An Ionic Breeze is an "Electrostatic Precipitator". A smaller version of the air filters used on US submarines. Dust, smoke, and bacteria are charged by the electric charge on wires, then they move to teh "collector" section where they stick to plates of the opposite charge. This part actually works. ( I know. I had to clean the filthy fucking things).
The concept that they move air by electrostatic charge is technically accurate, but I don't think they impart enough energy to the particles in the air to do much good. The ones on subs are installed at the inlets of the forced air circulation system. Prefilters are also used to trap larger particles...
Honestly, I think the Sharper Image things are junk based on my knowledge of operation alone (never actually used one). I'll bet a better device would be the ones used by taverns to remove smoke, but those use fans and filters too.
You want a good invention idea? This sounds good: (in this order) Prefilter --> Precipitator --> UV virus killer --> Fan --> HEPA filter. The fan obviously moves the air, the prefilter filters out big stuff, the Precip gets the small stuff and smoke, UV kills viruses, Hepa gets the rest.
Build it, sell it, make money, Buy me a new truck...