Sure, it's legal. Most states do not have laws as to how "no weapons" is to be posted. It is still privately owned property, and as such, the owner/tenant can say what is allowed or not. Just like 'No Shoes, No Shirt, No Service", or "We refuse the right to refuse service to anyone". Don't have to like it. In a state where there is a law, and the sign isn't perfect, is it worth a confrontation?
Just as if the owner/management asked you to leave politely, and you refused, is it trespass?
I do not usually 'see' those signs (BIG EXCEPTION: State or Federal buildings, Post Offices, etc), but if my carrying is noticed (cc = seldom) I politely leave. I have only had one 'incident' where the LEO asked me to leave, and I did, politely. I have not returned to that business, there are many more out there that don't care.
My second amendment rights do not over-ride your personal property rights.