Given a $C^*$-Algebra $A\subseteq\mathbb{B}(H)$ (for some Hilbert Space $H$), a linear mapping $V:\mathbb{B}(H)\rightarrow A^{\ast\ast}$ is called a generalized weak expectation if $\|V\|\leq 1$ and $V(a)=a$ for each $a\in A$. Such $A$ is said to have Lance's weak expectation property (or WEP). More details of WEP and properties of $V$ can be found in this paper.
Given a topological space $X$ and a subspace $A$, a continuous map $f:X\rightarrow A$ is a retraction if $f(a)=a$ for each $a\in A$. Since $A^{\ast\ast}$ is the weak-$\ast$ closure of $A$ and the weak-$\ast$ closure of $A$, say $\overline{A}^{w\ast}$, is contained in $\mathbb{B}(H)$, then there will be a mapping say $\Psi$ from $\mathbb{B}(H)$ to the $\overline{A}^{w\ast}$ such that for each $a\in\overline{A}^{w\ast}, \Psi(a)=a$.
My question are the following:
in a $C^*$-Algebra, is there a property that is similar to the existence of a deformation retract, or strong deformation retract?
Is there any relation between the concept of retraction and the concept of WEP (since their definitions are so similar)? Languages in topology and $C^*$-Algebra are different in general but I wish to see some concepts that look similar, and any examples or explanations will be highly appreciated.