Algebras and modules
Model category presentations
Geometry on formal duals of algebras
AQFT and operator algebra
States and observables
-coalgebras are like -algebras, but coalgebras. Their duals are -algebras.
Let be a Banach *-coalgebra over the ground field . Let be a bounded linear functional on , and let be the composite
(where is complex conjugation, trivial if is real?); although some of the maps in this composite may be only antilinear, the composite is linear (over all of ). Now consider the composite
since and are short, the norm of this functional is at most .
is a -coalgebra if the norm of the map in (1) is exactly , for every bounded linear functional .
Although there is an asymmetry in (1) (in the relative placement of and ), if we start with instead of , we see that the universally quantified definition of -coalgebra is symmetric.
If we take the formal dual of everything in the definition above, then becomes a Banach *-algebra and becomes (multiplication of scalars by) an element of . The formal dual of the composite (1) is (multiplication of scalars by) the element . The requirement that the norm of this be exactly the square of the norm of is the -identity that defines a -algebra.
So the definition of -coalgebra dualises everything in the definition of -algebra, down to using coelement?s (in this case linear functionals) instead of elements.
In general, the dual of a coalgebra is an algebra, in any closed monoidal category. In particular, the dual of a Banach coalgebra is a Banach algebra. The involution gets along with this just fine; the dual of a Banach *-coalgebra is a Banach *-algebra. Finally, the linear functional in (1) becomes the element in the -identity for the dual, so the dual of a -coalgebra is a -algebra.
But we have more! If is a -coalgebra, then (as we've just seen) is a -algebra; but since has a predual , this means that is actually a -algebra as well.
Which -algebras arise in this way?
Examples (and non-examples)
The sequence space is a -coalgebra, whose dual -algebra is the sequence space . (For details of the comultiplication on , see the examples in Banach coalgebra.)
Although is a Banach coalgebra (under ‘coconvolution’), it is not a -coalgebra (at least not under coconvolution).
(YC) Moreover, the TVS-isomorphism class of the predual of a -algebra is very restricted (let alone its isomorphism class in Bang). In particular doesn’t have a snowball’s chance in Texas of being a -coalgebra under any kind of choice of comultiplication, because it’s the wrong kind of Banach space.)
Although the dual of the Lebesgue space (on the real line with Lebesgue measure) is the -algebra , is not a -coalgebra, nor even a Banach coalgebra (at least not in the obvious way). Essentially, this is because the diagonal in has measure zero (so takes an element of , interpreted as an absolutely continuous measure on , to a measure on that is not absolutely continuous and so cannot be reinterpreted as an element of ).