The enveloping algebra of a covariant system

In an earlier work, Doplicher, Kastler and Robinson have examined a mathematical structure consisting of a pair (A, G), whereA is aC*-algebra andG is a locally compact automorphism group ofA. We call such a structure a covariant system. The enveloping von Neumann algebraA(A, G) of (A, G) is defined as a *-algebra of operator valued functions (called options) on the space of covariant representations of (A, G). The system (A, G) is canonically embedded in, and in fact generates, the von Neumann algebraA(A, G). Further we show there is a natural one-to-one correspondence between the normal *-representations ofA(A, G) and the proper covariant representations of (A, G). The relation ofA(A, G) to the covarainceC*-algebraC*(A, G) is also examined.     

Local normality in Quantum Statistical Mechanics

It is shown that K.M.S.-states are locally normal on a great number ofC*-algebras that may be of interest in Quantum Statistical Mechanics. The lattice structure and the Choquet-simplex structure of various sets of states are investigated. In this respect special attention is payed to the interplay of the K.M.S.-automorphism group with other automorphism groups under whose action K.M.S.-states are possibly invariant. A seemingly weaker notion thanG-abelianness of the algebra of observables, namelyG-abelianness, is introduced and investigated. Finally a necessary and sufficient condition (on aC*-algebra with a sequential separable factor funnel) for decomposition of a locally normal state into locally normal states is given.     

The C*-algebra of the electromagnetic field

We show that there is a unique C*-algebra for the transverse quantum electromagnetic field obeying the Maxwell equations with any classical charge-current. For nonzero charge, the representation of the C*-algebra differs from the representation with zero charge.     

Pure states as a dual object forC*-algebras

We consider the set of pure states of aC*-algebra as a uniform space equipped with transition probabilities and orientation, and show that the pure states with this structure determine theC*-algebra up to *-isomorphism.Partially supported by NSF grant MCS78-02455     

A remark on a theorem of Powers and Sakai

Given an abelian locally compact groupG and aC*-algebra with unit,U, the set of those continuous representations ofG by automorphisms ofU which fulfill a spectrum condition is closed.     

Some Remarks on Group Bundles and <Emphasis Type="Italic">C</Emphasis>* Dynamical Systems

We introduce the notion of fibred action of a group bundle on a C(X)-algebra. By using such a notion, a characterization in terms of induced C*-bundles is given for C*-dynamical systems such that the relative commutant of the fixed-point C*-algebra is minimal (i.e., it is generated by the centre of the given C*-algebra and the centre of the fixed-point C*-algebra). A class of examples in the setting of the Cuntz algebra is given, and connections with superselection structures with nontrivial centre are discussed. The author was partially supported by the European Network “Quantum Spaces - Noncommutative Geometry” HPRN-CT-2002-00280.     

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.     

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.     

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     

Invariant states and conditional expectations of the anticommutation relations

The groupG of unitary elements of a maximal abelian von Neumann algebra on a separable, complex Hilbert spaceH acts as a group of automorphisms on the CAR algebraA(H) overH. It is shown that the set ofG-invariant states is a simplex, isomorphic to the set of regular probability measures on aw*-compact setS ofG-invariant generalized free states. The GNS Hilbert space induced by an arbitraryG-invariant state onA(H) supports a *-representation ofC(S); the canonical map ofA(H) intoC(S) can then be locally implemented by a normal,G-invariant conditional expectation.     

C*-algebras and automorphism groups

Let (A,G, α) be aC*-dynamical system withG a topological group. Let π be a representation ofA. We will show that there exists a quasiequivalent representation \(\hat \pi \) to π which is a covariant representation, if and only if the folium of π is invariant under the action ofG and this action is strongly continuous.     

Unbounded Representations of q-Deformation of Cuntz Algebra

We study a deformation of the Cuntz–Toeplitz C ^*-algebra determined by the relations ${a_i^*a_i=1+q a_ia_i^*,\, a_i^*a_j=0}$ . We define its well-behaved unbounded *-representations and classify all irreducible ones up to unitary equivalence.     

A duality theorem for the automorphism group of a covariant system

In this paper we examine the covariant representation theory of a covariant system (A, G) introduced by Doplicher, Kastler and Robinson. (A is aC*-algebra andG is a locally compact group of automorphisms ofA.) We define the concept of left tensor product of two covariant representations. Loosely stated, our duality theorem says thatG is canonically isomorphic to the set of bounded operator valued maps on the set of covariant representations of the covariant system (A, G) which preserve direct sums, unitary equivalence and left tensor products. We further show that the enveloping von Neumann algebraA(A, G) of the covariant system (A, G) admits a (not necessarily injective) comultiplicationd such that (A(A, G),d) is a Hopf von Neumann algebra. The intrinsic group of this Hopf von Neumann algebra is canonically isomorphic and (strong operator topology) homeomorphic toG.     

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.     

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.     

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.     

Spectra of states,and asymptotically abelianC*-algebras

For each state of aC*-algebra its spectrum is defined and shown to coincide with the spectrum of the naturally associated modular operator. For strongly clustering states of asymptotically abelianC*-algebras the spectrum is minimal among the states in the same quasi-equivalence class, hence is a *-isomorphic invariant for the weak closure of the G.N.S.-representation. Furthermore, the non-zero elements in the spectrum of strongly clustering states form a multiplicative group.     

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.     

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.