About halfway through “Satchmo at the Waldorf,” a sharp and engrossing play about Louis Armstrong, the actor playing Armstrong at The Stage in San Jose walks toward the audience and shows off the Star of David pendant hanging around his neck.

“I wear it every day ’cause the Jews, they been so good to me,” he says in his gravelly, New Orleans lilt. “Maybe that’s why I trusted Mr. Glaser.”

That would be Joe Glaser, a fast-talking Jew from Chicago and an associate of Al Capone’s who managed Armstrong for much of his career. In “Satchmo,” set in March 1971, a few months before Armstrong died, the great jazz trumpeter and singer reminisces in a New York City dressing room about his life and career, often returning to stories involving Glaser. The complicated relationship between the two men lies at the heart of the play, which runs through Feb. 26.

(“Satchmo” was one of Armstrong’s nicknames; contrary to a meme that circulated online in 2021, the word does not mean “big cheeks” in Yiddish. It is derived from “satchel mouth,” a reference to the fact that Armstrong had a particularly large mouth.)

‘Satchmo’ plays with the idea that Armstrong and Glaser were the same person.

In a remarkable feat of vocal and physical dexterity, veteran Bay Area actor L. Peter Callender plays Armstrong, Glaser and Miles Davis. The latter appears mostly to denounce Armstrong for pandering to white audiences and allowing Glaser to be his boss, rather than the other way around. With a simple change in lighting, Callender snaps from character to character. The effect is magical, and there is something avant-garde about Callender, a Black non-Jewish actor, portraying on stage a real-life white Jewish man with a nasally, high-pitched voice. (Last year, John Douglas Thompson became the first Black actor to play Shylock in a production of “The Merchant of Venice” in New York.)

Written by the late drama critic and Armstrong biographer Terry Teachout and directed by Ted Lange, “Satchmo” plays with this idea that Armstrong and Glaser were the same person, at least when it came to business. Armstrong says Glaser once told him, “I’m you, Louie, and you’re me. You’re my boy.”

But, of course, in early 20th-century America, the two men were seen and treated very differently based on the color of their skin. Glaser recounts going on the road with Armstrong to the Jim Crow South. “Hadda pack two pistols on my belt,” Glaser says. “Used to go into restaurants and carry food back to the bus in paper bags ’cause they wouldn’t serve colored inside. They’d serve a Jew before they’d serve colored.” At this, he adds an expletive.

In the play, Glaser takes credit for turning Armstrong into a big star by encouraging him to play his trumpet less and sing more as a way of appealing to white audiences not hip to jazz music. His philosophy was: “They don’t care about jazz. It’s Louie they love.” Armstrong did as Glaser said, recording hits such as “Hello, Dolly!” And to hear each man tell it, the arrangement worked out well for both. Yet Armstrong was hurt that Glaser didn’t leave him part of his talent-booking business — a business he built on Armstrong’s back — in his will. “All he did was leave me a tip, and that hurt!” Armstrong says.

Many Jewish fans of Armstrong’s may already be familiar with one of the stories that Callender-as-Armstrong relates from the musician’s childhood. Growing up in New Orleans, Armstrong worked for a Jewish family of junk collectors, the Karnofskys, who had immigrated from Lithuania. They fed him and treated him “like family,” as he puts it. They even loaned him money to buy his first cornet. In one of the play’s more touching scenes, Armstrong leans back in his chair and sings a Shabbat song the Karnofskys taught him, “D’ror Yikra.”

“Satchmo” arrives in San Jose at a propitious moment, and not just because this is Black History Month. A few months ago, another Black entertainer — Ye, the rapper formerly known as Kanye West — unleashed a torrent of antisemitic remarks, including this one: “Jewish people have owned the Black voice, whether it’s through us wearing the Ralph Lauren shirt, or it’s all of us being signed to a record label, or having a Jewish manager ….” He might have been talking about Joe Glaser.

Meanwhile, “You People,” a Netflix rom-com released last month that revolves around a white Jewish-Black Muslim couple, has been roundly criticized for reinforcing stereotypes about both Jews and Black people, and for further driving a wedge between the two communities in the wake of the Ye ordeal.

“Satchmo” offers another opportunity for reflection on Black-Jewish ties. The character Joe Glaser is certainly crass. He uses a Yiddish slur for Black people several times, and he expresses racist views about them as a group. Yet he clearly has real affection for Armstrong, just as Armstrong has real affection for him. Does his partnership with Armstrong excuse his bigotry? Did he exploit Armstrong, or did he protect him, or both? Would the world have come to know and love Armstrong without Glaser’s help — or without the Karnofskys’?

These are just a few of the questions that the play raises. If you need to make a choice about how to spend two hours of leisure time this month, skip “You People” and see “Satchmo” instead.

“Satchmo at the Waldorf”

Through Feb. 26 at San Jose Stage Company, 490 S. 1st Street, San Jose. $34-$74.