The system
Ralph, pressurized CO<sub>2</sub> at room temperature exists as a supercritical liquid, meaning that it not only has some "oomph" like water (i.e., force to hit the dishes and dislodge crud), but also has a property similar to carbon tetrachloride fluid in that it can dissolve nonpolar molecules like fat and oil. So, it kicks but as "dishwasher fluid" in this design, because it not only performs like water (kind of), but also needs no detergent because it possesses an intrinsic chemical ability to dissolve things without additional cleaning agents.
Crud is then removed from the CO<sub>2</sub> cleaning fluid and is deposited into a greywater system (or, if you weren't environmentally correct, I guess down the drain).
Dry-cleaning for dishes! (In the same way that dry-cleaning uses fluid that dissolves nonpolar molecules and, frankly, isn't actually dry.)