B impleads C, who is also a citizen of New York. The court should dismiss the entire case for lack of complete diversity. The court should dismiss the claim between B and C because there is no diversity between them. A sues B in federal court, and B moves to dismiss for improper service of process. The court denies the motion.
B then moves to dismiss for lack of personal jurisdiction.
The court will rule on B's motion by looking at concepts of sovereignty, minimum contacts, long arm statutes and the like. If the claim arose in Georgia, the case may not be filed in Florida's federal courts. A may file such an action in federal court, but not in any Georgia state court. A sues B, and B moves to dismiss for improper service of process. As long as A can show that B actually received notice, B cannot prevail on his motion.