WILLIAMS, Circuit Judge.
After seven months of litigation in a California court concerning Zurich American Insurance Company’s duty under various insurance policies to defend Watts Industries, Inc. and James Jones Company, Zurich filed a petition in the Northern District of Illinois to compel arbitration. The state case continued forward, however, leading eventually to a clash between the district court and the California Superior Court over the scope of an order by the district court enjoining Watts and the California court from further proceedings pending the district court’s resolution of Zurich’s petition. That preliminary injunction is the subject of these cross appeals by Watts and Zurich. Watts asserts that the injunction violated the Anti-Injunction Act, 28 U.S.C. § 2283, while Zurich claims it should have been broader— that the district court was mistaken in holding that the Rooker-Feldman doctrine applied to some of the issues. We agree with Zurich that the Rooker-Feldman doctrine does not apply but hold that the preliminary injunction violates the Anti-Injunction Act and therefore reverse.
I. BACKGROUND
Zurich issued to Watts commercial general liability insurance policies, which provided for a duty to defend against lawsuits. James Jones Company, Watt’s subsidiary, was an insured under the policies (until it was later sold by Watts). In connection with these policies, Watts entered into de-ductibility agreements with Zurich, under which Watts agreed to reimburse Zurich for various defense and indemnity expenditures made pursuant to the general liability policies. The deductible agreements contain provisions requiring the arbitration of disputes arising out of those agreements.1 The insurance policies do not.
Watts and Jones were sued for fraud in two actions in California, Dep’t of Water and Power ex reí Armenia v. James Jones and Rothschild v. James Jones Co.2 Zurich refused to pay defense costs, so Watts and Jones separately sued Zurich in California Superior Court for coverage under the policies and the two coverage actions were consolidated. Several months [820]*820into that litigation, during settlement negotiations, Zurich asserted that, even if it were liable under the policies, Watts would be responsible under the deductible agreements to reimburse Zurich for the full defense costs. Watts responded that, by denying coverage, Zurich had repudiated the policies and the deductible agreements and, alternatively, that Zurich had waived any defense based on those agreements. Zurich responded with a demand for arbitration, which Watts rejected.
Zurich then filed a petition to compel arbitration in the district court and asked the California Superior Court to stay the state proceedings pending the district court’s consideration of Zurich’s petition. Shortly thereafter, the California Superior Court granted Watts’s pending motion for summary adjudication as to Zurich’s duty to defend Watts in the underlying Armen-ta matter. The Superior Court held that Zurich had a duty to defend Watts in that case and directed Zurich to reimburse it for reasonable defense costs. It also denied Zurich’s request for a stay, holding that the dispute under the deductible agreements was severable from the coverage dispute pending before it. Zurich appealed to the California Court of Appeal, which held that the order for summary adjudication and the denial of Zurich’s motion to stay were not appealable orders, and therefore dismissed the appeal.
Watts then sought enforcement of the Superior Court’s earlier order directing Zurich to pay defense costs, and after the Superior Court again ordered it to pay, Zurich filed a motion with the district court for a temporary restraining order enjoining further proceedings in California. The district court granted the motion, see Zurich Am. Ins. Co. v. Sup.Ct. for the State of Cal., 200 F.Supp.2d 929 (N.D.Ill.2002), and later granted Zurich’s request for a preliminary injunction.
The district court held that under the Rooker-Feldman doctrine, it lacked jurisdiction over issues already decided by the California court. Although noting the possible ambiguity regarding which issues were actually decided in the California court’s order denying Zurich’s motion for a stay, the district court interpreted that order as limited to the arbitrability of the duty to defend issue with respect to the Armenta case. The court found that Zurich was likely to succeed on the merits of its petition to compel arbitration of the other issues pending in California as to Watts and that Zurich would be irreparably harmed without the injunction. It therefore enjoined further proceedings in California concerning matters related to Watts other than the duty to defend in Armenta. It denied the motion as to Jones.
The Superior Court then issued a clarification, stating that its earlier order had held that there were presently no arbitra-ble issues in either of the underlying cases, Armenta or Rothschild. The district court refused to consider that clarification, considering it a violation of its injunction.
