# Stone Čech Compactification And Multiplier Algebras

26 Aug 2022 -

There is an isomorphism theorem of Gelfand, detailed in the next paragraph. It ends up giving an (anti)-equivalence of categories between $$C^\ast$$-algebras and the category of locally compact haussdorf spaces. It gives rise to the philosophy that $$C^\ast$$ algebras are really the study of "non-commutative topology". One of the ideas here is that phenomena in topology should have their counterparts in $$C^\ast$$-algebras. One of the most important examples of this is the extension of $$K$$-theory to $$C^\ast$$-algebras. Today, we do something simpler but still very much striking. We talk about compactifications and their analogues, and how we can use this interplay between topology and $$C^\ast$$-algebras to learn more about compactifications.

## The Gelfand Isomorphism

Recall a Banach space is a normed $$\bb{C}$$-vector space complete under the induced metric. A Banach algebra in turn is a Banach space with a multiplication, such that $$\norm{ab}\leq \norm{a}\norm{b}$$. The simplest such objects to study are the abelian Banach algebras. For each unital abelian Banach algebra $$A$$, take $$\Spec(A)$$ to be the space of characters, i.e homomorphisms $$A\lra{\gamma}\bb{C}$$. We can give $$\Spec(A)$$ the weak*-topology and its easy to check due after applying the Banach-Alaoglu theorem that this is infact a compact space. Now we get a representation $\Gamma: A \lra{} C(\Spec(A)),\quad \Gamma(a)(\gamma) = \gamma(a)$ The norm on $$C(X)$$ for compact haussdorf $$X$$, the space of continous functions $$X\lra{}\bb{C}$$, is $$\norm{f}_\infty= \mathrm{sup}_{x\in X} |f(x)|$$. We call this the Gelfand representation. Note we could have asked for $$A$$ to be non-unital, and then the character space would have been locally compact haussdorf. We would have to use $$C_0(\Spec(A))$$ then, the space of continous functions like before but they go to zero at infinity (think about infinity by thinking about the one point compactification.).

A $$C^\ast$$ algebra is an algebra with an involution $$\ast$$ such that $$\norm{a^\ast a} = \norm{a}^2$$. Note that if pointwise complex conjugation is the involution, both $$C(Y)$$ and $$C_0(X)$$ are $$C^\ast$$-algebras.Now the big theorem due to Gelfand is that for commutative $$C^\ast$$ algebras, the Gelfand representation is an isomorphism. This tells us each non-unital commutative $$C^\ast$$-algebra is dual to some LCH space, and that every unital commutative is dual to some compact haussdorf space. A lot of $$C^\ast$$ theory now is to see how much of this isomorphism theorem we can translate over to the non-commutative case. So in a sense $$C^\ast$$-algebra is the study of non-commutative topology. With that being said, we expect concepts in topology to carry over to $$C^\ast$$-algebras and vice versa. We will see that for compactification today.

I should mention that this really is an (anti-)equivalence of categories between commutative $$C^\ast$$ algebras and the category of LCH spaces with continuous maps. Send a space to its $$C_0$$ and a commutative algebra to its spectrum.

## Unitizations and Compactifications

We shall say compact $$Y$$ is a compactification of LCH $$X$$ (non-compact) if

• There is an embedding $$\iota: X \hookrightarrow Y$$
• The image $$\iota(X)$$ is dense in $$Y$$
• The easiest example of this is $$\opc{X}$$, the one point compactification. This is just adding a single point to $$X$$, call it $$\infty$$, to make it compact. We can also write it by its universal property, that any compactification of $$X$$ maps into $$\opc{X}$$, i.e $$\opc{X}$$ is the "smallest" compactificaiton.

As mentioned before, the philosophy is that every topological concept should have an analogue in operator theory. So lets look at $$C(\opc{X})$$ towards that end. Notice that we have a map $$C(\opc{X})\lra{\cong} C_0(X) \oplus \bb{C}$$ sending $$f\mapsto ((f-f(\infty))\mid_{X},f(\infty))$$. It is easy to see this is an isomorphism. So adjoining a unit like this is the analogue of the one-point compactification. Lets try to find the analogues for compactifications in general. There is a unique norm on each $$C^\ast$$-algebra that satisfies the $$C^\ast$$ identity, I leave it as an exercise to figure out the norm on $$C_0(X)\oplus \bb{C}$$ making it a $$C^\ast$$-algebra.

Let $$X\lra{\iota}Y$$ be a compactification of $$X$$. Consider $$f\in C_0(X)$$, then it extends to a $$\opc{f} \in C(\opc{X})$$. We can combine the map $$Y\lra{}\opc{X}$$ given by the universal property of $$\opc{X}$$ and $$\opc{f}$$ to get an element of $$C(Y)$$. Hence we have defined an embedding $$C_0(X) \lra{} C(Y)$$. Note that this entire arguement only required $$Y$$ to be compact, i.e we did not use the density of $$X$$ in $$Y$$. Here is how we translate that, take an ideal $$0\neq I\subset C(Y)$$, then notice that $$C_0(X)\cap C(Y)\neq 0$$. This is because if $$g\in C(Y)$$, then $$gC_0(X)=0$$ would mean $$g$$ is $$0$$ on $$X$$ and hence by density, on $$Y$$. So we collect these to get the dual concept of compactification for $$C^\ast$$ algebras.

