The L p Naturality Gap
F.E.J. LintonProposed for the International Category Theory Conference “CT04” at the
Middletown, CT 06459 USA
University of British Columbia (Vancouver, BC, Canada), 18-24 July 2004
The spaces l p(I) of I-indexed sequences with absolutely pth-power summable entries (1 ≤ p ≤ ∞) are covariantly functorial in I for p = 1, contravariantly functorial in I for p = ∞, but nothing much for other p.
By the same token, the l p-style “products” of I-tuples of Banach spaces serve as their coproducts for p = 1, their products for p = ∞, but nothing much (pace their Hilbert space structure when p = 2) for other p.
Similar observations pertain to the various l p-style “cross-norms” on tensor products of Banach spaces.
Again, the various spaces l p(I) behave somewhat like the components of a graded l ∞(I)-algebra, but not in an entirely satisfactory way.
The talk will fill in some of the details behind the remarks above, and, so far as they are available, behind their L p analogues.
Last modified: 29 May 2004.