Lifting endomorphisms to automorphisms

Normal endomorphisms of von Neumann algebras need not be extendable to automorphisms of a larger von Neumann algebra, but they always have asymptotic lifts. We describe the structure of endomorphisms and their asymptotic lifts in some detail, and apply those results to complete the identification of asymptotic lifts of unital completely positive linear maps on von Neumann algebras in terms of their minimal dilations to endomorphisms.

     

Radiated noise characteristics of a modern cargo ship

Extensive measurements were made of the radiated noise of M/V OVERSEAS HARRIETTE, a bulk cargo ship (length 173 m, displacement 25 515 tons) powered by a direct-drive low-speed diesel engine-a design representative of many modern merchant ships. The radiated noise data show high-level tonal frequencies from the ship's service diesel generator, main engine firing rate, and blade rate harmonics due to propeller cavitation. Radiated noise directionality measurements indicate that the radiation is generally dipole in form at lower frequencies, as expected. There are some departures from this pattern that may indicate hull interactions. Blade rate source level (174 dB re 1 microPa/m at 9 Hz, 16 knots) agrees reasonably well with a model of fundamental blade rate radiation previously reported by Gray and Greeley, but agreement for blade rate harmonics is not as good. Noise from merchant ships elevates the natural ambient by 20-30 dB in many areas; the effects of this noise on the biological environment have not been widely investigated.     

Perturbation theory for groups and lattices

A generalization of a theorem of Dan Voiculescu on perturbations of separable C¹-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.     

The Dirac Operator of a Commuting d-Tuple

Given a commuting d-tuple T=(T₁, …, T_d) of otherwise arbitrary operators on a Hilbert space, there is an associated Dirac operator D_T. Significant attributes of the d-tuple are best expressed in terms of D_T, including the Taylor spectrum and the notion of Fredholmness. In fact, all properties of T derive from its Dirac operator. We introduce a general notion of Dirac operator (in dimension d=1, 2, …) that is appropriate for multivariable operator theory. We show that every abstract Dirac operator is associated with a commuting d-tuple, and that two Dirac operators are isomorphic iff their associated operator d-tuples are unitarily equivalent. By relating the curvature invariant introduced in a previous paper to the index of a Dirac operator, we establish a stability result for the curvature invariant for pure d -contractions of finite rank. It is shown that for the subcategory of all such T that are (a) Fredholm and and (b) graded, the curvature invariant K(T) is stable under compact perturbations. We do not know if this stability persists when T is Fredholm but ungraded, although there is concrete evidence that it does.     

The Probability of Entanglement

We show that states on tensor products of matrix algebras whose ranks are relatively small are almost surely entangled, but that states of maximum rank are not. More precisely, let and be full matrix algebras with m ≥  n, fix an arbitrary state ω of N, and let E(ω) be the set of all states of that extend ω. The space E(ω) contains states of rank r for every r = 1, 2, . . . , m · rank ω, and it has a filtration into compact subspaces