Well, a downdraft will result in sucking the flame away from what you are cooking as well as a dirty ceiling, while nothing will result in a dirty ceiling above the cooking area as well as more dirt and grease throughout the house as aerosolized grease particles float around your house carrying odors and dirt to penetrate all of your soft furnishings.

Frying pan or fire.

Cooking is only 10% of the time you spend in a kitchen. Even when entertaining. 70% of your time is prep. Examine your layout to be sure you've got at least 48" next to that cooktop for a good prep zone if you want to be spending 80% of your time facing out. Otherwise, redo the layout to keep the prep there, and do the cooking on the perimeter when it's easier and cheaper and more effective to vent.

We had the same issue. First gas range/convection oven had downdraft (jennair). The downdraft really does pull the flame towards it so you are CONSTANTLY having to rotate the pans to insure even cooking. Six years later the electronic control panel failed so I had no oven. By then I was so sick of it, I didn't even attempt to have it repaired. I replaced it with an Electrolux gas range/convection stove and no downdraft. Our kitchen is in the center of the house and is 12 feet away from french doors that I could open if I needed to air the kitchen out. I have not regretted the decision EVER! I cook a lot and we entertain a fair amount and it is always in the kitchen. However, I do not fry much. I saute quite a bit but that uses minimal oil compared to frying so grease is not much of an issue for me. Our house is wide open with all of the public spaces flowing into each other, and I have not noticed any issues with smells through out the house or grease/odors into any of my soft furnishings. I do not have any carpet or drapes (hardwood/tile/interior shutters) so that may be why. Hope this helps you.

There are some downdraft ventilation systems that are pop-up and twist-over devices that might have some very modest effectiveness against modest temperature cooking.

If a hood over the island is out, is there no place on the ceiling where a vent to outside can be located?

How about on a wall? Moving even a portion of the effluent out before it precipitates beats letting it stay resident until it does.

Your local code may require some ventilation given that the cooktop is gas fed. Your body might require it also, but probably will be more subtle in its demands.

If your guests ask about ventilation, tell them that you modeled your island layout after poser kitchens seen in magazines. (Putting it in less snarky terms, I do not support letting appearance or socialization factors completely supersede needed functionality.)

kas

Ventilation is the #1 most neglected necessity in a kitchen redo. Lighting is #2. People run out of money spending it on the less important details that are often decorative in nature and not functional. And they regret it later. A redo is the chance to get it right. If cooking on a high powered range is an important part of your redo, then the necessary ventilation that goes hand in hand with that choice should be second on your list of priorities behind the range.

Home kitchens are also not Food Network shows. You really should have an impartial family member or friend perform your own time/motion study and you'll probably find that you fall into the norm that very little fire time is part of the creation of your meal as a whole. Most of your time will be spent as the commis not the chef de cuisine. Plan your kitchen around making that prep the most social activity, not just the bit of time stir frying.

Absolutely not. I agree with live. Prep on the island and cook against the wall if possible.

Not only will a greasy kitchen become a cleaning nightmare, but it may be a health hazard according to this story from the NY Times. It's been linked here before, but I will link it again.

Also, no vent may be a code violation where you live.

Here is a link that might be useful: NY Times Venting Article

I'd put in a soffit if I had to and ventilate properly. I DO have a downdraft right now and it is MUCH better than nothing. But I don't have gas where a downdraft impacts actual cooking.

If you really won't updraft, do a downdraft and get an induction cooktop so you can get SOME functionality out of it.