  We study tensor products of two structures situated, in a sense, between
normed spaces and (abstract) operator spaces. We call them Lambert and
proto-Lambert spaces and pay more attention to the latter ones. The considered
two tensor products lead to essentially different norms in the respective
spaces. Moreover, the proto-Lambert tensor product is especially nice for
spaces with the maximal proto-Lambert norm and in particular, for $L_1$-spaces.
At the same time the Lambert tensor product is nice for Hilbert spaces with the
minimal Lambert norm.
