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For matricially normed and
, one defines
Among the complete contractions, the complete isometries and the complete quotient mappings
play a special role. is called
completely isometric if all
are
isometric,5and a
complete quotient mapping if all
are quotient mappings.6
The set of all completely bounded mappings from
to
is denoted by
[Pau86, Chap. 7].
An operator space is called
homogeneous if each
bounded operator
is completely bounded with the same norm:
, and
[Pis96].