Concrete Realizations of Quotients of Operator Spaces
- Autores: M. A. Rieffel
- Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 114, Nº 2, 2014, págs. 205-215
- Idioma: inglés
- DOI: 10.7146/math.scand.a-17107
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Resumen
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.
