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\begin{document}

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\section*{Banach coalgebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{banach_coalgebras}{}\section*{{Banach coalgebras}}\label{banach_coalgebras}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{the_dual_algebras_of_a_coalgebra}{The dual algebras of a coalgebra}\dotfill \pageref*{the_dual_algebras_of_a_coalgebra} \linebreak
\noindent\hyperlink{operators_between_coalgebras}{Operators between coalgebras}\dotfill \pageref*{operators_between_coalgebras} \linebreak
\noindent\hyperlink{beyond_coalgebras}{Beyond coalgebras}\dotfill \pageref*{beyond_coalgebras} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Banach coalgebras (or cogebras) are like [[Banach algebras]], but [[coalgebras]]. The [[dual vector space|dual]] of a Banach coalgebra is a Banach algebra (but not conversely). We can also consider Banach [[bialgebras]] (or bigebras).

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

A \textbf{Banach coalgebra}, or \textbf{Banach cogebra}, is a [[comonoid object]] in the [[monoidal category]] $Ban$ of [[Banach spaces]] with [[short linear maps]] and the [[projective tensor product]]. (Recall that a [[Banach algebra]] is a [[monoid object]] in $Ban$.)

Explicitly, we have:

\begin{enumerate}%
\item a [[Banach space]] $A$
\item a [[short linear map]], the \textbf{comultiplication}:\begin{displaymath}
\Delta\colon A \to A {\displaystyle\hat{\otimes}_\pi} A
\end{displaymath}
to the [[projective tensor product]];


\item a short [[linear functional]], the \textbf{counit}:\begin{displaymath}
\epsilon\colon A \to K ,
\end{displaymath}
where $K$ is the [[ground field]];


\item an [[equation]], the \textbf{coassociativity}:\begin{displaymath}
(\Delta {\displaystyle\hat{\otimes}_\pi} \id_A) \Delta x = (\id_A {\displaystyle\hat{\otimes}_\pi} \Delta) \Delta x \in (A {\displaystyle\hat{\otimes}_\pi} A) {\displaystyle\hat{\otimes}_\pi} A \cong A {\displaystyle\hat{\otimes}_\pi} (A {\displaystyle\hat{\otimes}_\pi} A)
\end{displaymath}
for each $x\colon A$;


\item an equation, the \textbf{left coidentity}:\begin{displaymath}
(\epsilon {\displaystyle\hat{\otimes}_\pi} \id_A) \Delta x = x \in K {\displaystyle\hat{\otimes}_\pi} A \cong A
\end{displaymath}
for each $x\colon A$;


\item and an equation, the \textbf{right coidentity}:\begin{displaymath}
(\id_A {\displaystyle\hat{\otimes}_\pi} \epsilon) \Delta x = x \in A {\displaystyle\hat{\otimes}_\pi} K \cong K
\end{displaymath}
for each $x\colon A$.



\end{enumerate}
Technically, we've defined a \textbf{counital coassociative Banach coalgebra}. We can leave out (3,5,6) to get a \textbf{non-counital Banach coalgebra}, and (also) leave out (4) to get a \textbf{non-coassociative Banach coalgebra}. Warning: these terms are examples of the [[red herring principle]]. Note that (3) is a [[property-like structure]] (and 4--6 are obviously just [[properties]]).

On the other hand, we can \emph{add} the property of \textbf{cocommutativity}:

\begin{itemize}%
\item $\tau \Delta x = \Delta x$ for each $x\colon A$,

\end{itemize}
where the [[braiding]] $\tau\colon A {\displaystyle\hat{\otimes}_\pi} A \to A {\displaystyle\hat{\otimes}_\pi} A$ is generated by $\tau (u \otimes v) = v \otimes u$. Then we have a \textbf{cocommutative Banach coalgebra}.

To \textbf{freely adjoin a counit} to a non-counital Banach coalgebra $A$, take the Banach space $A \oplus_1 K$ (using the $l^1$-[[l-1-direct sum|direct sum]]), let $\Delta_{A \oplus K} (x,c)$ be $(\Delta_A x, 0, 0, c) \in (A {\displaystyle\hat{\otimes}_\pi} A) \oplus_1 A \oplus_1 A \oplus_1 K \cong (A \oplus_1 K) {\displaystyle\hat{\otimes}_\pi} (A \oplus_1 K)$, and let $\epsilon_{A \oplus_1 K} (x,c)$ be $c$. Then $A \oplus_1 K$ is a counital Banach coalgebra. (Freely forcing coassociativity or cocommutativity ---or even freely adjoining $\Delta$ in the first place--- is harder.)

