In the book: Higson N, Roe J. Analytic K-homology[M]. Oxford University Press, 2000. The Definition 5.1 is about dual of a C*-algebra, In simple term, embedding a C*-algebra a into a operator algebra B(H) of Hilbert space H, the dual of C*-algebra A is D(A)={T$\in$B(H); [T,a]~0 for any a$\in$A}, here [T,a]~0 means Ta-aT is compact operator.
Why do we call it dual C*-algebra? I think it is totally different with the spectrum of C*-algebras, maybe in some other meaning, but I do not find other books containing this name in the meaning.
