William Arveson

Interpolation problems in nest algebras

Abstract The main result of this paper is a theorem which allows one to determine when a finitely generated left ideal in certain reflexive operator algebras is trivial (i.e., contains the identity). This is based on a formula which expresses the distance from such an algebra to an arbitrary operator on the underlying Hilbert space. As an application, we are able to deduce an operator-theoretic variant of the Corona theorem. Some applications of the distance formula to quasitriangular operators are given, and we present some new “inner-outer” factorization theorems along the way to the main result.

Continuous analogues of Fock space IV: essential states

In this paper, the last of our series [1], [4], [5], we present a new procedure for constructing examples of E0-semigroups, and we show how these methods can be applied to settle a number of problems left open in [1], [4] and [5]. The central objects of study are semigroups a = {at: t~>0} of normal *-endomorphisms of the algebra ~(H) of all operators on a (separable) Hilbert space H, which are continuous in the sense that (at(A) ~, ~i) should be a continuous function of t for fixed A in ~(H) and fixed ~, r/in H. a is called an Eo-semigroup [11] if it is unital in the sense that at(1)=l, for every t>~0. At the opposite extreme, a is called singular if the projections Pt=at(1) decrease to zero as t---~oo. While it is E0-semigroups that are of primary interest, much of our analysis will concern singular semigroups. In particular, we will show that the generator of a singular semigroup is injective, and that its inverse is an unbounded completely positive linear map which can be represented in very explicit terms. Perhaps surprisingly, the results of this analysis of singular semigroups can be applied directly to E0-semigroups. This is accomplished by making appropriate use of the spectral C*algebra C*(E) associated with a product system E ([4], [6]). Recall that a product system is a measurable family of Hilbert spaces E= {Et: t>0} over the open interval (0, +oo), on which there is defined a (measurable) associative

Pure E0-Semigroups and Absorbing States

Abstract: An E0-semigroup acting on is called pure if its tail von Neumann algebra is trivial in the sense that We determine all pure E0-semigroups which have a weakly continuous invariant state ω and which are minimal in an appropriate sense. In such cases the dynamics of the state space must stabilize as follows: for every normal state ρ of there is convergence to equilibrium in the trace norm A normal state ω with this property is called an absorbing state for α. Such E0-semigroups must be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those perturbations which have absorbing states with prescribed finite eigenvalue lists.

Quantization and the uniqueness of invariant structures

We determine and classify certain algebraic structures, defined on the space of all complex-valued polynomials in 2n real variables, which admitaffine contact transformations as automorphisms. These are the structures which have the minimum symmetry necessary to define the basic linear and angular momentum observables of classical and quantum mechanics. The results relate to the so-called Dirac problem of finding an appropriate mathematical characterization of the canonical quantization procedure.

ON THE INDEX AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

It is known that every semigroup of normal completely positive maps P = {Pt:t≥ 0} of ℬ(H), satisfying Pt(1) = 1 for every t ≥ 0, has a minimal dilation to an E0 acting on ℬ(K) for some Hilbert space K⊇H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator \[ L: {\mathcal B} (H)\to {\mathcal B} (H) \] in terms of natural structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup P={exp tL:t≥ 0} to an E0-semigroup is is cocycle conjugate to a CAR/CCR flow.

Diagonals of normal operators with finite spectrum

Let X = {λ1, …, λN} be a finite set of complex numbers, and let A be a normal operator with spectrum X that acts on a separable Hilbert space H. Relative to a fixed orthonormal basis e1, e2, … for H, A gives rise to a matrix whose diagonal is a sequence d = (d1, d2, …) with the property that each of its terms dn belongs to the convex hull of X. Not all sequences with that property can arise as the diagonal of a normal operator with spectrum X. The case where X is a set of real numbers has received a great deal of attention over the years and is reasonably well (though incompletely) understood. In this work we take up the case in which X is the set of vertices of a convex polygon in ℂ. The critical sequences d turn out to be those that accumulate rapidly in X in the sense that We show that there is an abelian group ΓX, a quotient of ℝ2 by a countable subgroup with concrete arithmetic properties, and a surjective mapping of such sequences d ↦ s(d) ∈ ΓX with the following property: If s(d) ≠ 0, then d is not the diagonal of any such operator A. We also show that while this is the only obstruction when N = 2, there are other (as yet unknown) obstructions when N = 3.

Perturbation theory for groups and lattices

Abstract A generalization of a theorem of Dan Voiculescu on perturbations of separable C ∗ -algebras is proved. This is applied to solve two problems relating to the perturbation theory of unitary group representations, and of commutative subspace lattices. The latter generalizes a theorem of Niels Toft Andersen on compact perturbations of nests.

Dilation Theory Yesterday and Today

Paul Halmos’ work in dilation theory began with a question and its answer: Which operators on a Hilbert space H can be extended to normal operators on a larger Hilbert space K ⊇ H? The answer is interesting and subtle.

A note on extensions of semigroups of *-endomorphisms

We present a relatively elementary proof that every E 0 -semigroup acting on a type I ∞ factor can be extended to a one-parameter group of *-automorphisms acting on a larger type I ∞ factor

THE HEAT FLOW OF THE CCR ALGEBRA

Let Pf ( x ) =− if ′( x ) and Qf ( x ) = xf ( x ) be the canonical operators acting on an appropriate common dense domain in L 2 (ℝ). The derivations D P ( A ) = i ( PA − AP ) and D Q ( A ) = i ( QA − AQ ) act on the *-algebra [Ascr ] of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L = D 2 P + D 2 Q , as a linear mapping of [Ascr ] into itself. L generates a semigroup of normal completely positive linear maps on [Bscr ]( L 2 (ℝ)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E 0 -semigroup. In particular, the author shows that its minimal dilation is pure and has no normal invariant states, and he discusses the significance of those facts for the interaction theory introduced in a previous paper. There are similar results for the canonical commutation relations with n degrees of freedom, where 1 [les ] n