TY - JOUR AU - Rieffel, Marc A. PY - 2014/05/06 Y2 - 2024/03/29 TI - Concrete Realizations of Quotients of Operator Spaces JF - MATHEMATICA SCANDINAVICA JA - Math. Scand. VL - 114 IS - 2 SE - Articles DO - 10.7146/math.scand.a-17107 UR - https://www.mscand.dk/article/view/17107 SP - 205-215 AB - Let $\mathscr{B}$ be a unital C*-subalgebra of a unital C*-algebra $\mathscr{A}$, so that $\mathscr{A}/\mathscr{B}$ is an abstract operator space. We show how to realize $\mathscr{A}/\mathscr{B}$ as a concrete operator space by means of a completely contractive map from $\mathscr{A}$ into the algebra of operators on a Hilbert space, of the form $A \mapsto [Z, A]$ where $Z$ is a Hermitian unitary operator. We do not use Ruan's theorem concerning concrete realization of abstract operator spaces. Along the way we obtain corresponding results for abstract operator spaces of the form $\mathscr{A}/\mathscr{V}$ where $\mathscr{V}$ is a closed subspace of $\mathscr{A}$, and then for the more special cases in which $\mathscr{V}$ is a $*$-subspace or an operator system. ER -