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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].