Banach coalgebra

Banach coalgebras


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


A Banach coalgebra, or Banach cogebra, is a comonoid object in the monoidal category BanBan of Banach spaces with short linear maps and the projective tensor product. (Recall that a Banach algebra is a monoid object in BanBan.)

Explicitly, we have:

  1. a Banach space AA
  2. a short linear map, the comultiplication:
    Δ:AA^ πA \Delta\colon A \to A {\displaystyle\hat{\otimes}_\pi} A

    to the projective tensor product;

  3. a short linear functional, the counit:
    ϵ:AK, \epsilon\colon A \to K ,

    where KK is the ground field;

  4. an equation, the coassociativity:
    (Δ^ πid A)Δx=(id A^ πΔ)Δx(A^ πA)^ πAA^ π(A^ πA) (\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)

    for each x:Ax\colon A;

  5. an equation, the left coidentity:
    (ϵ^ πid A)Δx=xK^ πAA (\epsilon {\displaystyle\hat{\otimes}_\pi} \id_A) \Delta x = x \in K {\displaystyle\hat{\otimes}_\pi} A \cong A

    for each x:Ax\colon A;

  6. and an equation, the right coidentity:
    (id A^ πϵ)Δx=xA^ πKK (\id_A {\displaystyle\hat{\otimes}_\pi} \epsilon) \Delta x = x \in A {\displaystyle\hat{\otimes}_\pi} K \cong K

    for each x:Ax\colon A.

Technically, we've defined a counital coassociative Banach coalgebra. We can leave out (3,5,6) to get a non-counital Banach coalgebra, and (also) leave out (4) to get a 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 add the property of cocommutativity:

  • τΔx=Δx\tau \Delta x = \Delta x for each x:Ax\colon A,

where the braiding τ:A^ πAA^ πA\tau\colon A {\displaystyle\hat{\otimes}_\pi} A \to A {\displaystyle\hat{\otimes}_\pi} A is generated by τ(uv)=vu\tau (u \otimes v) = v \otimes u. Then we have a cocommutative Banach coalgebra.

To freely adjoin a counit to a non-counital Banach coalgebra AA, take the Banach space A 1KA \oplus_1 K (using the l 1l^1-direct sum), let Δ AK(x,c)\Delta_{A \oplus K} (x,c) be (Δ Ax,0,0,c)(A^ πA) 1A 1A 1K(A 1K)^ π(A 1K)(\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 ϵ A 1K(x,c)\epsilon_{A \oplus_1 K} (x,c) be cc. Then A 1KA \oplus_1 K is a counital Banach coalgebra. (Freely forcing coassociativity or cocommutativity —or even freely adjoining Δ\Delta in the first place— is harder.)

The category BanCoalgBan Coalg of Banach coalgebras has, as objects, Banach coalgebras and, as morphisms, short linear maps f:ABf\colon A \to B with equations

Δ Bfx=(f^ πf)Δ Ax \Delta_B f x = (f {\displaystyle\hat{\otimes}_\pi} f) \Delta_A x

and (unless we are allowing non-counital coalgebras)

ϵ Bfx=ϵ Ax \epsilon_B f x = \epsilon_A x

for all x:Ax\colon A. Warning: the term ‘homomorphism’ is used more generally; see below.

If AA and BB are Banach coalgebras, then their projective tensor product A^ πBA {\displaystyle\hat{\otimes}_\pi} B is a Banach coalgebra, generated by

Δ A^ πB(xy)=Δ AxΔ By(A^ πA)^ π(B^ πB)(A^ πB)^ π(A^ πB) \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)


ϵ A^ πB(xy)=(ϵ Ax)(ϵ By)K. \epsilon_{A {\displaystyle\hat{\otimes}_\pi} B} (x \otimes y) = (\epsilon_A x) (\epsilon_B y) \in K .

Similarly (but more simply), the ground field KK is itself a Banach coalgebra, with Δ\Delta and ϵ\epsilon both essentially the identity map. In this way, BanCoalgBan Coalg becomes a symmetric monoidal category.

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

(f,g)(x)(f^ πg)ΔxA^ πB (f,g)(x) \coloneqq (f {\displaystyle\hat{\otimes}_\pi} g) \Delta x \in A {\displaystyle\hat{\otimes}_\pi} B

(given f:ΓAf\colon \Gamma \to A, g:ΓBg\colon \Gamma \to B, and x:Γx\colon \Gamma) is a morphism of BanCoalgBan Coalg only when AA and BB are cocommutative. (I believe that BanCoalgBan Coalg does have a product, but it must be more complicated.)

The dual algebras of a coalgebra

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

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

(λμ)x=m(λ^ πμ)Δx, (\lambda \mu) x = m (\lambda {\displaystyle\hat{\otimes}_\pi} \mu) \Delta x ,

where m:B^ πBBm\colon B {\displaystyle\hat{\otimes}_\pi} B \to B is (generated by) the multiplication operation on BB. [A,B][A,B] is associative, unital, or commutative if AA and BB are (with ‘co‑’ in the names of AA's properties). In particular, A *A^* has one of these properties iff AA has the corresponding property.

Note that [B,A][B,A] (or even B *B^*) is not, in general, a Banach coalgebra. (That's because BanBan is closed, not coclosed?.)

Operators between coalgebras

Let AA and BB be Banach coalgebras.

