Polar decompositions of bounded linear functionals on operator subalgebras

Abstract:

Let $M$ be a von Neumann algebra and let $\varphi$ be a normal linear functional on a strongly closed ${C^{\ast }}$-subalgebra $N$ of $M$. Denote by ${\mathcal {F}_\varphi }$ the set of normal linear functionals $\psi$ on $M$ extending $\varphi$ with $||\psi || = ||\varphi ||$. It is shown that there exists a partial isometry $v$ in $N$ such that \[ \varphi = |\varphi |(v \cdot ),\qquad |\varphi | = \varphi ({v^{\ast }} \cdot ),\qquad ||\varphi || = |||\varphi |||\] and \[ \psi = |\psi |(v\cdot ),\qquad |\psi | = \psi ({v^{\ast }}\cdot ),\qquad ||\psi || = |||\psi |||\] for all $\psi$ in ${\mathcal {F}_\varphi }$, where $|\varphi |$ and $|\psi |$ denote the absolute values of $\varphi$ and $\psi$ respectively. Let $A$ be a ${C^{\ast }}$-algebra and let $B$ be a ${C^{\ast }}$-subalgebra of $A$. As a consequence of this result, we obtain that every state on $B$ has a unique state extension to $A$ if and only if every bounded linear functional on $B$ has a unique norm-preserving extension to a bounded linear functional on $A$.