In [1, 5.6.3] Pedersen 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?
* Is it possible to characterize algebras with this property?




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[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](https://zbmath.org/?q=an:0416.46043).



Is there a C*-algebra whose Pedersen ideal is not proper?
Examples include all non-unital algebraically simple $C^\ast$-algebras. By [Blackadar, Bruce E.; Cuntz, Joachim The structure of stable algebraically simple C∗-algebras. Amer. J. Math. 104 (1982), no. 4, 813–822.] a simple, stable $C^\ast$-algebra is algebraically simple if and only if it contains an infinite projection. So an explicit example would be a stabilised Cuntz algebra $\mathcal O\_n \otimes \mathcal K$.


Addon: in his original paper [Pedersen, Gert Kjaergȧrd, Measure theory for C∗ algebras. Math. Scand. 19 (1966), 131–145], Pedersen gives the following example of a non-unital $C^\ast$-algebra $A$ with $\mathrm{Ped}(A) = A$: Let $H$ be a non-separable Hilbert space, and let $A$ be the $C^\ast$-subalgebra of $B(H)$ of operators whos range projection project onto a separable subspace of $H$. This $C^\ast$-algebra is non-unital and $\mathrm{Ped}(A) = A$. To see this, let $a\in A$ and $p$ be the range projection of $a$. Then $p\in \mathrm{Ped}(A)$ and $a = pa \in \mathrm{Ped}(A)$ since $\mathrm{Ped}(A)$ is an ideal.


