No one can use the CrossFit name unless they are a licensed affiliate with CrossFit Inc. Any globo gym that uses the name “CrossFit” in any way is using the trademarked name without permission, and any CrossFit trainer that uses the name outside of a registered affiliate is in violation of his/her license agreement. CrossFit has an email to report any misuses of their name.
LA Fitness, BAlly’s and big box clubs first of all would not make the financial commitment to purchase equipment or commit space, this is not what they are about, just training, some group fitness, that is about it. I personally don’t think they should anyway, most of the trainers there would not be crossfit or properly certified and trained, would be a recipe for disaster.
I am a CrossFit Coach and a Trainer at a Corporate Gym. The issue is mainly liability as both of these gentlemen have stated. With that said, it is very possible to still do a CrossFit style workout that involves a ton of different exercises and modalities. The main things you will not be able to do are drop weights (no bumper plates), also no rope climbing, and a lack of space will create some limits.