Dilations and constrained algebras

It is well known that unital contractive representations of the disk algebra are completely contractive. Let A denote the subalgebra of the disk algebra consisting of those functions f whose first derivative vanishes at 0. We prove that there are unital contractive representations of A which are not completely contractive, and furthermore provide a Kaiser and Varopoulos inspired example for A and present a characterization of those contractive representations of A which are completely contractive. In the positive direction, for the algebra of rational functions with poles off the distinguished variety V in the bidisk determined by (z-w)(z+w)=0, unital contractive representations are completely contractive.Comment: New to version 2 is a proof of rational dilation for the distinguished variety in the bidisk determined by (z-w)(z+w)=

The Multitude of Molecular Hydrogen Knots in the Helix Nebula

We present HST/NICMOS imaging of the H_2 2.12 \mu m emission in 5 fields in the Helix Nebula ranging in radial distance from 250-450" from the central star. The images reveal arcuate structures with their apexes pointing towards the central star. Comparison of these images with comparable resolution ground based images reveals that the molecular gas is more highly clumped than the ionized gas line tracers. From our images, we determine an average number density of knots in the molecular gas ranging from 162 knots/arcmin^2 in the denser regions to 18 knots/arcmin^2 in the lower density outer regions. Using this new number density, we estimate that the total number of knots in the Helix to be ~23,000 which is a factor of 6.5 larger than previous estimates. The total neutral gas mass in the Helix is 0.35 M_\odot assuming a mass of \~1.5x10^{-5} M_\odot for the individual knots. The H_2 intensity, 5-9x10^{-5} erg s^{-1} cm^{-2} sr^{-1}, remains relatively constant with projected distance from the central star suggesting a heating mechanism for the molecular gas that is distributed almost uniformly in the knots throughout the nebula. The temperature and H_2 2.12 \mu m intensity of the knots can be approximately explained by photodissociation regions (PDRs) in the individual knots; however, theoretical PDR models of PN under-predict the intensities of some knots by a factor of 10.Comment: 26 pages, 3 tables, 10 figures; AJ accepte