Convergent vs divergent ducts, nozzles vs diffusers, these do nothing but trade dynamic pressure (velocity) for static pressure. Total pressure remains constant (minus any losses). The entire point of a compressor is to increase total pressure, which can only be done by using a work input. A divergent duct will not increase total pressure.
You ask why is a compressor convergent from front to back. This is simply answered that since mass flow is constant, but density increases, then the exit area must be smaller compared to the inlet area. Now as DC-10Tech pointed out, if you look at the individual compressor stages, the flow path actually does expand, just not in the axial direction.
Compressors work by using alternating rows of rotor blades and stator vanes.
Every row of stators will turn the flow away from the axial direction. Because the flow is turned to the side tangentially, the exit area of each stator is actually larger than the inlet area, which means you get a rise in static pressure and a drop in velocity. However, stators don't move so no work is put into the flow, so total pressure remains constant (once again, minus any losses).
Rotors then turn the flow back to the axial direction. But because they are moving, work is put into the flow. Two things now happen in the rotors. The rotating blades increase the dynamic pressure of the flow. Rotors also turn the flow in a way that like the stators, relative to the flow they have exit areas larger than inlet areas, which increases the static pressure. The sum of the these two pressure rises is your increase in total pressure.
Here's a diagram comparing the diffuser that you are thinking of to the cascade type of diffuser in a row of rotor blades. Both have exit areas larger than inlet areas and will increase the static pressure.
The alternating rows of rotor blades and stator vanes work by allowing you to put work in to increase total pressure (rotor), then convert this total pressure rise to an increase in static pressure (stator). Net result of the compressor is an increase in pressure and a decrease flow velocity.
[Edited 2006-06-13 16:59:39]