Unitisation of $C^{*}$-algebras via double centralizers In most of the books I read about $C^{*}$-algebras, the author usually embeds the algebra, say, $A$, as an ideal of $B(A)$, the algebra of bounded linear operators on $A$, by identifying $a$ and $M_a$, the left multiplication of $a$.
However, in Murphy's $C^{*}$-algebras and operator theory, $A$ is embedded as an ideal of the space of 'double centralizers'. See p39 of his book.
I do not quite understand why we need this complicated construction since the effect is almost the same as the usual embedding. The author remarked that this construction is useful in certain approaches to K-theory, which further confuses me.
Can somebody give a hint?
Thanks!

The set of double centralizers of a $C^*$-algebra $A$ is usually also called the multiplier algebra $\mathcal{M}(A)$. It is in some sense the largest $C^*$-algebra containing $A$ as an essential ideal and unital. If $A$ is already unital it is equal to $A$. (Whereas in your construction of a unitalisation we have that for unital $A$ the unitalisation is isomorphic as an algebra to $A\oplus \mathbb{C}$.
Multiplier algebras can also be constructed as the algebra of adjointable operators of the Hilbert module $A$ over itself. Since in $KK$ theory Hilbert modules are central object, and $K$ theory is special case of $KK$ theory this could be one reason why it is good to introduce these multiplier algebras quite early.
But if you want to learn basic theory I think this concept is not that important yet.
