When does idolization cross over into obsession? Cary Hoffman, a shy Jewish kid growing up in postwar Queens, admired Frank Sinatra so much that he dreamed of becoming the singer himself. In Hoffman’s thought-provoking one-man show, “My Sinatra,” now playing Off-Broadway with musical direction by Alex Nelson, the performer interweaves the story of his infatuation with the singing of two dozen of the singer’s standards. His voice is so uncannily similar to Sinatra’s that few can tell them apart.

Anti-Semitism was never far below the surface of the notorious blacklist of the 1950s. Did sexism play a role as well? In Julie S. Halpern’s new play, “Diminished Fifth,” two women with Jewish roots, writers Lillian Hellman (Stacey Scotte) and Dorothy Parker (Jacquelyn Poplar), along with three non-Jewish women, broadcaster Jean Muir (Mary McGloin), actress Margaret Webster (Elaine LeGaro) and civil rights activist Eslanda Robeson (Ronalda Ay Nicholas), grapple with the shattering experience of being blacklisted.