After these appeals were filed, the California court held that, based on its clarification of the scope of its earlier order and the district court’s holding that Rooker-Feldman barred federal jurisdiction over issues already decided, the injunction did not bar its continued proceedings and directed counsel for Watts and Jones to “proceed forthwith with any and all motions that would effectuate an adjudication of coverage issues.” We granted Zurich’s and Watts’s request for judicial notice of that order, reminded those subject to the injunction that they were bound until it was modified or reversed, and after oral argument of these appeals, reversed the [821]*821injunction for the reasons we now explain.3
II. ANALYSIS
On appeal, Zurich argues that the district court erred in holding that it lacked jurisdiction, under the Rooker-Feldman doctrine, over those issues already determined by the California court. Watts argues that the district court construed the state court’s order too narrowly and should have found that it lacked jurisdiction over any of the issues raised in Zurich’s petition. Watts further argues that the court’s preliminary injunction over the California action was barred by the Anti-Injunction Act. Because the Rooker-Feld-man doctrine is jurisdictional, see Lewis v. Anderson, 308 F.3d 768, 771-72 (7th Cir.2002); Long v. Shorebank Dev. Corp., 182 F.3d 548, 554-55 (7th Cir.1999), we address that question first before determining whether the injunction was appropriate.
A. Rooker-Feldman doctrine
The Rooker-Feldman doctrine bars review by lower federal courts of state court judgments. See Rooker v. Fidelity Tr. Co., 263 U.S. 413, 415-16, 44 S.Ct. 149, 68 L.Ed. 362 (1923); District of Columbia Ct.App. v. Feldman, 460 U.S. 462, 482-86, 103 S.Ct. 1303, 75 L.Ed.2d 206 (1983). It “rests on the principle that district courts have only original jurisdiction; the full appellate jurisdiction over judgments of state courts in civil cases lies in the Supreme Court of the United States.” GASH Assocs. v. Vill. of Rosemont, 995 F.2d 726, 728 (7th Cir.1993). We review the district court’s application of the Rooker-Feldman doctrine de novo. Remer v. Burlington Area Sch. Dist., 205 F.3d 990, 996 (7th Cir.2000).
Because the doctrine comes into play when there is a prior state court judgment,4 it is often confused with res judicata, “[b]ut the two are not coextensive.” GASH Assocs., 995 F.2d at 728. As the district court recognized, in determining application of the doctrine the pivotal question is whether the plaintiff’s federal claim is independent of the state court’s action; if so, the relevant doctrine is res judicata, not Rooker-Feldman:
The Rooker-Feldman doctrine asks: is the federal plaintiff seeking to set aside a state judgment, or does he present some independent claim, albeit one that denies a legal conclusion that a state court has reached in a case to which he was a party? If the former, then the district court lacks jurisdiction; if the latter, then there is jurisdiction and state law determines whether the defendant prevails under principles of preclusion.
[822]*822GASH Assocs., 995 F.2d at 728; see also Brokaw v. Weaver, 305 F.3d 660, 664-65 (7th Cir.2002); Remer, 205 F.3d at 998; Long, 182 F.3d at 555. Stated differently, “the fundamental and appropriate question to ask is whether the injury alleged by the federal plaintiff resulted from the state court judgment itself or is distinct from that judgment. If the injury alleged resulted from the state court judgment itself, Rooker-Feldman directs that the lower federal courts lack jurisdiction.” Garry v. Geils, 82 F.3d 1362, 1365-66 (7th Cir.1996); see also Johnson v. De Grandy, 512 U.S. 997, 1005-06, 114 S.Ct. 2647, 129 L.Ed.2d 775 (1994) (“a party losing in state court is barred from seeking what in substance would be appellate review of the state judgment ... based on the losing party’s claim that the state judgment itself violates the loser’s federal rights”); Lewis, 308 F.3d at 772; Durgins v. City of East St. Louis, 272 F.3d 841, 844 (7th Cir.2001); GASH Assocs., 995 F.2d at 728-29.
Although the district court identified the correct test of when Rooker-Feldman applies, it arrived at the wrong answer. If it applied the analysis we just described, the district court must have determined that Zurich’s petition to compel arbitration was at least partly dependent on the state court’s prior adjudication when it held that it lacked jurisdiction over issues the California court had already decided. But Zurich’s federal claim arises under the Federal Arbitration Act (FAA) and would exist even if the state court had determined the duty to defend or arbitrability issues in Zurich’s favor or if these issues had never been before it. The federal claim does not therefore seek to set aside the state court’s orders and does not depend on a determination that the court erred. See Remer, 205 F.3d at 998 (distinguishing cases in which, “but for the state court determinations, the federal plaintiffs would have had no complaint”).
