  As is known, there exists an alternative, "non-matricial" way to present
basic notions and results of quantum functional analysis (= operator space
theory). This approach is based on considering, instead of matrix spaces, a
single space, consisting, roughly speaking, of vectors from the initial linear
space equipped with coefficients taken from some good operator algebra. It
seems that so far there was no systematical exposition of the theory in the
framework of the non-matricial approach. We believe, however, that in a number
of topics the non-matricial approach gives a more elegant and transparent
theory.
  In this paper we introduce, using only the non-matricial language, both
quantum versions of the classical (Grothendieck) projective tensor product of
normed spaces. These versions correspond to the "matricial" Haagerup and
operator-projective tensor products. We define them in terms of the universal
property with respect to some classes of bilinear operators, corresponding to
"matricial" multiplicatively bounded and completely bounded, and then produce
their explicit constructions. Among the relevant results, we shall show that
both tensor products are actually quotient spaces of some "genuine" projective
tensor products. Moreover, the Haagerup tensor product is itself a "genuine"
projective tensor product, however not of just normed spaces but of some normed
modules.
