Extensions of completely positive mappings - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:37:18Z http://mathoverflow.net/feeds/question/104373 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104373/extensions-of-completely-positive-mappings Extensions of completely positive mappings Masayoshi Kaneda 2012-08-09T19:04:19Z 2012-08-09T19:04:19Z <p>I would like to ask the following two questions.</p> <p>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 $\phi^2=\phi$.</p> <blockquote> <strong>Question 1.</strong> 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?<br><strong>Question 2.</strong> 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.) </blockquote> <p>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.</p>