Heh! There was once a Hot Rod maxim: The only thing that beats cubic inches is cubic money.

I can point you to myriad threads here where relatively quiet and effective kitchen cooktop ventilation has been installed, or at least specified. In all of these cases the cost of the equipment required and its installation may be described as non-trivial. Non-trivial is not necessarily unaffordable; that is a personal issue. For larger cooktops and ranges used with high temperature cooking, the cost can rival the high-end ranges that are or could have been ventilated by the hood system and its essential partner, the make-up air system.

I can't help further if your problem is specified initially to have no solution. One can't keep grease, moisture, and entrained odors out of the household without capturing and containing it as it is generated. This requires a hood over the cooktop. It requires either external exhaust or a sufficient filtering system for recirculation. Both approaches require expenditure, and both approaches result in some obstruction of site lines over islands. There is also complexity involved in specifying the parts, but dealing with it in this response is not worthwhile if architectural arguments, cost arguments, or aesthetic arguments against doing what is needed preclude a solution.

If you are willing to intermittently leave the house for some period of time, and then flush the air when you return, ozone treatment will generally destroy sources of odor, but may have adverse effects on fabrics and other materials.

I'm very curious about air purifiers and how well they really capture particles. A few articles state that they have to be run constantly to to keep the air clean. Noise and energy expense apparently prevent many people from using them correctly.