The [[category]] \textbf{$Ban Coalg$} of Banach coalgebras has, as [[objects]], Banach coalgebras and, as [[morphisms]], [[short linear maps]] $f\colon A \to B$ with equations

\begin{displaymath}
\Delta_B f x = (f {\displaystyle\hat{\otimes}_\pi} f) \Delta_A x
\end{displaymath}
and (unless we are allowing non-counital coalgebras)

\begin{displaymath}
\epsilon_B f x = \epsilon_A x
\end{displaymath}
for all $x\colon A$. Warning: the term `homomorphism' is used more generally; see below.

If $A$ and $B$ are Banach coalgebras, then their \textbf{[[projective tensor product]]} $A {\displaystyle\hat{\otimes}_\pi} B$ is a Banach coalgebra, generated by

\begin{displaymath}
\Delta_{A {\displaystyle\hat{\otimes}_\pi} B} (x \otimes y) = \Delta_A x \otimes \Delta_B y \in (A {\displaystyle\hat{\otimes}_\pi} A) {\displaystyle\hat{\otimes}_\pi} (B {\displaystyle\hat{\otimes}_\pi} B) \cong (A {\displaystyle\hat{\otimes}_\pi} B) {\displaystyle\hat{\otimes}_\pi} (A {\displaystyle\hat{\otimes}_\pi} B)
\end{displaymath}
and

\begin{displaymath}
\epsilon_{A {\displaystyle\hat{\otimes}_\pi} B} (x \otimes y) = (\epsilon_A x) (\epsilon_B y) \in K .
\end{displaymath}
Similarly (but more simply), the [[ground field]] $K$ is itself a Banach coalgebra, with $\Delta$ and $\epsilon$ both essentially the [[identity map]]. In this way, $Ban Coalg$ becomes a [[symmetric monoidal category]].

The [[full subcategory]] $Cocomm Ban Coalg$ of cocommutative Banach coalgebras becomes a [[cartesian monoidal category]] under the projective tensor product. Actually, $K$ is the [[terminal object]] even in $Ban Coalg$ (with the unique coalgebra morphism to $K$ being $\epsilon$ itself), but the [[pairing]]

\begin{displaymath}
(f,g)(x) \coloneqq (f {\displaystyle\hat{\otimes}_\pi} g) \Delta x \in A {\displaystyle\hat{\otimes}_\pi} B
\end{displaymath}
(given $f\colon \Gamma \to A$, $g\colon \Gamma \to B$, and $x\colon \Gamma$) is a morphism of $Ban Coalg$ only when $A$ and $B$ are cocommutative. (I believe that $Ban Coalg$ \emph{does} have a [[product]], but it must be more complicated.)

\hypertarget{the_dual_algebras_of_a_coalgebra}{}\subsection*{{The dual algebras of a coalgebra}}\label{the_dual_algebras_of_a_coalgebra}

If $A$ is a Banach coalgebra, then the [[dual vector space]] $A^*$ is a [[Banach algebra]]. Actually, this is more general than $A^* = [A,K]$; if $B$ is any Banach algebra, then so is $[A,B]$ (the Banach space of [[bounded linear maps]] from $A$ to $B$).

This result is nothing special about Banach (co)algebras; it holds in any [[closed monoidal category]]. The multiplication operation in $[A,B]$ is given by

\begin{displaymath}
(\lambda \mu) x = m (\lambda {\displaystyle\hat{\otimes}_\pi} \mu) \Delta x ,
\end{displaymath}
where $m\colon B {\displaystyle\hat{\otimes}_\pi} B \to B$ is (generated by) the multiplication operation on $B$. $[A,B]$ is associative, unital, or commutative if $A$ and $B$ are (with `co' in the names of $A$'s properties). In particular, $A^*$ has one of these properties iff $A$ has the corresponding property.

