nLab algebraic theories in functional analysis

At the moment, this is a “place holder” page. I (Andrew Stacey) want to learn about the appearance of algebraic theories in functional analysis and shall record what I learn here. A preliminary outline is to find out about the following statements:

1. The category of Banach spaces with linear short maps is not monadic over Set. The “nearest” algebraic theory is that of totally convex spaces.
2. The category of Banach algebras is also not algebraic.
3. The category of $C^*$-algebras is algebraic.

Banach Spaces

We consider the category of Banach spaces with linear short maps. That is, this is the category $\operatorname{Ban}$ with:

• Objects: Banach spaces over $\mathbb{R}$
• Morphisms $E \to F$: Linear short maps. That is, bounded linear transformations $T \colon E \to F$ such that $\|T\| \le 1$

We define a functor $B \colon \operatorname{Ban} \to \operatorname{Set}$ sending a Banach space to its unit ball. Since linear short maps $E \to F$ take the unit ball of $E$ into the unit ball of $F$, this is well-defined.

There is a functor in the opposite direction which assigns to a set the “free” Banach space on that set. That is, it assigns to a set $X$ the Banach space $\ell^1(X)$ of all absolutely summable sequences indexed by elements of $X$. It is a standard result that such a sequence must have countable support, no matter how large $X$ is.

Lemma

$\ell^1$ is left adjoint to $B$.

Proof

We need to define the adjunction natural transformations: $\eta_X \colon X \to B \ell^1(X)$ and $\epsilon_E \colon \ell^1(B E) \to E$. The first is the map which assigns to $x$ the sequence $(\delta_{x y})$ which is $1$ at $x$ and $0$ elsewhere. The second is the summation map which assigns to an absolutely summable sequence $(a_e)$ indexed by $e \in B E$ its sum, $\sum a_e e$.

This adjunction defines a monad over $\operatorname{Set}$. Let us spell out the details. The functor $T \colon \operatorname{Set} \to \operatorname{Set}$ sends a set $X$ to the unit ball of $\ell^1(X)$. That is, an element of $T(X)$ is a weighted (formal) sum of elements of $X$, $\sum a_x$, such that $\sum |a_x| \le 1$. The unit for the monad sends an element $x \in X$ to the delta sequences in $T(X)$. The product, $\mu$, takes a “sum of sums” and evaluates them. That is, given a formal sum $\sum a_s$ where each $s$ is of the form $\sum s_x$, $\mu(\sum a_s) = \sum b_x$ where $b_x = \sum_s s_x$.

AS: I think! I need to check exactly how the product works in this example but I’m just getting the basic sketch down first.

The key question is whether or not $\operatorname{Ban}$ is (equivalent to) the category of algebras for this monad. That is, is $B \colon \operatorname{Ban} \to \operatorname{Set}$ tripleable? If not (as it will turn out), how close is it?

Beck's tripleability theorem gives three conditions for a functor to be tripleable. We already have one (the adjunction), let us show that the second also holds.

Lemma

$B \colon \operatorname{Ban} \to \operatorname{Set}$ reflects isomorphisms.

Proof

Let $T \colon E \to F$ be a linear short map which induces an isomorphism on the unit balls of $E$ and $F$. It is evident that it is therefore a bijection from the underlying set of $E$ to that of $F$. Hence, by the open mapping theorem, it is a linear homeomorphism. It remains to show that $\|T(x)\| = \|x\|$ (so that its inverse is a short map as well). This is simple to show: if we had some $x \in E$ with $\|x\| = 1$ but $\|T(x)\| \lt 1$ (if it fails, it must fail that way as $T$ is short) then there would be some $\lambda \gt 1$ such that $\|T(\lambda x)\| \le 1$. As $B(T) \colon B E \to B F$ is surjective, there is some $y \in B E$ such that $T(y) = T(\lambda x)$. But as $\lambda \gt 1$, $\lambda x \notin B E$ so $\lambda x \ne y$, contradicting the injectivity of $T$. (Incidentally, this argument is valid constructively; it is a property of located real numbers that any number that is neither greater nor smaller than $1$ must equal $1$.)

AS: To be continued …

References:

The above is essentially my “notes” on reading the following (and whatever necessary to understand the following):

Section 4.4 of Toposes, Triples, and Theories by Barr and Wells (TAC reprint)

References

• On the equational theory of $C^*$-algebras, Pelletier, J. Wick and Rosický, J., MR1223636
• Any more suggested by this question on mathoverflow