From Judge Stephen V. Wilson's opinion in Tulsi Now, Inc. v. Google, LLC, which strikes me as quite correct:

Plaintiff's essential allegation is that Google violated Plaintiff's First Amendment rights by temporarily suspending its verified political advertising account for several hours shortly after a Democratic primary debate. Plaintiff's claim, however, "runs headfirst into two insurmountable barriers—the First Amendment and Supreme Court precedent." Prager Univ. v. Google LLC, No. 18- 15712, 2020 WL 913661, at *1 (9th Cir. Feb. 26, 2020). The First Amendment provides: "Congress shall make no law … abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble …." "The First Amendment, applied to states through the Fourteenth Amendment, prohibits laws abridging the freedom of speech." In effect, "the First Amendment means that government has no power to restrict expression because of its message, its ideas, its subject matter, or its content." Google is not now, nor (to the Court's knowledge) has it ever been, an arm of the United States government. "The text and original meaning of those Amendments, as well as this Court's longstanding precedents, establish that the Free Speech Clause prohibits only governmental abridgment of speech. The Free Speech Clause does not prohibit private abridgment of speech." Manhattan Cmty. Access Corp. v. Halleck, 139 S. Ct. 1921, 1926 (2019) (emphasis in original). Plaintiff alleges Google has become a state actor by virtue of providing advertising services surrounding the 2020 presidential election.

"Under [the Supreme] Court's cases, a private entity can qualify as a state actor in a few limited circumstances—including, for example, (i) when the private entity performs a traditional, exclusive public function; (ii) when the government compels the private entity to take a particular action; or (iii) when the government acts jointly with the private entity." Plaintiff's argument is that, by regulating political advertising on its own platform, Google exercised the traditional government function of regulating elections. "To draw the line between governmental and private, this Court applies what is known as the state-action doctrine. Under that doctrine, as relevant here, a private entity may be considered a state actor when it exercises a function 'traditionally exclusively reserved to the State.'" Traditional government functions are defined narrowly. "It is not enough that the federal, state, or local government exercised the function in the past, or still does. And it is not enough that the function serves the public good or the public interest in some way. Rather, to qualify as a traditional, exclusive public function within the meaning of our state-action precedents, the government must have traditionally and exclusively performed the function." "Under the Court's cases, those functions include, for example, running elections and operating a company town." There is no argument that webservices or online political advertising are traditionally exclusive government functions. Plaintiff argues that, by providing some restriction on political advertising on its platform, Google is in effect regulating elections.

Disclosure: I have represented Google as a lawyer, including in writing a white paper arguing that the First Amendment protects search engine results, though that is a different question than the one I'm discussing here; I have not been asked to blog about this, and I am speaking entirely for myself here.