That's the topic of The (Surprisingly) Prevalent Role of States in an Era of Federalized Class Actions by law prof Linda Mullenix. Here is the abstract:
In enacting the Class Action Fairness Act of 2005, Congress intended to expand access to federal courts for interstate class actions by creating minimal diversity and removal jurisdiction. Congress stated that a purpose of CAFA was to “restore the intent of the framers of the United States Constitution by providing for Federal court consideration of interstate cases of national importance under diversity jurisdiction.”
Despite CAFA, states have retained a role in addressing complex litigation aided by Supreme Court decisions recognizing the independent role of state courts in enforcing local legal norms.
An historical examination of dual system complex litigation illustrates the extent to which federal courts have successfully (or unsuccessfully) intervened in pending parallel state court proceedings through application of abstention, the Anti-Injunction Act, preclusion, and Erie doctrines. Thus, the Court has upheld the right of state courts to maintain state class litigation notwithstanding federal court repudiation of certification of the same litigation. In so doing, the Court has recognized principles of federalism and comity, signaled a “non-interference” stance with state class proceedings, and strengthened the independent role of state courts in complex litigation. Moreover, several federal courts have rejected the primacy of federal courts in applying Rule 23 class certification standards in derogation of countervailing state statutes that would prohibit prosecution of the same class litigation in state court. CAFA additionally recognized a role for state court adjudication of complex litigation by carving out local controversy exceptions to its removal provisions.
The Court also has recognized the role of state attorneys general in their parens patriae capacity under CAFA to pursue complex litigation on behalf of state citizenry, in spite of defense attempts to evade state court jurisdiction. In addition, state attorneys generals have the right to receive notice of federal class action settlements and to lodge comments or objections to pending settlements that might affect state constituents. Thus, CAFA and the Court have given state attorneys general a relatively robust role in addressing complex litigation and afforded significant protection to state enforcement efforts. The Court also has held that the Securities Litigation Uniform Standards Act of 1998 did not strip state courts of their longstanding jurisdiction to adjudicate class actions alleging only § 1933 Securities Act violations. In enacting SLUSA, Congress did nothing to deprive state courts of jurisdiction over class actions based on federal law.
In sum, although the received understanding of CAFA was to federalize class litigation, state courts nonetheless have continued to perform a role in addressing complex cases. To a significant extent, state courts have been insulated from federal judicial encroachment on states’ ability to handle complex litigation in its own courts, and state attorneys generals have in various ways been empowered to pursue aggregate relief on behalf of state citizenry.