It looks like a nice piece of craftsmanship around a couple of fluorescent tubes.

It boils down to what you want to do with the space.
Do you want to keep the period look? Or do you want to gut and replace the entire kitchen?
Does the fixture provide enough light?

If you're determined to replace it, what options would you be willing to consider?
e.g. - cove lighting, recessed lighting, modifying the enclosure (eg - cutting down the depth?)
Would you be changing the rest of the kitchen then?

David, you've asked some very good questions.
There isn't nearly enough light but also the bulbs aren't
all fully working.
We plan to paint the cabinets white, add hardware, replace appliances with SS,
new backsplash and granite with undermount sink.

You can see pictures of the rest of the kitchen here

Here is a link that might be useful: Kitchen

From the pictures, it seems that you can put lighting on the top of the cabinets.

A careful inspection is warranted to make sure that things are fine and also map out the structures (inside the ceiling, below the floor) - joists, insulation, ...

Personally, I would take the time to figure out the family needs (especially the main cook) and what doesn't work in the current kitchen. Bad hinges and doors could be the impetus to get new cabinets instead of painting and/or refacing.