$J$ is an ideal of $B$, then the exact sequence $0 \rightarrow J \rightarrow B \rightarrow B/J \rightarrow 0$ locally splits. Namely, for every finite dimensional operator system $V \subseteq B / J$, there exists a unital completely positive map $\Phi \colon V \rightarrow B$ such that $\Phi(a) + J = a$, $a \in V$.

For a neat description on local reflexivity, the reader is referred to the book [@OzawaBook Chapter 9] by Brown–Ozawa.

Amenable trace {#SubsectionAmeTr}
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We recall the definition of amenable trace.

\[DefinitionAmeTr\] Let $B \subseteq \mathbb{B}(\mathcal{H})$ be a unital C$^*$-algebra. A tracial state $\theta$ on $B$ is said to be amenable if there exists a state $\rho$ on $\mathbb{B}(\mathcal{H})$ such that $\rho |_B = \theta$ and $\rho(uau^*) = \rho(a)$ for every $a \in \mathbb{B}(\mathcal{H})$ and unitary $u \in B$.

By Arveson’s extension theorem, this definition does not depend on the choice of the faithful representation. Kirchberg defined the notion of a liftable trace ([@KirchbergT Definition 3.1]) and proved that it is equivalent to amenability in the above sense ([@KirchbergT Proposition 3.2]). Instead of Definition \[DefinitionAmeTr\], we use other equivalent conditions. Let $\pi_\theta \colon B \rightarrow \mathbb{B}(\mathcal{H}_\theta)$ be the GNS-representation for the tracial state $\theta$.

\[TheoremAmenableTrace\] Let 