Pops, the much-admired 2009 Louis Armstrong biography by Wall Street Journal drama critic Terry Teachout, was essentially a work of redemption. While even Armstrong’s harshest critics acknowledged the originality of his trumpet playing, he didn’t become beloved by being a genius. He got there by being an entertainer. By performing that beatific eye-roll and smile. By letting Joe Glaser—his powerful, mob-connected manager, whom Armstrong called “my Daddy” in Glaser’s 1969 New York Times obituary—keep him on the road 300 nights per year, playing to increasingly white audiences. By recording material white audiences loved, even if Armstrong thought it was shit: Most notably, the title song of the musical “Hello, Dolly!”Armstrong thought so little of Jerry Herman’s tune that neither he nor his band could recall it soon after the session where they’d cut it. But their version was huge, ending The Beatles’ 14-week reign of the pop charts in 1964.

In other words, Armstrong achieved immortality by being—in the estimation of younger peers like Dizzy Gillespie and Miles Davis—an Uncle Tom. And over the decades, Teachout argued, increasing discomfort with the grinning, non-threatening persona Armstrong cultivated made critics undervalue his contributions to music and culture.

Teachout’s one-man-play Satchmo at the Waldorf, which followed his book by a couple of years and had a long off-Broadway run in 2014, compresses this line of thinking into a series of monologues set mostly in Armstrong’s dressing room. (Set designer Andrew Cohen evokes the luxury of a vanished era in Mosaic Theater Company’s sturdy new production.) It’s 1971, and Armstrong is in the last weeks of his life, having traded his brutal touring schedule for a cushy residency at the Waldorf Astoria Hotel. Backstage huffing oxygen from a can, he is moved to defend his legacy. 

Craig Wallace, the ever-reliable D.C. actor who embodies Armstrong, also slips into the personas of two other figures: Glaser, the manager who engineers his client’s transformation into a mainstream entertainer, and Davis, who acknowledges Armstrong’s gift while condemning his pandering. Wallace sells these quick-changes easily, straightening his posture and adopting a more nasal tone when performing as Glaser, and donning sunglasses and a quicker cadence when he’s Davis. Teachout’s decision to have a single actor play all three parts has more to do with practicality than psychology: This Glaser tells us things Armstrong never knew.

It’s only so helpful to compare Satchmo with Signature Theatre’s concurrent show about a pioneering black musician with a complicated sense of his identity, Jelly’s Last Jam.For one thing, this is a play, with no singing or dancing and just a smattering of Armstrong recordings played over scene changes. But if it gives us fewer examples of its subject’s genius, it offers more insight into how its subject thought of himself, as supplied by … Louis Armstrong. 

He began making audiotapes of his performances, but also of his conversations and private reflections, in 1947. Teachout was the first biographer to take advantage when these tapes became available in 2002. The play quotes at least one passage from the tapes verbatim, when Armstrong observes, “Every white man in the world got one [n-word] at least—they just love his dirty drawers.” A little later, he laments that the black audience that used to come see him in New York City has moved on, now preferring to hear Davis (“a stuck-up doctor’s son”) play the Village Vanguard. But he can still count on white crowds to turn out. 

Wallace lets his mostly-white audience absorb that and laugh before continuing. This is not the first production of Satchmo where this fourth-wall-breaking joke has operated in this way, and it won’t be the last.

At the Atlas Performing Arts Center to Sept. 25.1333 H St. NE. $20-$40. (202) 399-7993. atlasarts.org.