A unitization of a $$C^\ast$$-algebra $$A$$ is:

• An embedding into an unital $$C^\ast$$-algebra $$B$$, i.e $$A\lra{}B$$
• the image of $$A$$ in $$B$$ is a (two-sided) essential ideal, i.e for every ideal $$I\subset B$$, $$A\cap I \neq 0$$.
• I have already shown compactifications lead to unitizations.I leave it as an exercise to show the converse.

## Multiplier Algebras.

The Stone-Čech compactification of a LCH space $$X$$, $$\beta X$$, is the biggest compactification of $$X$$. As we saw above, we expect this to correspond to the biggest unitization of a non-unital $$C^\ast$$-algebra $$A$$. The advantage of thinking in these terms is that for $$C^\ast$$ algebras this is quite explicit. Indeed, the biggest unitization of $$A$$, called $$M(A)$$(its multiplier algebra) is given as follows:

$$M(A)$$ is the set of double centralizers $$(L,R)$$ on $$A$$. This means:

• $$L,R$$ are bounded linear maps $$A\lra{} A$$
• $$aL(b) = R(a)b$$
• $$\lambda(L,R) +(L',R') = (\lambda L+L', \lambda R+R')$$ for $$\lambda\in \bb{C}$$
• $$(L,R)(L',R')=(LL',R'R)$$
• Its unit is $$(1_A,1_A)$$
• One can check from this that under the operator norm, $$\norm{L}=\norm{R}$$, so naturally we define the norm $$\norm{(L,R)} = \norm{L}$$
• $$(L,R)^\ast= (R^\ast, L^\ast)$$, where $$T^\ast(a) = T(a^\ast)^\ast$$ for any bounded operator $$T$$ on $$A$$ and $$a\in A$$.
• For $$c\in A$$, we can define the double centralizer $$(L_c,R_c)$$, where $$L_c$$ is left translation by $$c$$ and $$R_c$$ right translation. This gives an isometric embedding $$A\lra{} M(A)$$. I leave it as an exercise to show that $$A$$ is essential in $$M(A)$$.

Suppose $$B$$ contains $$A$$ as an ideal. Then notice we have a map $$\psi:B\lra{}M(A)$$, by sending $$c\mapsto (L_c,R_c)$$ again and using the fact that $$A$$ is an ideal. This extends the cannonical map $$A\lra{} M(A)$$. This is infact unique, for say $$\phi:B\lra{} M(A)$$ also extended it. Then for $$b\in B, a\in A$$, we would have $\psi(b) (L_a,R_a) = \psi(ba) = (L_{ba},R_{ba}) =\phi(ba) = \phi(b) (L_a,R_a)$ So that $$(\phi(a)-\psi(a)) A =0$$ in $$M(A)$$, but $$A$$ is essential in $$M(A)$$ so $$\phi(a)=\psi(a)$$. Finally suppose $$A$$ is essential in $$B$$, then if $$b\in \ker(\psi)$$ then $$L_b(A) = bA=0$$, so that $$b=0$$. Hence the map is injective.

To summarize, whenever $$B$$ contains $$A$$ as an essential ideal, $$B$$ has a unique embedding into $$M(A)$$. So $$M(A)$$ is the biggest unitization as promised. We will get to the commutative case in the next section, but a good example for now is that if $$A=K(H)$$, compact operators on some hilbert space, then $$M(A) = B(H)$$, bounded operators on that hilbert space.

## Stone-Čech Compactification

The Stone-Čech Compactification, $$\beta X$$,is the biggest compactification of a space, i.e if $$Y$$ is a compactification than $$\beta X$$ uniquely maps into $$Y$$ making the obvious diagram commute. This is literally the dual condition of the multiplier algebra, so we just need to figure out what $$M(C_0(X))$$ is. Luckily for us, the answer is easy: $$C_b(X)$$, bounded continuous functions on $$X$$ with sup norm. One can easily check that $$C_0(X)$$ is essential in $$C_b(X)$$, so that there is a injective map $$\psi: C_b(X) \lra{} M(C_0(X))$$. All we need to do is show surjection, which is easily done but the proof I know is a bit annoying and uses some other tools (like approximate units). I will hence refer the user to Gerard Murphy's book on the topic, page 83.

By the Gelfand Isomorphism, we know that $$C_b(X)\cong C(\beta X)$$ for some $$\beta X$$, and by the universal property discussed before, this $$\beta X$$ is indeed the Stone-Čech compactification of $$X$$. This proves that such a compactification exists in the first place but also gives a very nice explicit form for what the space should be.

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