Similarly, Zurich’s injury was not caused by the state court, but by its adversary’s conduct (Watts’s refusal to arbitrate); its only gripe with the state court is that it failed to remedy that conduct. But that court’s alleged erroneous application of federal law is not itself a cognizable injury, and Zurich has not alleged any other possible federal claims against the California court. See Homola v. McNamara, 59 F.3d 647, 650 (7th Cir.1995) (“a decision by a state court, however erroneous, is not itself a violation of the Constitution actionable in federal court”); Durgins, 272 F.3d at 844 (holding that Rooker-Feldman does not apply when the only claim is that the state court failed to remedy an injury by some other actor); see also Long, 182 F.3d at 556 (holding that Rooker-Feldman did not apply to plaintiffs claim under the Fair Debt Collection Practices Act, which “was independent of and complete prior to the entry of [the state court order]”). Zurich’s federal claim simply seeks to bypass the state court’s order, and does not directly attack it, so Rooker-Feldman does not apply. See Nesses v. Shepard, 68 F.3d 1003, 1004 (7th Cir.1995); see also Rizzo v. Sheahan, 266 F.3d 705, 714 (7th Cir.2001); Garry, 82 F.3d at 1367-68.
The district court may have been led astray by two opinions from other circuits discussing Rooker-Feldman in the context of petitions to compel arbitration under the Federal Arbitration Act. In Brown & Root, Inc. v. Breckenridge, 211 F.3d 194 (4th Cir.2000), a case relied upon by Watts, the Fourth Circuit held that Rooker-Feldman barred jurisdiction over the plaintiffs petition to compel arbitration because a state court had previously denied its motion seeking the same relief. The court quoted with approval our decision in Long, which identified the appropriate inquiry as whether the federal plaintiff sought to set aside the state decision or was instead [823]*823presenting an independent claim. Brown & Root, 211 F.3d at 202 (citing Long, 182 F.3d at 555). The Fourth Circuit concluded that the plaintiffs claim was dependent on the state court’s decision denying the petition to compel arbitration, and therefore that Rooker-Feldman barred the federal claim. Id. But like Zurich’s federal claim, the plaintiffs petition to compel arbitration in Brown & Root was not based on any injury by the state court but instead sought to bypass the state court’s adverse ruling and was therefore independent of that decision. See id. Because the Fourth Circuit’s conclusion in Brown & Root is inconsistent with our circuit’s well-settled understanding of the limits of Rooker-Feldman, we respectfully decline to follow it.
By contrast, the Second Circuit, in Doctor’s Associates, Inc. v. Distajo, 107 F.3d 126 (2d Cir.1997), held that Rooker-Feld-man did not apply to the plaintiffs petition to compel arbitration because the federal action was filed before any ruling on the merits in the state court. We need not comment on the district court’s efforts at distinguishing Doctor’s Associates because the Second Circuit’s conclusion in that case was based on its understanding that Rook-er-Feldman “‘at a minimum’ is coextensive with preclusion principles,” 107 F.3d at 137 (quoting Moccio v. New York State Office of Ct. Admin., 95 F.3d 195, 199-200 (2d Cir.1996)), a view that is fundamentally at odds with this court’s interpretation of the doctrine. See, e.g., Durgins, 272 F.3d at 844; Rizzo, 266 F.3d at 713; Centres, Inc., 148 F.3d at 701; GASH Assocs., 995 F.2d at 728 (explaining that “[ejquating the Rooker-Feldman doctrine with preclusion is natural ... [b]ut the two are not coextensive”).
Nevertheless, Watts argues that Rook-er-Feldman not only bars claims that attack the state court decision, but also those that are “inextricably intertwined” with that decision. In determining whether a claim is so intertwined with the judgment as to be barred by Rooker-Feldman, our cases say that we must decide whether the “ ‘district court is in essence being called upon to review the state-court action.’ ” Ritter v. Ross, 992 F.2d 750, 754 (7th Cir.1993) (quoting Feldman, 460 U.S. at 483-84 n. 16, 103 S.Ct. 1303); see also Centres, Inc., 148 F.3d at 702. According to Watts, because Zurich’s petition to compel arbitration might require the district court to reach a conclusion on arbitrability different than the California court’s, it would “in essence” be reviewing the California court’s order.
