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I would like to know how close one can get to realizing the category of C*-algebras as a category of monoid objects. Related (almost, but not quite, duplicate) questions are:

J.M. Egger attempted to do this in the category of operator spaces using the Haagerup tensor product: http://www.mscs.dal.ca/~jegger/ioa.pdf (also available at: http://cheng.staff.shef.ac.uk/pssl85/egger.pdf) . The $*$-operation complicates things a bit; one must deal with "involutive monoids". He writes down a nice category of involutive monoids that C*-algebras are are included in, but not every such involutive monoid is C*-algebraic (c.f. example 4.6).

This reference is from 2007, and I'm not familiar with the operator systems literature, so perhaps someone else has something written down. Here is a subquestion:

Is it possible to reproduce what Egger does in the category of operator systems and completely positive maps instead of operator spaces and completely bounded maps?

My superficial understanding of operator spaces is of even higher degree in superficiality when it comes to operator systems; so I don't even know if that question makes sense. However, Egger's counterexample is not a concrete operator system, and I feel like working with operator systems may rid of the complications of involutive monoids as, roughly speaking, the involution is already implicit in the data of an object of the underlying category.

However, I also believe the question as stated in the title is a pipe-dream: getting the C*-identity on the nose doesn't seem likely, so it's probably a better to ask if one can realize the category of C*-algebras as a reflective subcategory of some category of monoid objects: e.g. consider the full subcategory $\mathbf{A}^{*}$ of the category of Banach algebras and $*$-homomorphisms given by those Banach spaces that admit faithful $*$-homomorphisms into $B \mathcal{H}$ for some Hilbert space $\mathcal{H}$ (Takesaki calls these A*-algebras---an ungooglable term---in Def 9.19 of Op. Algs. I). There is a universal enveloping C*-algebra functor $\mathcal{C}: \mathbf{A}^{*} \rightarrow \mathbf{C}^{*}$ into the category $\mathbf{C}^{*}$ of C*-algebras, which explicitly realizes $\mathbf{C}^{*}$ as a reflective subcategory.

In this sense perhaps its more realistic to ask if $\mathbf{A}^{*}$ can be realized as a category of monoid objects. (Egger's counterexample doesn't appear to be an $A^{*}$-algebra either, but I might be missing something).

I imagine if one understands the situation for C* algebras, then the W* algebra case could be understood with the proper modifications and decorations with the prefix "co".


Optional Motivation : The question is interesting in its own right, but I'll supply some motivation for why I care: I'm working with a construction where the natural output involves cosimplicial W*-algebras; for various reasons I much rather deal with something like (non-negatively supported) "cochain dg"-objects by applying a dual version of the normalized Moore complex functor and using the Alexander-Whitney map to define multiplication. Of course, to make sense of these sorts of things I need to realize $W^{*}$-algebras as monoid (or comonoid if working with the predual) objects in some sufficiently nice category.

In finite dimensions everything is straightforward: the category of finite-dimensional W*-algebras (forget the $\dagger$-category structure) is equivalent to the category of semisimple finite-dimensional $\mathbb{C}$-algebras.

Of course I can probably just find a "hack" for my particular problem without understanding monoidal structures, but I don't think that gives a very deep understanding.

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    $\begingroup$ This doesn't address your main points or questions, but in your category ${\bf A}^*$ are you assuming that the morphisms are contractive (or "short" in the lingo I've seen on the nLab)? $\endgroup$ – Yemon Choi Oct 3 '17 at 0:29
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    $\begingroup$ Regarding the reflective subcategory question (which I may have misunderstood): I think I once convinced myself that the forgetful functor from (unital $C^*$-algebras and *-HMs) to (operator spaces and completely contractive linear maps) has a left adjoint -- certainly Pisier makes frequent use in his book of $C^*\langle E\rangle$ where $E$ is an operator space. Anyway, perhaps this left adjoint functor factorizes through some category like your ${\bf A}^*$? $\endgroup$ – Yemon Choi Oct 3 '17 at 0:39
  • $\begingroup$ Yes it may be best to take the morphisms in $\mathbf{A}^{*}$ as contractive *-homomorphisms. I think what I said about reflective subcategories works whether you take the morphisms to be arbitrary or contractive, but if you want to realize the objects of $\mathbf{A}^{*}$ as monoid objects you'll need multiplication to be contractive anyway --- so contractions all around. $\endgroup$ – Tom Mainiero Oct 3 '17 at 3:07
  • $\begingroup$ Also, thanks for your comment about $C^{*} \langle E \rangle$. It might factor through something interesting, but probably not $\mathbf{A}^{*}$. I say this because the universal morphism $E \rightarrow C^{*} \langle E \rangle$ is an isometric embedding. On the other hand the universal morphism $A \mapsto \mathcal{C}(A)$ is a contraction unless $A$ is already $C^{*}$ (here $\mathcal{C}(A)$ is the completion of $A$ with respect to $\| - \|_{*} := \sup \{ \|\pi(-) \|:\text{$\pi$ is a $*$-rep of $A$} \}$). Would an intermediate category for $C^{*} \langle - \rangle$ be helpful? $\endgroup$ – Tom Mainiero Oct 3 '17 at 3:28
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    $\begingroup$ At least in finite dimension you can characterise C*-algebras as certain (so-called Frobenius) monoids in the category of Hilbert spaces, see arxiv.org/abs/0805.0432. In infinite dimension this works too, but the problem is that the carrier object of the monoid has to be a Hilbert space, so you only recover so-called H*-algebras, see arxiv.org/abs/1011.6123. It also works for Hilbert modules over a commutative C*-algebra, see arxiv.org/abs/1704.05725. $\endgroup$ – Chris Heunen Oct 3 '17 at 9:56

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