Can I take American equipment to Europe?

That gizmo would be a voltage converter. $9.99 at Walmart.

Probably there's an audiophile voltage converter that will cost 10 x as much, and do exactly the same.

there will be no difference in sound, but no doubt someone will come up with some reason why it would sound different.

You need a heavy duty step down transformer that will be able to handle the power required for your system. I personally would not trust the cheap travel converters to be used on expensive audio equipment.

There is a sonic difference between different converters.
Just like different amps and power cords.
The Walmart converter might sound great, thats a broad stroke to say there's no difference in the sonic signature.

My wife (who is European) got one of those converters when she moved here so she could use her favorite hairdryer. The hairdryer fried on the first attempt, and the converter's fuse blew.

I would not hook any of my gear up to a cheapie unit.

Rbrowne is correct, unless your equipment can be easily converted to European voltage. When I was stationed in Europe we used step down transformers of various ratings. They are fairly large pieces depending on what you are running. The small voltage converters won't survive long.