This circuit has already commented on the difficulty in identifying which claims are “inextricably intertwined” with the judgment. See Brokaw, 305 F.3d at 664; Remer, 205 F.3d at 996; Centres, Inc., 148 F.3d at 702. We have used the phrase to refer to different situations, including those involving claims that are related to the judgment the plaintiff is attacking, but which were not raised before the state court, see, e.g., Levin v. Attorney Registration and Disciplinary Comm’n of the Sup. Ct. of Ill., 74 F.3d 763, 766-67 (7th Cir.1996); Garry, 82 F.3d at 1369; see also Feldman, 460 U.S. at 483 n. 16, 103 S.Ct. 1303, and to claims that were in fact dependent on the judgment, such as if there would be no injury but for the state court’s action. See, e.g., Epps v. Creditnet, Inc., 320 F.3d 756, 759-61 (7th Cir.2003); Long, 182 F.3d at 557; Garry, 82 F.3d at 1366-67; Johnson v. Sup.Ct. of Ill., 165 F.3d at 1142. Zurich’s federal claim is not dependent in any way on the state’s judgment, nor is it related to a claim that is dependent on that judgment. A mere assertion that a district court, in considering a claim that is independent of the state court judgment, might negate a legal conclusion that the state court reached is insufficient to [824]*824trigger application of Rooker-Feldman. See GASH Assocs., 995 F.2d at 728.5
B. Anti-Injunction Act
The Anti-Injunction Act, 28 U.S.C. § 2283, bars a district court from enjoining pending state litigation unless the injunction falls within one of the Act’s three stated exceptions:
A court of the United States may not grant an injunction to stay proceedings in a State court except as expressly authorized by Act of Congress, or where necessary in aid of its jurisdiction, or to protect or effectuate its judgments.
28 U.S.C. § 2288. The Act rests on the “fundamental constitutional independence of the States and their courts,” and its purpose is to make the dual system of state and federal courts work without “ ‘needless friction.’ ” Atlantic Coast Line R.R. Co. v. Broth, of Locomotive Eng’rs., 398 U.S. 281, 286-87, 90 S.Ct. 1739, 26 L.Ed.2d 234 (1970) (quoting Okla. Packing Co. v. Okla. Gas & Elec. Co., 309 U.S. 4, 9, 60 S.Ct. 215, 84 L.Ed. 537 (1940)). We review the district court’s decision to enjoin the state action for abuse of discretion, see In Re VMS Sec. Litig., 103 F.3d 1317, 1323 (7th Cir.1996); Commonwealth Edison Co. v. Gulf Oil Corp., 541 F.2d 1263, 1274 (7th Cir.1976), evaluating its factual findings for clear error, and its legal conclusions, including its interpretation of the scope of the Anti-Injunction Act, de novo. See Winkler v. Eli Lilly & Co., 101 F.3d 1196, 1203-04 (7th Cir.1996).
Because of the Act’s constitutional foundation, its three exceptions are to be applied narrowly, and “[a]ny doubts as to the propriety of a federal injunction against state court proceedings should be resolved in favor of permitting the state courts to proceed in an orderly fashion to finally determine the controversy. The explicit wording of § 2283 itself implies as much, and the fundamental principle of a dual system of courts leads inevitably to that conclusion.” Atlantic Coast Line, 398 U.S. at 297, 90 S.Ct. 1739; see Ramsden v. AgriBank, FCB, 214 F.3d 865, 869 (7th Cir.2000). And even if the injunction is authorized by one of the exceptions, a district court must still determine whether an injunction is an appropriate exercise of its authority, Winkler, 101 F.3d at 1203; see also Chick Kam Choo v. Exxon Corp., 486 U.S. 140, 151, 108 S.Ct. 1684, 100 L.Ed.2d 127 (1988) (“The fact that an injunction may issue under the Anti-Injunction Act does not mean that it must issue.”), recognizing the respect due the courts of a sovereign state. See Mitchum v. Foster, 407 U.S. 225, 243, 92 S.Ct. 2151, 32 L.Ed.2d 705 (1972) (noting that the Court’s holding that the injunction was authorized by an exception did not “qualify in any way the principles of equity, comity, and federalism that must restrain a federal court when asked to enjoin a state court pro[825]*825ceeding”). The extraordinary relief of an injunction of state court proceedings must also be supported by the traditional equitable requirements such as irreparable harm for which there is no adequate remedy at law. See Owens-Corning Fiberglas Corp., 959 F.2d at 635 (citing Pennzoil Co. v. Texaco Inc., 481 U.S. 1, 10-11, 107 S.Ct. 1519, 95 L.Ed.2d 1 (1987)); see also Ramsden, 214 F.3d at 868 (7th Cir.2000).
