First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and Figureoa "Elements of noncommutative geometry". This theory resembles the Gelfand–Najmark theory for commutative $C^*$-algebras but there are some points which are different.
I will follow the aproach from this book. The setting is the following: they start with the Hopf algebra $H$ over $\mathbb{R}$ and consider the set of all algebra homomorphisms $\mathcal{G}(H)$. This set could be equipped with the so called convolution product and it turns out to be a group. One can define the topology of pointwise convergence on $\mathcal{G}(H)$ and $\mathcal{G}(H)$ becomes a topological group. Then a suitable functional $J$ is introduced: as far as I understood, this is a prototype of a Haar integral. It must satisfy certain axioms: a commutative Hopf algebra together with this functional is called a commutative skewgroup. With this extra structure on $H$ it it shown that $\mathcal{G}(H)$ becomes a compact group.
Then comes a theorem concerning Hopf algebras: starting from the commutative skewgroup $H$ one can form a compact group $\mathcal{G}(H)$ and then form an algebra of representative functions $\mathcal{R}(\mathcal{G}(H))$. The theorem states that:
$H$ is isomorphic to $\mathcal{R}(\mathcal{G}(H))$
The isomorphism is the same as in Gelfand–Najmark and some steps in the proof are similar. Using Stone–Weierstrass theorem one shows that the image (call it $V$) of the canonical "to be" isomorphism is dense in $\mathcal{R}(\mathcal{G}(H))$ and one needs to check that it is closed. Then the authors claim that it is enough to prove that $V$ is a $\mathcal{G}(H)$-module (meaning that there is an action, say from the left, of $\mathcal{G}(H)$ on $V$ such that $g \triangleright ( \cdot)$ is linear) and they refer to the book of Brocker and Dieck "Representations of compact Lie group". I've checked this theorem and have two questions:
Q1. It is about compact Lie groups. In Gracia–Bondia's book there is nothing about the Lie group structure on $\mathcal{G}(H)$. How to fix this problem?
Q2. This theorem provides equivalent conditions for a $G$-submodule of $\mathcal{R}(G)$ to be closed. I don't see why these conditions are (if really they are!) automatically satisfied in out situation. Is this really the case?
Further, in Gracia–Bondia's book the authors formulate a second theorem:
If $G$ is a compact Lie group then $\delta \colon x \mapsto \delta_x$ (where $\delta_x(f)=f(x)$) is an isomorphism between compact Lie groups $G$ and $\mathcal{G}(\mathcal{R}(G))$.
I've checked the proof and as far as I understood the steps the situations is the following:
a) the fact that it is an group homomorphism uses only algebraic structures
b) injectivity follows from Peter-Weyl thm. and continuity is immediate.
The problem is with surjectivity: let us denote $\mathcal{G}:=\mathcal{G}(\mathcal{R}(G))$ and $\hat{}:\mathcal{R}(G) \to \mathcal{R}(\mathcal{G})$ is an isomorphism from the previous theorem. One shows that $\delta^*:F \mapsto F \circ \delta$ is right inverse of $\hat{}$ and therefore is also an isomorphism. Since $\mathcal{R}(G)$ and $\mathcal{R}(\mathcal{G})$ are dense in $C(G)$ and $C(\mathcal{G})$ resp. it follows that $\delta^*$ extends to the isomomorhism between $C(G)$ and $C(\mathcal{G})$ and thus $\delta$ is surjective. A continuous bijection beetween compact spaces is a homeomorphism, and a continuous homomorphism of Lie groups is smooth so we are done. But here is my third question:
- Q3. Where in the proof we used the fact that $G$ is a Lie group? Do we really need this asumption?
The proof uses the previous theorem but in the formulation of the previous theorem there is nothing about Lie structure. It seems to me that this Lie structure is somehow hidden. So, to summarize let me ask:
- Q4. Which of this two theorems has to be reformulated (if any) and how should it look like? Are the proofs which I've sketched here correct?
And the final question:
- Q5. why it doesn't work for $\mathbb{C}$? Where is a key point of the argument?