Of course, AA and BB are Banach spaces, so we may consider the whole panoply of linear operators from AA to BB. In general, a linear operator is only a partial function, defined on a linear subspace of AA (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 of AA), the linear mappings (each defined on all of AA), the bounded operators (each defined on all of AA and bounded or equivalently continuous), and the short operators (each bounded with a norm at most 11).

A comultiplicative linear operator from AA to BB is a linear operator T:ABT\colon A \to B such that the following hold for all xdomTx \in \dom T: * Δx(domT)^ π(domT)\Delta x \in (\dom T) {\displaystyle\hat{\otimes}_\pi} (\dom T), * Δ BTx=(T^ πT)Δ Ax\Delta_B T x = (T {\displaystyle\hat{\otimes}_\pi} T) \Delta_A x (which exists by the previous line), and * ϵ BTx=ϵ Ax\epsilon_B T x = \epsilon_A x (which always exists).

We can also consider densely-defined comultplicative linear operators. A coalgebra homomorphism, or cohomomorphism, is a comultiplicative linear mapping; we can also consider bounded homomorphisms and short homomorphisms. The last of these are, as above, the morphisms in BanCoalgBan 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?.)

Beyond coalgebras

A Banach bialgebra, or Banach bigebra, is a bimonoid in BanBan: a Banach space AA equipped with the structures of both a Banach algebra and a Banach coalgebra, such that Δ\Delta and ϵ\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: * Δ(xy)=(Δx)(Δy)\Delta (x y) = (\Delta x) (\Delta y) (with the induced multiplication on A^ πAA {\displaystyle\hat{\otimes}_\pi} A), * Δ1=11\Delta 1 = 1 \otimes 1 (which is the identity in A^ πAA {\displaystyle\hat{\otimes}_\pi} A), * ϵ(xy)=(ϵx)(ϵy)\epsilon (x y) = (\epsilon x) (\epsilon y) (with the multiplication on the right in KK), and * ϵ1=1\epsilon 1 = 1 (with the 11 on the right in KK).

The category BanBialgBan 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 Banach Hopf algebra is a Hopf object? in BanBan: a Banach bialgebra AA with a (necessarily unique) short linear map (the antipode) S:AAS\colon A \to A such that

m(S^ πid)Δx,m(id^ πS)Δx=1ϵx m (S {\displaystyle\hat{\otimes}_\pi} \id) \Delta x, m (\id {\displaystyle\hat{\otimes}_\pi} S) \Delta x = 1 \epsilon x

(where mm is the multiplication with identity 11) for all x:Ax\colon A.

The category BanHopfAlgBan Hopf Alg of Banach Hopf algebras has, as objects, Banach Hopf algebras and, as morphisms, short linear maps f:ABf\colon A \to B that are morphisms of Banach bialgebras and preserve antipodes:

f(S Ax)=S B(fx) f (S_A x) = S_B (f x)

for all x:Ax\colon A.

A Banach **-coalgebra is a **-monoid object? in BanBan: a Banach coalgebra AA equipped with an antilinear map (the adjoint) xx *:AAx \mapsto x^*\colon A \to A such that

Δx *=(τΔx) * \Delta x^* = (\tau \Delta x)^*

(where τ\tau again is the braiding and ** on A^ πAA {\displaystyle\hat{\otimes}_\pi} A is generated by (xy) *=x *y *(x \otimes y)^* = x^* \otimes y^*) for all x:Ax\colon A and

ϵx *=ϵx¯ \epsilon x^* = \overline {\epsilon x}

(where a bar indicates complex conjugation) for all x:Ax\colon A.

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

f(x *)=f(x) * f(x^*) = f(x)^*

for all x:Ax\colon A.

There are also C *C^*-coalgebras, which have their own page.


It’s well known that the sequence space l 1l^1 of absolutely summable infinite sequences, thought of as l 1()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 1l^1 is a Banach coalgebra, its dual space l 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 l^\infty is also a Banach coalgebra, and indeed a Banach Hopf **-algebra too.

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

( i+j=ka i,j) k (\sum_{i + j = k} a_{i,j})_k

(summing along antidiagonals); comultiplication takes (a k) k(a_k)_k to the diagonal matrix

Δa=( i=j=ka k) i,j \Delta a = (\sum_{i = j = k} a_k)_{i,j}

(which is not quite the origin of the symbol ‘Δ\Delta’ but might as well be). The tensor square l ^ πl 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 l^\infty takes the matrix (a i,j) i,j(a_{i,j})_{i,j} to the sequence

( i=j=ka i,j) k=(a k,k) k (\sum_{i = j = k} a_{i,j})_k = (a_{k,k})_k

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(a_k)_k to

( i+j=ka k) i,j=(a i+j) i,j (\sum_{i + j = k} a_k)_{i,j} = (a_{i + j})_{i,j}

(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 l^\infty, but (at least in classical mathematics) l 1l^1 is only a subspace of that. So from this perspective, what’s lucky is that l 1l^1 is closed under convolution. (In dream mathematics, l 1l^1 is the entire dual of l l^\infty, so no luck is required.) Of course, l 1l^1 is the dual of c 0c_0 (the space of sequences with limit 00), but c 0c_0 is not closed (coclosed?) under coconvolution (try any non-zero example), so we are still lucky.

Revised on March 15, 2016 09:38:19 by Ingo Blechschmidt (