I would like to lay some slate tiles next to a fireplace and have a laminate floor come up to the tiles without a transition strip. One dealer talked to me about using a Schluter System strip and glueing the laminate down right next to this strip. The rest of the floor would still be floating. Has anyone heard of gluing down one strip of a floating floor?

Sure you can. You could glue the entire floor down if you wanted to, it just isn't necessary.

Before I read the entire post, I was going to recommend a Schluter strip! If you are VERY careful with the cut, you don't need to glue down the laminate, the Schluter strip has a lip on it and the laminate can slip under it. Nail down the Schluter strip but not the laminate. You can, as Joypulv, notes, glue down a laminate floor.