  This work performs a study of the category of complete matrix-normed spaces,
called matricial Banach spaces. Many of the usual constructions of Banach
spaces extend in a natural way to matricial Banach spaces, including products,
direct sums, and completions. Also, while the minimal matrix-norm on a Banach
space is well-known, this work characterizes the maximal matrix-norm on a
Banach space from the work of Effros and Ruan as a dual operator space.
  Moreover, building from the work of Blecher, Ruan, and Sinclair, the Haagerup
tensor product is merged with the direct sum to form a Haagerup tensor algebra,
which shares the analogous universal property of the Banach tensor algebra from
the work of Leptin.
