Reference Request and Category Theoretic Interpretation of a Result on Banach Spaces

Question

Today I learned in class the following result, which my professor stated without proof:

Given a Banach space $V$, there exists a compact Hausdorff space $X$ such that $V$ embeds into $C(X)$ as a closed subspace.

Recall that $C(X)$ is the space of all continuous complex valued functions on $X$, which is a unital $C^*$-algebra (the only ones in fact).

First, does anyone know where I can find a proof of this result? Secondly, is there some category theoretic interpretation of this result? I won't even presume to know much category theory, but it seems to say something like the unital commutative $C^*$-algebras are universal objects in some sense...?

Answer 1

Here is a proof of the result:

Let $X$ be the closed unit ball of $V^*$, with the weak$^*$-topology. This is a compact Hausdorff space, by the Banach-Alaouglu theorem. Then the map $T:V\to C(X)$ given by $(Tv)(x)=x(v)$ is an isometry (this follows from the Hahn-Banach theorem), hence $T(V)$ is a closed subspace of $C(X)$.

As far as a categorical interpretation, I doubt there is one. Given a bounded linear map, $T:V\to W$ between Banach spaces, it doesn't necessarily follow that $T^*$ maps to ball of $W*$ into the ball of $V^*$.

Answer 2

For the categorical interpretation see proposition 5.2 (and everything around it) in The Hitchhiker Guide to Categorical Banach Space Theory. Part I. by Jesus M. F. Castillo

Answer 3

The categorical interpretation is as follows: the embedding defined in Awegan's answer is the unit of an adjunction. The details of the definitions are as follows.

Define $\newcommand{\Ban}{\mathbf{Ban}_1}\Ban$ to be the category with Banach spaces as objects, and linear contractions (i.e. maps with operator norm $\leq 1$) as morphisms, and define $\newcommand{\CC}{\mathbf{CC}^*}\CC$ to be the category of commutative unital C$^*$-algebras, having unital $*$-homomorphisms as morphisms. There's a forgetful functor $U : \CC \rightarrow \Ban$, since unital $*$-homomorphisms have operator norm $\leq 1$. If $E$ is a Banach space, I will use $E_1$ to mean the unit ball of $E$. We can then define $F : \Ban \rightarrow \CC$ on objects by taking $F(E)$ to be $C(E^*_1)$, where $E^*_1$, the unit ball of $E^*$, is made into a compact Hausdorff space using the weak-* topology. On maps $f : E \rightarrow F$, we take $$ F(f)(a)(\phi) = a(\phi \circ f), $$ where $a \in C(X)$ and $\phi \in F^*_1$.

The statement of the main result is:

$F$ is a left adjoint to $U$, and the embedding of each Banach space $E$ in a space of the form $C(X)$ is given by the unit of this adjunction.

When proving this, it is helpful to redefine $\CC$ to be its full subcategory on spaces of the form $C(X)$. This is equivalent to all of $\CC$ by Gelfand duality, and the range of $F$ is inside this category, so we don't lose anything by doing this.

For each Banach space $E$ we define $\eta_E : E \rightarrow F(E)$ by $$ \eta_E(x)(\phi) = \phi(x), $$ where $x \in E$ and $\phi \in E^*_1$. This is the embedding mentioned in the question and Aweygan's answer, and is easily proved to be a natural transformation.

To define the counit, we will make use of the functor $\newcommand{\CHaus}{\mathbf{CHaus}}\newcommand{\op}{^\mathrm{op}}C : \CHaus\op \rightarrow \CC$, defined as expected on objects and as $C(f)(b) = b \circ f$ on maps, where $f : X \rightarrow Y$ is a map in $\CHaus$ and $b \in C(Y)$. We will also use the function, defined for each compact Hausdorff space $\delta_X : X \rightarrow C(X)^*_1$ by $\delta_X(x)(a) = a(x)$, where $x \in X$ and $a \in C(X)$. These definitions are known from the theory of Gelfand duality. Then we define $\varepsilon_{C(X)} : F(C(X)) \rightarrow C(X)$ to be $C(\delta_{X})$. This is well-typed because $F(C(X)) = C(C(X)^*_1)$.

To prove that these definitions make $F$ into a left adjoint to $U$, we only need to prove that the unit-counit triangle diagrams commute. This is equivalent to showing that $\newcommand{\id}{\mathrm{id}} \varepsilon_{C(X)} \circ \eta_{C(X)} = \id_{C(X)}$ for all compact Hausdorff spaces $X$ and $\varepsilon_{F(E)} \circ F(\eta_E) = \id_{F(E)}$ for all Banach spaces $E$. Each of these is proved simply by expanding out the definitions, with everything getting swapped back and forth in a somewhat confusing way, so I won't write out the proofs. However, I will mention that it helps in the second case to prove that $\delta_{E^*_1}(\phi) \circ \eta_E = \phi$ for all $\phi \in E^*_1$ first.

---

Another important general fact is that every Banach space $E$ is the quotient of some space $\ell^1(X)$. In this case we may take $X = E_1$, and in fact the mapping is the counit of the adjunction where $\newcommand{\Set}{\mathbf{Set}}\ell^1 : \Set \rightarrow \Ban$ is the left adjoint to the functor $\Ban \rightarrow \Set$ that takes $E$ to the set $E_1$.

However, in practice it seems that the existence of these mappings at all is a more important thing than their realization as units or counits of adjunctions. These mappings are an important part of the theoretical background of tensor products of Banach spaces, nuclear and integral operators and so on. The embedding of a Banach space in some $C(X)$ is also an important motivator for the definition of an operator space, a closed subspace of $B(\mathcal{H})$, where the analogy is

Banach space : commutative C$^*$-algebra :: operator space : noncommutative C$^*$-algebra