Disclaimer: I am not from Chicago. My family is however, and I've brought them here before and they all were impressed with Luke's.
Here's the deal. I became addicted to Luke's back when I worked at their other location (now defunct) in Scottsdale many years ago as a teenager. The beef sandwiches were all awesome, and the sweet peppers were my favorite.
At this location, I stop in for a combo (beef and sausage) with cheese and sweet peppers. I then finish my order off with a Hires root beer and a large fries and I am in heaven. It really is the best beef in Phoenix.
The ambiance of the place screams Chicago, like many others have noted on Yelp. The service also screams Chicago, in the sense that the service isn't friendly. On my last visit, the guy had literally nothing to say to me and he took my order at the pace of a snail.
It is what it is. Authentic. In a good and bad way. Can't have one without the other.
Yes, Luke's will get my money on occasion. It will remain that way.