Abstract:
Lambert discovered a new type of structures situated, in a sense, between
normed spaces and abstract operator spaces. His definition was based on the
notion of amplifying a normed space by means of the spaces $\ell_2^n$.
Later, several mathematicians studied more general structures
(`$p$-multi-normed spaces') introduced by means of the spaces
$\ell_p^n$, $1\le p\le\infty$. We pass from $\ell_p$ to $L_p(X,\mu)$
with an arbitrary measure. This becomes possible in the framework of
the non-coordinate approach to the notion of amplification.
In the case of a discrete counting measure, this approach is equivalent
to the approach in the papers mentioned.
Two categories arise. One consists of amplifications by means of an arbitrary
normed space, and the other consists of $p$-convex amplifications by means
of $L_p(X,\mu)$. Each of them has its own tensor product of objects
(the existence of each product is proved by a separate explicit construction).
As a final result, we show that the `$p$-convex' tensor product has
an especially transparent form for the minimal $L_p$-amplifications
of $L_q$-spaces, where $q$ is conjugate to $p$. Namely, tensoring $L_q(Y,\nu)$
and $L_q(Z,\lambda)$, we obtain $L_q(Y\times Z,\,\nu\times\lambda)$.
Keywords:$\mathbf{L}$-space, $\mathbf{L}$-boundedness, general $\mathbf{L}$-tensor product,
$p$-convex tensor product.
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