Zurich does not argue that the exception for injunctions “expressly authorized” by Congress applies, nor does it argue, at least not directly, that the court had authority to enjoin the state proceedings “to protect or effectuate its judgments.” 6 Instead, Zurich asserts that the district court was correct in holding that the exception for injunctions “necessary in aid of its jurisdiction” applies. We disagree.
“Necessary in aid of its jurisdiction” means that “federal injunctive relief may be necessary to prevent a state court from so interfering with a federal court’s consideration or disposition of a case as to seriously impair the federal court’s flexibility and authority to decide that case.” Atlantic Coast Line, 398 U.S. at 295, 90 S.Ct. 1739. Historically, this exception applied only to in rem rather than in personam proceedings, see Vendo Co. v. Lektro-Vend Corp., 433 U.S. 623, 641, 97 S.Ct. 2881, 53 L.Ed.2d 1009 (1977) (plurality opinion), the general rule being that parallel in personam proceedings should be allowed to continue concurrently. Id. at 641-42, 97 S.Ct. 2881; Commonwealth Edison Co., 541 F.2d at 1274. Thus, the possibility that a parallel state proceeding might “interfere with a protected federal right” or erroneously apply federal law does not make an injunction “necessary” to aid the court’s jurisdiction:
[T]he state and federal courts had concurrent jurisdiction in this case, and neither court was free to prevent either party from simultaneously pursuing claims in both courts.... Therefore the state court’s assumption of jurisdiction over the state law claims and the federal preclusion issue did not hinder the federal court’s jurisdiction so as to make an injunction necessary to aid that jurisdiction.
Atlantic Coast Line, 398 U.S. at 294-96, 90 S.Ct. 1739; Texas v. United States, 837 F.2d 184, 186 n. 4 (5th Cir.1988). Instead of an injunction by the federal district court, the aggrieved party’s recourse is by appeal through the state court system and, ultimately, to the Supreme Court. Atlantic Coast Line, 398 U.S. at 296, 90 S.Ct. 1739.
There has been some limited expansion of this exception beyond in rem actions, most notably in the context of school desegregation cases, see Swann v. Charlotte-Mecklenburg Bd. of Ed., 501 F.2d 383 (4th Cir.1974), and consolidated multidistrict litigation, see Winkler, 101 F.3d at 1202. In Winkler, for example, we held that a federal court managing multidistrict litigation was authorized to enjoin parallel state proceedings in order to protect the integrity of its pretrial discovery orders. Id. at [826]*8261203. In this case, by contrast, there were no federal court orders to protect, and Zurich has cited no circuit court case that expands the “in aid of its jurisdiction” exception under similar circumstances.7
Nevertheless, Zurich asks us to extend the exception to this case because of the important federal interest favoring arbitration represented in the Federal Arbitration Act. It asserts that, absent an injunction, the state court could (before the district court has a chance to rule on the petition for arbitration) make rulings that, by operation of preclusion principles, might bind Zurich in later arbitration. This would be contrary, according to Zurich, to the federal interest supporting the parties’ selection of an arbitral rather than judicial forum. A similar argument for such an expansion of the “in aid of its jurisdiction” exception was rejected by the Eleventh Circuit, which held that the exception does not apply if the federal court has not yet ordered arbitration. See TranSouth Financial Corp. v. Bell, 149 F.3d 1292, 1297 (11th Cir.1998).
Furthermore, Zurich’s argument is based on an assumption that the state court cannot competently protect the parties’ federal rights. This assumption, however, strikes at the heart of the Anti-Injunction Act, which “evidences confidence in state courts.” See Commonwealth Edison Co., 541 F.2d at 1274; see also Chick Kam Choo, 486 U.S. at 149-50, 108 S.Ct. 1684 (state courts “are presumed competent to resolve federal issues”). Zurich’s argument is also undermined by the structure of the FAA, which provides concurrent jurisdiction to states to enforce arbitration agreements, with a federal forum available only when diversity or federal question jurisdiction is otherwise established. See 9 U.S.C. § 4; Moses H. Cone Mem. Hosp. v. Mercury Constr. Corp., 460 U.S. 1, 25 n. 32, 103 S.Ct. 927, 74 L.Ed.2d 765 (1983). Although the FAA represents “federal policy to be vindicated by the federal courts where otherwise appropriate,” Moses H. Cone, 460 U.S. at 25 n. 32, 103 S.Ct. 927, it does not suggest that state courts are less competent to give effect to its provisions. Ultracashmere House, Ltd. v. Meyer, 664 F.2d 1176, 1180 (11th Cir.1981) (noting that Congress “viewed state courts as fully capable of effectuating those policies”); CIGNA Healthcare of St. Louis, Inc. v. Kaiser, 294 F.3d 849, 854 (7th Cir.2002) (The FAA “requires both federal and state courts to apply a federal common law of arbitrability to the arbitration of disputes arising under contracts in commerce, a common law applicable equally in federal and state courts.”) (citing Allied-Bruce Terminix Cos. v. Dobson, 513 U.S. 265, 272, 115 S.Ct. 834, 130 L.Ed.2d 753 (1995); Southland Corp. v. Keating, 465 U.S. 1, 14-15, 104 S.Ct. 852, 79 L.Ed.2d 1 (1984)).
