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Let
be an index set and
for each
an operator space. Then there are an
operator space
and complete contractions
with the following
universal mapping property: For each family of complete contractions
there is exactly one complete contraction
such that
for all
.
is called
-direct sum of the
and is denoted by
.The
are completely isometric.
One can construct a
-direct sum for instance as the closure of the sums of the
images of the mappings
, where
is the projection from
onto
.
The equation
holds isometrically.
Prof. Gerd Wittstock
2001-01-07