Finite diagram tower inclusion and normalized right closure

Use the diagram algebra ${\mathrm{T}\mathrm{L}}_n(\mathbb{C},\delta )$ and its stacking convention from Temperley–Lieb planar diagram data and endomorphism modules and Stacking is associative and horizontal juxtaposition is strict monoidal. For $n\ge 1$, append a rightmost vertical strand to every $(n,n)$ diagram and extend linearly; denote this map by

For a basis diagram $D:(n+1)\to (n+1)$, join its rightmost top and bottom boundary points by an arc outside the rectangle, remove every resulting closed component with the factor $\delta$, and retain the induced $(n,n)$ matching. Extend this right closure linearly as $c_n$. When $\delta \in {\mathbb{C}}^{\times }$, define the normalized right-closure map

The normalized closure traces ${\tau }_n$ are those of Normalized Temperley–Lieb trace from diagram closure. No positivity of ${\tau }_n$ is included in this definition. The next result proves the algebraic expectation identities for $E_n$ and states precisely when it is a positive conditional expectation.

When $\delta \in \mathbb{R}$, equip each diagram algebra with the conjugate-linear involution that reflects a diagram across a horizontal line and conjugates coefficients. Reflection reverses stacking and fixes the loop weight, so $(xy{)}^{\ast }=y^{\ast }{x}^{\ast }$ and $1^{\ast }=1$. The positivity assertion below assumes this involution and a compatible finite-dimensional $C^{\ast }$-algebra structure.

For $n=2$, closing the rightmost strand of $d_2\in {\mathrm{T}\mathrm{L}}_3$ removes the cap and cup on the last two points and leaves the identity diagram on the first two points; the extra factor ${\delta }^{-1}$ is the normalization. The diagram below records the endpoint convention only. The proof uses the finite matching graph, not the drawing.