidden variables is given two formulations in terms ofC ^*-algebras. In particular, Bell's inequality holds for all states onAB wheneverA andB are unitalC ^*-algebras at least one of which is Abelian, i.e., at least one corresponds to a classical physical system.     

Stable equivalence of the weak closures of free groups convolution algebras

We prove in this paper that the von Neumann algebras associated to the free non-commutative groups are stably isomorphic, i.e. that they are isomorphic when tensorized by the algebra of all linear bounded operators on a separable, infinite dimensional Hilbert space. This gives positive evidence for an old question, due to R.V. Kadison (see also S. Sakai's book on W^*-algebras), whether the von Neumann algebras associated to free groups are isomorphic or not.     

Recurrence in Quantum Mechanics

We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C^*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouville's theorem is then proposed. We proceed to study Poincaré recurrence in C^*-algebras by mimicking the measure theoretic setting. The results are interpreted as recurrence in quantum mechanics, similar to Poincaré recurrence in classical mechanics.     

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.     

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