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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!

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    Have you done the exercise of identifying unitisations in the commutative case with compactifications of the spectrum? For each compactification of the spectrum you obtain a unitization and vice versa. In the proper formulation (which is part of the exercise) you have that the minimal unitization $A \oplus \mathbb{C}$ corresponds to the one-point compactification while the compactification via double centralizers corresponds to the Stone-Cech compactification.2012-07-05
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    @t.b. Oh, I should have done that. I will definitely look at the exercise. Thanks, t.b.!2012-07-05
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    On a completely different (maybe slightly silly) note: I noticed that you asked 47 questions and you only cast 6 votes in total, while you often acknowledged that the answers you got were helpful. On this site the idea is that votes are a means to express appreciation and to "reward" the answerers symbolically for their efforts. It's just a mouse click after all.2012-07-06
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    @t.b. I totally understand that. I often use my iPad to ask questions and view the answers, but the up vote button does not work quite well on an iPad- it's too small and sometimes not responding to clicks. Anyway, I will up vote those helpful comments and answers when I get back today with my computer.2012-07-06

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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.

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    +1 I wonder, why can't people write books the same way you write answers?2012-07-05
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    Amazingly informative answer! Thanks, Jonas! It seems you have read all books on operator algebra.2012-07-05
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    @Norbert: Thank you. I think that the books mentioned above (and many other books) are written very well.2012-07-06
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    @HuiYu: Thank you. If only I had even read in their entirety all three of the books mentioned in this answer, I'd be better educated.2012-07-06
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    That's a very nice answer indeed. I myself am partial to the double centralizer viewpoint since it is purely algebraic in that it needs no involution nor anything else: it's a natural generalization of (the image of) the map $(\lambda_a,\rho_a) : A \to {\rm End}_k(A) \times {\rm End}_k(A)^{\rm op}$ obtained from the left and right regular representations of a $k$-algebra $A$ and as such goes back to [Hochschild](http://projecteuclid.org/euclid.dmj/1077474484), 3.1 (cont.)2012-07-06
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    (cont) The universal property can be phrased algebraically (if $A$ is non-degenerate) and the map $B \to D(A)$ granted by the universal property (which is $b \mapsto (\lambda_b|_A, \rho_b|_A)$) just happens to be a \*-homomorphism (and is unique) if $A$ is an (essential) \*-ideal of $B$. I also think the [original paper](http://dx.doi.org/10.1090/S0002-9947-1968-0225175-5) by Busby is very nice. Finally, I'd like to mention Wendel's identification of $D(L^1(G))$ with $M(G)$ for a locally compact group, that can be pushed further to show that $L^1(G)$ and $M(G)$ determine $G$ and its topology.2012-07-06
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    @t.b.: I am more grateful for your comments than comment upvotes can express. Thank you! I look forward to learning more about this.2012-07-06
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    You're very welcome. If you would be interested I could post an excerpt (maybe two or three pages) of some old lecture notes of mine here where I wrote a short summary (with proofs) of some of the basic facts on double centralizers. Most of this is extracted from Busby's paper and Johnson's article Busby cites and should also be in Palmer's or Dales's huge books on Banach algebras, so I'm only going to do this this if you're really interested (I'm pretty busy these days and it would take some time to adapt the LaTeX to the answer format). It's not going to be a very polished or deep thing.2012-07-08
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    @t.b.: Thank you for the offer. I am ambivalent: I'm sure I would learn much from them, but I do not feel so strong a desire as to ask you to go through all of that work, especially at an inopportune time.2012-07-09
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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.