Unbounded derivations ofC*-algebras II

It is demonstrated that a closed symmetric derivation δ of aC?-algebra \(\mathfrak{A}\) generates a strongly continuous one-parameter group of automorphisms of aC?-algebra \(\mathfrak{A}\) if and only if, it satisfies one of the following three conditions

1. (αδ+1)(D(δ))= \(\mathfrak{A}\) , α∈?\{0}.
2. δ possesses a dense set of analytic elements.
3. δ possesses a dense set of geometric elements.

Together with one of the following two conditions

1. ∥(αδ+1)(A)∥≧∥A∥, α∈IR,A∈D(δ).
2. If α∈IR andA∈D(δ) then (αδ+1)(A)≧0 impliesA≧0.

Other characterizations are given in terms of invariant states and the invariance ofD(δ) under the square root operation of positive elements.     

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.     

Amenability of crossed products ofC*-algebras

It is shown that the class of amenable (resp. strongly amenable)C*-algebras is closed under the process of taking crossed products with discrete amenable groups. Under certain circumstances, amenability is also preserved under taking a crossed product with an amenable semigroup of linear endomorphisms. These facts are used to show that certain simpleC*-algebras studied by J. Cuntz are amenable but not strongly amenable (thus answering a question of B. E. Johnson), yet are stably isomorphic to strongly amenable algebras.Partially supported by NSF     

Nets ofC*-algebras and classification of states

The concept of locality in quantum physics leads to mathematical structures in which the basic object is an operator algebra with a net of distinguished subalgebras (the local subalgebras). Such nets provide a classification of the states of this algebra in equivalence classes determined by local or asymptotic properties. The corresponding equivalence relations are natural generalizations of the (more stringent) standard quasiequivalence relation (they are also useful for classifying states by their properties with respect to automorphism groups). After discussing general nets from this point of view we investigate in the last section more specialized nets (funnels of von Neumann algebras) with special emphasis on their locally normal states.The research in this paper was supported in part by the N.S.F. and the Ministère de l'Education Nationale.     

Abelian faces of state spaces ofC*-algebras

LetF be a closed face of the weak* compact convex state space of a unitalC*-algebraA. The class ofF-abelian states, introduced earlier by the author, is studied further. It is shown (without any restriction onA orF) thatF is a Choquet simplex if and only if every state inF isF-abelian, and that it is sufficient for this that every pure state inF isF-abelian. As a corollary, it is deduced that an arbitraryC*-dynamical system (A, G, ) isG-abelian if and only if every ergodic state is weakly clustering. Nevertheless the set of allF-abelian (or evenG-abelian) states is not necessarily weak* compact.     

On weak and monotone σ-closures ofC*-algebras

It is proved that the monotone -closure of the self-adjoint part of anyC*-algebraA is the self-adjoint part of aC*-algebra . IfA is of type I it is proved that is weakly -closed, i.e. is a*-algebra. The physical importance of*-algebras was explained in [1] and [7].     

Superderivations and symmetric Markov semigroups

Unbounded superderivations are used to construct non-commutative elliptic operators on semi-finite von Neumann algebras. The method exploits the interplay between dynamical semigroups and Dirichlet forms. The elliptic operators may be viewed as generators of irreversible dynamics for fermion systems with infinite degrees of freedom.     

Quantum ergodicity ofC * dynamical systems

We establish phase transitions for a class of continuum multi-type particle systems with finite range repulsive pair interaction between particles of different type. This proves an old conjecture of Lebowitz and Lieb. A phase transition still occurs when we allow a background pair interaction (between all particles) which is superstable and has sufficiently short range of repulsion. Our approach involves a random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. In the course of our argument, we establish the existence of a percolation transition for Gibbsian particle systems with random edges between the particles, and also give an alternative proof for the existence of Gibbs measures with supperstable interaction.To the memory of Roland DobrushinResearch partially supported by the Isaac Newton Institute Cambridge.Research supported by the Swedish Natural Science Research Council and the Deutsche Forschungsgemeinschaft.     

Small perturbations ofC*-dynamical systems

It is shown that if is the generator of a strongly continuous oneparameter group of *-automorphisms of aC*-algebraA and is an unbounded *-derivation ofA with the same domain as , then + is also a generator for all sufficiently small real numbers .     

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.     

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.     

Unbounded derivations of commutativeC*-algebras

It is shown that an unbounded *-derivation of a unital commutativeC*-algebraA is quasi well-behaved if and only if there is a dense open subsetU of the spectrum ofA such that, for anyf in the domain of , (f) vanishes at any point ofU wheref attains its norm. An example is given to show that even if is closed it need not be quasi well-behaved. This answers negatively a question posed by Sakai for arbitraryC*-algebras.It is also shown that there are no-zero closed derivations onA if the spectrum ofA contains a dense open totally disconnected subset.     

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.     

Unbounded derivations of C*-algebras

We study unbounded derivations ofC*-algebras and characterize those which generate one-parameter groups of automorphisms. We also develop a functional calculus for the domains of closed derivations and develop criteria for closeability. Some specialC*-algebras are considered \(\mathfrak{B}\mathbb{C}(\mathfrak{H}),\mathfrak{B}(\mathfrak{H})\) , UHF algebras, and in this last context we prove the existence of non-closeable derivations.     

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.     

Dynamical entropy ofC* algebras and von Neumann algebras

The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.     

The order structure of topological -algebras of unbounded operators I

In this paper we begin to study the order structure of topological -algebras of unbounded operators in Hilbert space with the investigation of the normality and the bounded decomposition property of the cones. We prove that for a large class of topological -algebras the normality of the wedge of positive elements is necessary and sufficient for a topological -algebra to be algebraically and topologically isomorphic to a -algebra of unbounded operators equipped with the uniform topology. From this theorem we obtain some corollaries, so for instance, well-known results of Lassner, Brooks and Grothendieck.