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Tannaka-Krein duality shows how to recover a group $G$ from its category $\mathbf{Rep}(G)$ of finite-dimensional complex representations and the forgetful functor $F:\mathbf{Rep}(G)\to \mathbf{Vect}_{\mathbb{C}}$.

On the other hand, there is Takesaki's theorem which shows how to recover a (separable) $C^*$-algebra $A$ from its representation theory. We have just started looking into the details of this and trying to reformulate this in categorical terms. It seems tantalizingly similar to the Tannaka-Krein reconstruction theorem. In particular, it seems that Takesaki secretly also considers natural transformations from the forgetful functor $F:\mathbf{Rep}(A)\to\mathbf{Hilb}$ to itself.

So, the question is:

Can Takesaki's duality theorem indeed be formulated in categorical terms similar to Tannaka-Krein duality? Where can we read about it?

We're mostly interested in the unital case. Follow-up question:

Consider the category of unital $C^*$-algebras and unital completely positive maps. Is there a "nice" description of its opposite as a concrete category?

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1 Answer 1

This isn't an answer but a long comment. Tannaka-Krein duality will mislead you about how hard you should expect this result to be. "Tannaka-Krein duality for rings" is actually very easy and looks like this.

Theorem: Let $R$ be a ring. Then $R$ is isomorphic to the endomorphism ring of the forgetful functor $\text{Mod-}R \to \text{Ab}$. (Recall that the category of functors from a category into an $\text{Ab}$-enriched category is $\text{Ab}$-enriched, so endomorphism monoids are rings.)

Proof. This functor is represented by the right $R$-module $R_R$, so its endomorphism ring is the endomorphism ring of $R_R$ by the Yoneda lemma. By a second application of the Yoneda lemma (!), the endomorphism ring of $R_R$ is $R$. $\Box$

(And of course if you forget the forgetful functor then you cannot recover $R$ at all, only its Morita equivalence class.)

"Set-theoretic Tannaka-Krein duality for monoids" is also very easy and says that a monoid $M$ is isomorphic to the endomorphism monoid of the forgetful functor $\text{Set-}M \to \text{Set}$; the proof is identical. Tannaka-Krein duality itself is hard because we are considering linear representations of groups, which a priori should only let us at best recover the group algebra; we need more structure (e.g. the tensor product) to recover the group itself.

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Thanks for this, that's a good start. Those Yoneda lemma applications are incredibly sneaky! –  Tobias Fritz Apr 21 '13 at 21:15
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Have you checked to see how far the nLab entry on Tannaka Duality (ncatlab.org/nlab/show/Tannaka+duality) takes you? –  David Corfield Apr 22 '13 at 11:49
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The same construction works for C*-algebras, except that one should work in the bicategory of C*-categories, C*-functors, and natural transformations. Specifically, Hilbert C*-modules over a given C*-algebra form a C*-category. Hilbert spaces also form a C*-category and the C*-algebra of endomorphisms of the forgetful functor is canonically isomorphic to the original C*-algebra. –  Dmitri Pavlov Apr 27 '13 at 16:28

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