Arens product

[Anders Kock <kock@imf.au.dk>]

In reply to Yemon Choi:

The situation you describe has been studied in the context of
symmetric monoidal closed categories, in some articles by me in the
early 1970s (references below). The main point about double
dualization in Banach spaces is that it is part of a V-enriched
("strong") monad on V (for suitable  symmetric monoidal closed
category V); and the two "Arens extensions" are special cases of the
two canonical monoidal structures which any V-enriched monad on V
admits. Commutative monads are those where the two structures agree.

[1] Monads on symmetric monoidal closed categories, Archiv der Math.
21 (1970), 1-10.
[2] On double dualization monads, Math. Scand 27 (1970), 151-165.
[3] Bilinearity and Cartesian closed monads, Math. Scand 29 (1971), 161-174.
[4] Strong functors and monoidal monads, Archiv der Math. 23 (1972), 113-120.
[5] Closed categories generated by commutative monads, J. Austral.
Math. Soc. 12 (1971), 405-424.

The V-enrichment ("strength") of an endofunctor T on V can be encoded
without reference to the closed structure of V as a transformation
T(A)@B-->T(A@B) ("tensorial strength", introduced in [4]).

Strong monads applied in functional-analytic contexts are also considered in my

[6]  Some problems and results in synthetic functional analysis , in
Category Theoretic Methods in Geometry, Proceedings Aarhus 1983,
Aarhus Various Publication Series 35 (1983) 168-191.

All these papers, except [5], can be downloaded from my home page (go
to the bottom of it),
http://home.imf.au.dk/kock/

I hope the above references can be useful. Best wishes.

Anders Kock