This case is therefore not so much about preserving the federal interest in arbitration, but in preserving Zurich’s choice of a federal forum to enforce the arbitration agreement. Under similar circumstances, we have held that a district court may abstain from consideration of the arbitration question when the issue was already before, and much closer to resolution in, the state court. See CIGNA Healthcare, 294 F.3d at 855. Although not controlling on the question before us, CIGNA Health[827]*827Care’s holding that the plaintiffs choice of a federal forum may give way to principles of judicial economy takes away much of the force of Zurich’s argument that its interest justifies expansion of the “in aid of its jurisdiction” exception. If the ordinary rule that parallel proceedings should be allowed to continue can be overcome in appropriate circumstances by reasons of judicial economy, allowing the district court to abstain from consideration of the arbitration question, see id. at 853, it is hard to see why concurrent state proceedings could be seen as such a serious threat to the district court’s authority that they must be halted by an injunction. Without, at the very least, some evidence that the state court was indifferent to or hostile to the parties’ rights under the FAA, see Diane P. Wood, Fine-Tuning Judicial Federalism: A Proposal for Reform of the Anti-Injunction Act, 1990 BYU L.Rev. 289, 316 (suggesting that injunction might be warranted if the state law was hostile to arbitration and refused a stay); cf. CIGNA HealthCare, 294 F.3d at 855, a state court’s erroneous application of federal law does not justify the extraordinary remedy of an injunction and is not authorized by the “in aid of its jurisdiction” exception. See Atlantic Coast Line, 398 U.S. at 295-96, 90 S.Ct. 1739; Texas v. United States, 837 F.2d at 186 n. 4 (“In no event may the ‘aid of jurisdiction’ exception be invoked merely because of the prospect that a concurrent state proceeding might result in a judgment inconsistent with the federal court’s decision.”).
In this case, the California Superior Court rejected Zurich’s request for a stay, but we find nothing to suggest that, in reaching that decision, it was indifferent to its responsibilities under the FAA. The court’s decision was based on its conclusion that the dispute under the deductible agreements was severable from the' dispute pending before it concerning Zurich’s duties under the insurance policies. The court also held that no arbitrable dispute had yet arisen and expressed its willingness to consider the question again if defense costs were paid and reimbursement under the deductible agreements was disputed. The state court’s conclusions differed from the district court’s preliminary determination — in evaluating Zurich’s likelihood of success on the merits — that all issues as to Watts (not already decided in the California litigation) were subject to arbitration. Whether or not the state court’s conclusions are ultimately correct (an issue upon which we offer no opinion), they were not so unreasonable as to suggest hostility to the requirements of the FAA.8 And we have no reason to believe that, if presented with a federal judgment compelling arbitration of issues it had thought were not arbitrable, the Superior Court would refuse to give effect to that judgment or to stay further' proceedings. See CaLCode of Civ. P. § 1281.4 (requiring court in which action is pending to stay proceedings if a “court of competent jurisdiction” has ordered arbitration); cf. Moses H. Cone, 460 U.S. at 26, 103 S.Ct. 927 (“[Sjtate courts, as much as federal courts, are obliged to grant stays of litigation under § 3 of the Arbitration Act.”). If it did refuse, then an injunction might be warranted under the “protect or effectuate its judgments” rationale. See We Care Hair Dev., Inc. v. Engen, 180 F.3d 838, 844 (7th Cir.1999); Commonwealth Edison Co., 541 F.2d at 1273; D. Wood, supra, at 316. But given principles of comity, we [828]*828cannot lightly assume that a state court would disregard federal law.9
III. CONCLUSION
The district court abused its discretion in issuing the preliminary injunction, and the injunction is therefore Reveíised. The case is Remanded to the district court for further proceedings. Each party will bear its own costs on appeal.