categorical literature on Arens products?

categorical literature on Arens products?

[Yemon Choi]

Dear categorists (and anyone else reading),

I have a question that's been bugging me for some time (in genesis,
all the way back to 2002 I think). It concerns a certain construction
of interest in the world of (Banach) algebras: given an algebra over a
field one can equip its double dual with two natural algebra
structures, in general distinct, which extend the original algebra
structure. More generally any bilinear map from a pair of vector
spaces to a third vector space admits two natural extensions to a
bilinear map on the bidual spaces.

Functional analysts know this through work of Arens, but reading
through his original paper (Monatsh Math 1955) it becomes clear that
most of the calculations
work in any symmetric closed monoidal category and so I wondered if
this kind of construction has been well-studied or is
well-known/well-understood in the categorical community.
(Arens comes close to this level of abstraction but requires his
objects to be sets with structure.)

Here are the details (parphrased by me into the SCMC setting), with
apologies for the dodgy attempts at notation.

We work in a fixed symmetric closed monoidal category V. Fix a
distinguished object C in V (we are thinking of V as complex Banach
spaces and C as the ground field). Denote the `tensor' in V by @, and
isomorphism in V by ~ .

For A in ob(V) define A' using the Hom-tensor adjunction in V, i.e.

 Hom_V ( __ @ A, C) ~ Hom_V( __, A').

As usual we have a natural transformation with components A --> A'',
call this K.


Given objects R, S, T in V let's write r(R,S,T) for the isomorphism

R @ S @ T ~> S @ T @ R


Now: given an arrow m: E@F --> G  in V, we compose with the natural
map K_G to get E@ F --> G'', then use the Hom-tensor adjunction to get
an arrow E@F@G' --> C. Composing with the isomorphism r(G',E,F) gives
us an arrow

G' @ E @ F --> C

and using Hom-tensor adjunction again gives us, finally, an arrow

L(m) : G' @ E --> F'


Note that in the case V=vector spaces, E an algebra, and F=G a left
E-module with m the module action, L(m) is just the adjoint (right)
action of E on the dual of F.

Iterating this construction we get L^2(m): F'' @ G' --> E'  and

L^3(m): E'' @ F'' --> G''

which we might call the left Arens extension, or left Arens bidual, of m.

The right Arens extension is constructed similarly: as before we
produce from the original arrow m: E @ F --> G an arrow E @ F @ G' -->
C. This time we compose with the isomorphism r(E,F,G')^{-1} to get an
arrow

F@ G' @ E --> C

and apply Hom-tensor adjunction to get

R(m): F @ G' --> E'

Iterating this construction gives R^2(m): G' @ E'' --> F' and

R^3(m): E'' @ F'' --> G''

which we call the right Arens extension, or right Arens bidual, of m.

Then left and right Arens extensions have been studied by functional
analysts, mainly in the particular case where E=F=G is a Banach
algebra and m is the multiplication map; in this case both L^3(m) and
R^3(m) define associative multiplication maps on the double dual A'',
but in general these multiplication maps are not the same. (They
coincide if A is a C^*-algebra, and it would be interesting if there
was a categorical interpretation of the proof.)

Two noticeable features of the work functional analysts have done on
this area (in the case where V=Ban is Banach spaces and continuous
linear maps) is that

1) the notation used to prove things is horrible, and usually looks a
little like a proof by commutative diagrams would if it were written
out as a line-by-line equational argument;

2) a lot of the proofs use analytic tools (Hahn-Banach theorem, weak
compactness of various mappings) but look as if they should have
`purely algebraic proofs', i.e. the desired equations can often be
derived purely from the closed structure on Ban.

So if anyone can point me to any existing categorical/logical
literature in this vein I'd be very grateful! In particular I would
like to know why there are two equally `canonical' extensions, and
whether there are universal properties underlying them. Also: in view
of point 1) above, is there a better kind of graphical calculus for
doing calculations?

Best wishes
Yemon

--
Dr. Y. Choi
519 Machray Hall
Department of Mathematics
University of Manitoba
Winnipeg. Manitoba
Canada R3T 2N2
Tel: (204)-474-8734


-- 
Dr. Y. Choi
519 Machray Hall
Department of Mathematics
University of Manitoba
Winnipeg. Manitoba
Canada R3T 2N2
Tel: (204)-474-8734