In (5.6.3) of Pedersen’s book (1) the author states without proof or reference that there are non-unital C*-algebras whose Pedersen
ideal is the whole algebra. Does anyone know where can I find such an example?
(1) Pedersen, Gert K., C*-algebras and their automorphism groups, London Mathematical Society Monographs. 14. London – New York -San Francisco: Academic Press. X, 416 p. $ 60.00 (1979). ZBL0416.46043.