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Is there a coalgebra structure $\Delta_{n}$ on $M_{n}(\mathbb{C})$ which is compatible with the natural embedding $i_{n:}M_{n}(\mathbb{C})\to M_{n+1}(\mathbb{C})$ with $i_{n}(A)= A\oplus 0$. That is $(i_{n}\otimes i_{n})\circ \Delta_{n}=\Delta_{n+1}\circ i_{n}$.

If the answer is yes, can this method be used to equip the space of compact operators, the (topological) direct limit of $M_{n}(\mathbb{C}),s$, with a coalgebraic structure?

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    $\begingroup$ Just to clarify: when you pass to the "limit" and look at $K(\ell_2)$, would you be satisfied with a comultiplication that took values in some completion of the usual tensor product, such as the spatial tensor product $K(\ell_2)\overline{\otimes} K(\ell_2)$? $\endgroup$
    – Yemon Choi
    Commented Nov 9, 2014 at 20:44
  • $\begingroup$ @YemonChoi Very interesting point. However the usual definition of coalgebra requires the comultiplication took value in the algebraic tensor product, but your interesting comment is a motivation to ask about calgebraic structure on $C^{*}$ algebras with comultiplication in $A\bar{\otimes} A$. Is there some standard definition on this generalized coalgebraic structure? Any way both algebraic and topological comultiplication values are interesting. $\endgroup$
    – Ali Taghavi
    Commented Nov 10, 2014 at 6:24
  • $\begingroup$ And after a reasonable coalgebraic structure on compact operators the next question would be about closed coideals and cosubalgebras. Is it a cosimple coalgebra? $\endgroup$
    – Ali Taghavi
    Commented Nov 10, 2014 at 6:27

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For $V$ finite-dimensional, the algebra $\hom(V, V) \cong V^\ast \otimes V$ is naturally self-dual, and this duality may be used to transfer an algebra structure on $\hom(V, V)$ to a coalgebra structure on $\hom(V, V)$, and vice-versa. (Notice that the duality functor $\text{Vect}^{op} \to \text{Vect}$ on finite-dimensional spaces induces a functor from coalgebras to algebras, and vice-versa.)

The map $M_n(\mathbb{C}) \to M_{n+1}(\mathbb{C})$ mapping $A \mapsto A \oplus 0$ is dual to the map $M_{n+1}(\mathbb{C}) \to M_n(\mathbb{C})$ that takes an $(n+1) \times (n+1)$ matrix to the $n \times n$ matrix made from the first $n$ rows and columns. If we choose the algebra structure on the spaces $M_n(\mathbb{C})$ to be not the usual matrix multiplication but the one given by entrywise multiplication, then these are algebra maps and they dualize to coalgebra maps. This gives an affirmative answer to the question. However, this is clearly highly dependent on basis and doesn't strike me as terribly interesting.

The forgetful functor $\text{Coalg} \to \text{Vect}$ creates colimits, and so the colimit of a chain of coalgebra maps $M_n(\mathbb{C}) \to M_{n+1}(\mathbb{C})$ is created from the colimit in $\text{Vect}$. This answers the second question affirmatively.

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  • $\begingroup$ thank you for your interesting answer. Could you please explain how we extend the comultiplication to the space of compact operators(The closure of the algebraic direct limit)? $\endgroup$
    – Ali Taghavi
    Commented Nov 9, 2014 at 19:59
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    $\begingroup$ Ali's recent edit inserted the word "topological", which might introduce some delicacy since there are various tensor products one might consider on TVS. But for any symmetric monoidal category $M$, the forgetful functor $\text{Coalg}(M) \to M$ still creates colimits. $\endgroup$
    – Todd Trimble
    Commented Nov 9, 2014 at 20:00
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    $\begingroup$ @AliTaghavi Keeping in mind that I am not a functional analyst, the coalgebra comultiplication $\text{colim}\; M_n \to (\text{colim}\; M_n) \otimes (\text{colim}\; M_n)$ would be induced (via the universal property of colimits) from the cocone $M_n \to M_n \otimes M_n \to (\text{colim}\; M_n) \otimes (\text{colim}\; M_n)$ where $M_n \to \text{colim}\; M_n$ is the canonical inclusion. Here colimits = direct limits would be computed in a suitable TVS sense. Does that help? $\endgroup$
    – Todd Trimble
    Commented Nov 9, 2014 at 20:09
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    $\begingroup$ Todd, if we work in the symmetric monoidal category Bang of Banach spaces and maps of norm at most 1, then doesn't one have to tweak your initial remarks about hom(V,V) being self-dual? It seems one would need to do a bit more work to create a coalgebra structure on ${\rm End}_{\rm Bang}(\ell_2^n)$ a.k.a. $M_n({\bf C})$ with its usual norm. But perhaps I am missing something. $\endgroup$
    – Yemon Choi
    Commented Nov 9, 2014 at 20:28
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    $\begingroup$ @YemonChoi Yes, I think you are probably right. I composed my answer ignoring the topological aspects, which were brought more forcefully to attention by Ali's edit. So what he is really after no doubt deserves more attention. $\endgroup$
    – Todd Trimble
    Commented Nov 9, 2014 at 20:34

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