Note that $[B,A]$ (or even $B^*$) is \emph{not}, in general, a Banach coalgebra. (That's because $Ban$ is closed, not [[coclosed category|coclosed]].)

\hypertarget{operators_between_coalgebras}{}\subsection*{{Operators between coalgebras}}\label{operators_between_coalgebras}

Let $A$ and $B$ be Banach coalgebras.

Of course, $A$ and $B$ are [[Banach spaces]], so we may consider the whole panoply of [[linear operators]] from $A$ to $B$. In general, a linear operator is only a [[partial function]], defined on a [[linear subspace]] of $A$ (and otherwise only required to be a [[linear map]]); but in particular we consider the [[densely-defined operator]]s (each defined on a [[dense subspace|dense]] subspace of $A$), the [[linear mappings]] (each defined on all of $A$), the [[bounded operators]] (each defined on all of $A$ and [[bounded map|bounded]] or equivalently [[continuous map|continuous]]), and the [[short operators]] (each bounded with a [[norm]] at most $1$).

A \textbf{comultiplicative linear operator} from $A$ to $B$ is a linear operator $T\colon A \to B$ such that the following hold for all $x \in \dom T$: * $\Delta x \in (\dom T) {\displaystyle\hat{\otimes}_\pi} (\dom T)$, * $\Delta_B T x = (T {\displaystyle\hat{\otimes}_\pi} T) \Delta_A x$ (which exists by the previous line), and * $\epsilon_B T x = \epsilon_A x$ (which always exists).

We can also consider \textbf{densely-defined comultplicative linear operators}. A \textbf{coalgebra homomorphism}, or \textbf{cohomomorphism}, is a comultiplicative linear mapping; we can also consider \textbf{bounded homomorphisms} and \textbf{short homomorphisms}. The last of these are, as above, the [[morphisms]] in $Ban Coalg$; of course, any of these classes of operators (except the densely-defined ones, which are not closed under [[composition]]) could be taken to be morphisms of a different category with the same objects, but then we would have [[isomorphisms]] that are not [[isometries]]. (See also [[isomorphism of Banach spaces]].)

\hypertarget{beyond_coalgebras}{}\subsection*{{Beyond coalgebras}}\label{beyond_coalgebras}

A \textbf{Banach [[bialgebra]]}, or \textbf{Banach bigebra}, is a [[bimonoid]] in $Ban$: a Banach space $A$ equipped with the structures of both a [[Banach algebra]] and a Banach coalgebra, such that $\Delta$ and $\epsilon$ are both morphisms of Banach algebras, or equivalently such that the multiplication and unit of the Banach algebra are both morphisms of Banach coalgebras. Explicitly, this requirement is: * $\Delta (x y) = (\Delta x) (\Delta y)$ (with the induced multiplication on $A {\displaystyle\hat{\otimes}_\pi} A$), * $\Delta 1 = 1 \otimes 1$ (which is the identity in $A {\displaystyle\hat{\otimes}_\pi} A$), * $\epsilon (x y) = (\epsilon x) (\epsilon y)$ (with the multiplication on the right in $K$), and * $\epsilon 1 = 1$ (with the $1$ on the right in $K$).

The [[category]] \textbf{$Ban Bialg$} of Banach bialgebras has, as objects, Banach bialgebras and, as morphisms, short linear maps that are morphisms of both Banach algebras and Banach coalgebras.

A \textbf{Banach [[Hopf algebra]]} is a [[Hopf object]] in $Ban$: a Banach bialgebra $A$ with a (necessarily unique) short linear map (the \textbf{antipode}) $S\colon A \to A$ such that

\begin{displaymath}
m (S {\displaystyle\hat{\otimes}_\pi} \id) \Delta x, m (\id {\displaystyle\hat{\otimes}_\pi} S) \Delta x = 1 \epsilon x
\end{displaymath}
(where $m$ is the multiplication with identity $1$) for all $x\colon A$.

The [[category]] \textbf{$Ban Hopf Alg$} of Banach Hopf algebras has, as objects, Banach Hopf algebras and, as morphisms, short linear maps $f\colon A \to B$ that are morphisms of Banach bialgebras and preserve antipodes:

\begin{displaymath}
f (S_A x) = S_B (f x)
\end{displaymath}
for all $x\colon A$.

