The L ^p Naturality Gap

F.E.J. Linton
Wesleyan University
Middletown, CT 06459 USA

Proposed for the International Category Theory Conference “CT04” at the
University of British Columbia (Vancouver, BC, Canada), 18-24 July 2004

    The spaces l ^p(I) of I-indexed sequences with absolutely p^th-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.

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Last modified: 29 May 2004.