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?

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There is a minimal way to imbed a nonunital $C^*$-algebra $A$ into a unital $C^*$-algebra. As a $*$-algebra this is $A\oplus \mathbb C$ with componentwise addition, with multiplication $(a,s)(b,t)=(ab+ta+sb,st)$, and with involution $(a,s)^*=(a^*,\overline s)$. But to give this algebra a $C^*$ norm, one method is to identify it with $\{L_a:a\in A\}+\mathbb C\mathrm{id}_A\subset B(A)$, where $L_a:A\to A$ is defined by $L_ab=ab$. One can then check that the operator norm of this algebra as a subspace of $B(A)$ is a $C^*$ norm.

There is also a maximal way to imbed a nonunital $C^*$-algebra $A$ into a unital $C^*$-algebra as an ideal in an “essential” way. The essentialness is captured by stipulating that every nonzero ideal in the unitization intersects $A$ nontrivially. This is equivalent to the condition that $bA=\{0\}$ implies $b=0$. As mland mentioned, this maximal unitization is the multiplier algebra of $A$, $M(A)$. The double centralizer approach is one particular concrete description, but $M(A)$ has other decriptions and is characterized by a universal property: For every imbedding of $A$ as an essential ideal in a $C^*$-algebra $B$, there is a unique $*$-homomorphism from $B$ to $M(A)$ that is the identity on $A$.

t.b. has already mentioned that in the commutative case this runs parallel to one-point versus Stone–Čech compactification.

Here is another example. The algebra $K(H)$ of compact operators on an infinite dimensional Hilbert space $H$ has minimal unitization (isomorphic to) $K(H)+\mathbb CI_H$, and multiplier algebra (isomorphic to) $B(H)$.

One reason we may want to go all the way to $M(A)$ is to better understand automorphisms of $A$. Conjugation by a unitary element of $M(A)$ is an automorphism of $A$. In the case of $K(H)\subset B(H)\cong M(K(H))$, every automorphism is of this form, and you couldn’t get most of these automorphisms by only conjugating by unitaries in the minimal unitization $K(H)+\mathbb C I_H$.

The approach mentioned by mland of identifying $M(A)$ with the algebra of adjointable operators on $A$ can be found in Lance’s Hilbert C*-modules or in Raeburn and Williams’s Morita equivalence and continuous trace C*-algebras with a lot more useful introductory information in each. I agree with mland that for the basics of K-theory you do not need to get into multiplier algebras, but you can learn more about their importance in K-theory from Blackadar’s K-theory for operator algebras. Chapter VI is described as a collection of “all the results needed for Ext-theory and Kasparov theory,” and it starts with a review of multiplier algebras and examples.

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.