CHAPTER 1
T H E CONSTRUCTION S O F £
P
-SPACES
In [S] Schechtman used a simple tensor product to construct infinitely many
isomorphically distinct £p-spaces. This tensor is defined only for subspaces of
Jbp.
DEFINITION
1.1. Let X and Y be subspaces of Lp[0,1] and define X gY to
be the closed linear span of {x(t)y(s) : x G X and y G Y} in Lp([0,1] x [0,1])
with the usual product measure.
This tensor product depends on the representation of X and Y, a priori.
Note that if T and S are bounded operators on Lp[0,1] then we may define
T g S on Lp([0,1] x [0,1]) by [T g S](x ® y) = (Tx) g (Sj/) for all z,y e
Lp[0,1]. A straight-forward calculation using the Fubini Theorem shows that
T®S is well-defined and that \\T®S\\ ||T|| ||S||. It also follows from standard
techniques (integration against the Rademacher functions) that if (xi) and (yi)
are unconditional basic sequences in Lp[0,1] then (xi^y^ij is an unconditional
basis for [x^ ® [yj]. (See [S] or the proof of Lemma 1.2 below.)
Schechtman defines spaces ®nXp Xp®Xp®- -g)Xp (n-times). The remarks
above show that &nXp is complemented in Lp([0, l]n) where the projection is
P ® P g ) g P (n-times) and P is a projection from Lp[0,1] onto Xp. Thus
it is immediate that
®nXp
is a £p-space. Unfortunately in this representation
the norm of the projection goes to oo with n. Indeed, it was communicated to
me by Schechtman from Pisier, that the norm of the projection must be at least
as large as the product of the smallest norms of projections onto the factors.
To see this we have by [TJ,Lemma 32.3] that for X C Y the relative projection
constant
A(®nX, g)nY))
= sup{|tr(ix : ®
n
X - 0
n
Y) | : u G
P(®nY, (g)nY),
v{u) 1 and
u(®nX)
C 0
n
X } ,
where tr denotes the trace and v is the nuclear norm. Because tr((g)™P) =
Y\™ tr(P) and i/(0]fP) Hi" ^C^)* ^ follows that the relative projection constant
of
gnXp
in Lp([0,
l]n)
is no better than the nth power of the relative projection
constant for Xp in Lp[0,1].
Because of Rosenthal's Inequality it is possible to represent ®nXp as a se-
quence space relative to the unconditional basis (xkx ®Xk2 ®'' •®£fcn)fcjEN,i 7'n
and explicitly compute a formula for an equivalent sequence space norm. (In [S]
the case n = 2 is given.) The next lemma will allow us to do the computation
inductively.
5
Previous Page Next Page