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