A \textbf{Banach $*$-[[star-algebra|coalgebra]]} is a $*$-[[star-monoid object|monoid object]] in $Ban$: a Banach coalgebra $A$ equipped with an [[antilinear map]] (the \textbf{adjoint}) $x \mapsto x^*\colon A \to A$ such that

\begin{displaymath}
\Delta x^* = (\tau \Delta x)^*
\end{displaymath}
(where $\tau$ again is the [[braiding]] and $*$ on $A {\displaystyle\hat{\otimes}_\pi} A$ is generated by $(x \otimes y)^* = x^* \otimes y^*$) for all $x\colon A$ and

\begin{displaymath}
\epsilon x^* = \overline {\epsilon x}
\end{displaymath}
(where a bar indicates [[complex conjugation]]) for all $x\colon A$.

The [[category]] \textbf{$Ban {*} Coalg$} of Banach $*$-coalgebras has, as objects, Banach $*$-coalgebras and, as morphisms, short linear maps $f\colon A \to B$ that are morphisms of Banach bialgebras and preserve adjoints:

\begin{displaymath}
f(x^*) = f(x)^*
\end{displaymath}
for all $x\colon A$.

There are also $C^*$-[[C-star-coalgebra|coalgebras]], which have their own page.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

It's well known that the [[sequence space]] $l^1$ of absolutely summable [[infinite sequences]], thought of as $l^1(\mathbb{Z})$ (where $\mathbb{Z}$ is the [[abelian group]] of [[integers]] under addition), is a [[Banach algebra]] under [[convolution]]; however, it is also a Banach coalgebra, and these structures together make it a Banach bialgebra, in fact a Banach Hopf $*$-algebra. Since $l^1$ is a Banach coalgebra, its [[dual vector space|dual space]] $l^\infty$ (the sequence space of absolutely bounded sequences) is a Banach algebra (which is also well known); and although there is no guarantee that it should work, in this case $l^\infty$ is also a Banach coalgebra, and indeed a Banach Hopf $*$-algebra too.

Explicitly: The projective tensor square $l^1 {\displaystyle\hat{\otimes}_\pi} l^1$ is the space of absolutely summable infinite [[matrix|matrices]]; convolution takes the matrix $(a_{i,j})_{i,j}$ to the sequence

\begin{displaymath}
(\sum_{i + j = k} a_{i,j})_k
\end{displaymath}
(summing along antidiagonals); comultiplication takes $(a_k)_k$ to the [[diagonal matrix]]

\begin{displaymath}
\Delta a = (\sum_{i = j = k} a_k)_{i,j}
\end{displaymath}
(which is not quite the origin of the symbol `$\Delta$' but might as well be). The tensor square $l^\infty {\displaystyle\hat{\otimes}_\pi} l^\infty$ is the space of infinite matrices with absolutely bounded entries;

(comment added 26-08-2012 by YC: I am not convinced; Grothendieck's inequality, anyone?)

the dual multiplication on $l^\infty$ takes the matrix $(a_{i,j})_{i,j}$ to the sequence

\begin{displaymath}
(\sum_{i = j = k} a_{i,j})_k = (a_{k,k})_k
\end{displaymath}
of its diagonal entries; the dual comultiplication (which part of me wants to call `nvolution', but let's say coconvolution instead) takes $(a_k)_k$ to

\begin{displaymath}
(\sum_{i + j = k} a_k)_{i,j} = (a_{i + j})_{i,j}
\end{displaymath}
(so each antidiagonal is constant).

We are lucky that coconvolution exists, since the dual of a Banach algebra need not be a Banach coalgebra; but arguably coconvolution is easier to describe than convolution, so let us shift perspective and take coconvolution as basic. Then convolution necessarily exists on the dual of $l^\infty$, but (at least in [[classical mathematics]]) $l^1$ is only a subspace of that. So from this perspective, what's lucky is that $l^1$ is closed under convolution. (In [[dream mathematics]], $l^1$ is the entire dual of $l^\infty$, so no luck is required.) Of course, $l^1$ \emph{is} the dual of $c_0$ (the space of sequences with limit $0$), but $c_0$ is \emph{not} closed (coclosed?) under coconvolution (try any non-zero example), so we are still lucky.

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