You could use an island hood.

I agree, Island Hood

I hope the hood is a RECIRCULATING variety ( not all that effective, better than nothing ) ............and I HOPE you have a framing/ drywall guy?! as that wall will not be tall enough, and it will look hideous on the back side, facing whatever is back there in the dark.. Why the gap anyway?

If it doesn't look too weird on the other side, you could drywall behind the hood.

I would either extend the wall to the ceiling, and use the hood you have, or get a recirculating hood instead. A chimney hood will need access to the outside world anyway -- is that an exterior wall?

Why can't you extend the wall to the ceiling?

Thank you for the response. For reference, here is the hood we have:

https://www.whirlpool.com/kitchen/cooking/hoods/canopy/p.WVW53UC0FS.html

It will be vented outside through the roof. is the back of the vent stack covered with steel too? I was told the back is open due to it being on a wall. Are all 4 aidss covered in steel? If so, then I guess it wouldn't be so much of an issue.

It is sort of a weird gap and I guess we could drywall up to the ceiling but it brings in some light from the dining which is on the other side and has a big sliding glass door.

You may be able to use this hood and get additional ductwork that is designed for an island hood for the exposed part. However, the gap is pretty strange, and design-wise I would recommend bringing the wall up to the ceiling.

It is a strange looking wall, but instead of building the whole thing up to the ceiling, I think just build up the section of the wall behind the ductwork to the ceiling. It may add some interest and allow you to mount a regular venting duct hood.

Eric, people are suggesting an island hood vent because it attaches to the ceiling not the wall and is covered in metal on all four sides. You have a wall mount vent, which is meant to install on a wall and may look odd with that open gap on top. If you could exchange what you have for a ceiling mount island vent that is what you need.

Just get the wall to the ceiling! It looks totally weird. Add lighting in the kitchen, it can't be giving that much from the other side. Daylight does not penetrate very far: )

It's not a "half" wall , it's like an eighty percenter lol

Ok. Looks like we will build up the wall. Thank you all for the opinions!