I would like to know which $C^\*$-algebras are ideals in their second duals?


There is a [paper by S. Watanabe](https://projecteuclid.org/download/pdf_1/euclid.nihmj/1273778977) that claims in introduction that it is well known that a $C^\*$-algebra is an ideal in its second dual iff it is a dual $C^\*$-algebra. But I do not know what does he mean by term "dual $C^\*$-algebra".


For the general case of normed algebras there is a criterion (see Banach algebras and the general theory of \*-algebras. Volume I. Algebras and Banch algebras. Theodore W. Palmer, theorem 1.4.13): $A$ is a two-sided ideal in $A^{\*\*}$ iff the maps $L\_a:A\to A$, $b\mapsto ab$ and $R\_a:A\to A$, $b\mapsto ab$ are weakly compact for all $a\in A$.


May be this criterion could be improved for the case of $C^\*$-algebras. You may even assume that $A$ is commutative.



When a $C^*$-algebra is an ideal in its second dual?
Warning: the following is just what I found from some work on MathSciNet, following Yemon's hint in the comments. It's not meant to be accurate historical notes.


A "dual" $C^\ast$-algebra is defined as follows. Let $A$ be an algebra and for a subset $M\subseteq A$ let $R(M) = \{ x\in A : Mx=\{0\}\}$; similarly define $L(M) = \{ x\in A : xM=\{0\}\}$. Then a $C^\ast$-algebra is *dual* if for each closed left ideal $I$ we have that $L(R(I))=I$. (The involution can be used to show that the analogous definition with right ideals gives the same notion).


An early paper which studied these is [Kaplansky, The structure of certain operator algebras](https://www.ams.org/journals/tran/1951-070-02/S0002-9947-1951-0042066-0/), see section 2. It seems that [Berglund, Ideal $C^\ast$-algebras](https://projecteuclid.org/euclid.dmj/1077309715) was the first to obtain the equivalence you seek (why is the Duke journal archive behind a paywall??) A short proof is in [McCharen, A characterization of dual $B^{\ast}$-algebras](https://www.ams.org/journals/proc/1973-037-01/S0002-9939-1973-0306927-7/) These give the following:


**Claim:** A $C^\ast$-algebra $A$ is an ideal in its bidual if and only if $A$ is dual.


We now combine this with known characterisations of dual $C^\ast$-algebras:


**Claim:** A $C^\ast$-algebra $A$ is dual if and only if $A$ is isomorphic to a $C^\ast$-subalgebra of $K(H)$ for some $H$ if and only if $A$ is the $c\_0$-direct sum of algebras of the form $K(H)$ for some (finite or infinite dimensional) $H$.


Thus $K(H)$ and $c\_0$ really are the archetypal examples. These results are quoted in the paper of Kaplansky I linked above, and in Dixmier's book (English edition, section 4.7.20) though in both cases it seems only further references are given, not proofs. I am afraid that I don't know of a modern, self-contained treatment.


