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Dec 30, 2022 at 11:57 comment added J. De Ro @t.c. Per your last paragraph: I don't think you don't need unitality of the $C^*$-algebra for this.
Dec 8, 2022 at 23:41 comment added Andromeda Thanks! For those interested, it is theorem 15.4.2 and remarks following this in the book that is relevant here.
Dec 8, 2022 at 23:40 vote accept Andromeda
Dec 8, 2022 at 22:59 history answered user85913 CC BY-SA 4.0