I would like to ask the following two questions.
Let $1_{\mathcal{H}}\in \mathcal{A}\subset\mathcal{B}\subset\overline{\mathcal{A}}^{SOT}\subset\mathbb{B}(\mathcal{H})$ be a sequence of $C^{\ast}$-subalgebras and $\phi:\mathcal{B}\to\mathcal{A}$ a $*$-epimorphism such that $\phi^2=\phi$.
Question 1. Suppose that $\mathcal{A}$ is an $AW^*$-algebra and that $\{x_{\alpha}\}\subset\mathcal{B}_{sa}$ is a norm-bounded monotone increasing pairwise commuting net with strong limit $x\in\overline{\mathcal{A}}^{SOT}\setminus\mathcal{B}$. Since $\mathcal{A}$ is an $AW^*$-algebra, there exists a least upper bound $y$ for $\{\phi(x_{\alpha})\}$ in a maximal abelian $C^*$-subalgebra of $\mathcal{A}$ containing $\{\phi(x_{\alpha})\}$. Define $\tilde{\phi}:\operatorname{span}(\mathcal{B}\cup\{x\})\to\mathcal{A}$ to be the linear extension of $\phi$ such that $\tilde{\phi}(x)=y$, where $\operatorname{span}(\mathcal{B}\cup\{x\})$ is just the linear span (NOT taking products between $x$ and elements of $\mathcal{B}$). Is $\tilde{\phi}$ 2-positive? If not, how about if $\mathcal{A}$ is a monotone complete $C^*$-algebra?
Question 2. Suppose that $\mathcal{A}$ is a monotone complete $C^*$-algebra and that $\{x_{\alpha}\}\subset\mathcal{B}_{sa}$ is a norm-bounded monotone increasing pairwise commuting net with strong limit $x\in\overline{\mathcal{A}}^{SOT}\setminus\mathcal{B}$, Define $\tilde{\phi}:\operatorname{span}(\mathcal{B}\cup\{x\})\to\operatorname{span}(\mathcal{A}\cup\{z\})$ to be the linear extension of $\phi$ such that $\tilde{\phi}(x)=z$, where $z$ is the strong limit of $\{\phi(x_{\alpha})\}$ in $\overline{\mathcal{A}}^{SOT}$. Is $\tilde{\phi}$ 2-positive? (Assuming $\mathcal{A}$ being monotone complete will not be essential in Question 2, but I did so since I am trying to solve the problem in this context.)
My feeling is that the answer to Question 2 is negative, while the answer to Question 1 could be positive at least in the monotone complete case.
