A ternary ring of operator (TRO) between two complex Hilbert spaces $H$ and $K$ is defined to be a norm closed subspace $V$ of $B(H, K)$ and satisfies $xy^*z \in V$ for all $x, y, z \in V$. Let $V$ and $W$ be TROs. A linear map $\pi: V \to W$ is called TRO-homomorphism provided $\pi(xy^*z) =\pi(x) \pi(y) ^*\pi(z) $ for all $x, y, z \in V$.
Motivated by $C^{\ast}$-algebras, in section $5$ of Kaur and Ruan - Local Properties of Ternary Rings of Operators and Their Linking $C^*$-Algebras, tmax norm $\|.\|_{\text{tmax}}$ on $V \otimes W$ is defined as follows.
For $u= \sum_{i=1}^r v_i \otimes w_i \in V \otimes W$, the tmax norm $\lVert u\rVert_{\text{tmax}}$ is the supremum of the norm $\| \pi.\sigma(u)\|_{B(H)}=\lVert\sum_{i=1}^{r} \pi(v_i) \sigma(w_{i})\rVert_{B(H)}$ over all pairs of TRO-homomorphism $\pi: V \to B(H)$, $\sigma: W \to B(H)$ satisfying following commuting conditions
$ \pi(v) \sigma(w)=\sigma(w) \pi(v)$ and $ \pi(v) \sigma(w)^*=\sigma(w)^* \pi(v)$.
In the paper Section $5$, equation $5.2$ following is stated without proof
$\| u\|_{\vee} \leq \|u\|_{\text{tmax}}\leq \|u\|_{\wedge}$
where $\|.\|_{\vee}$ and $\|.\|_{\wedge}$ denotes the operator space injective and projective norm respectively. In the paper it is not explained that why $\| u\|_{\vee} \leq \|u\|_{\text{tmax}}$. I am unable to see this. Can someone please explain this to me?
P. S: This question has been posted on mathoverflow also and can be found here.
This really comes down to the appropriate perspective on the injective operator space tensor product.
If $E\subset B(H)$ and $F\subset B(K)$ are operator spaces, then one has embeddings $$ \pi\colon E\to B(H\otimes K),\,x\mapsto x\otimes 1,\quad \sigma\colon F\to B(H\otimes K),\,y\mapsto 1\otimes y, $$ and the injective operator space tensor product of $E$ and $F$ is the closure of $\pi.\sigma(E\odot F)$ in $B(H\otimes K)$. In particular, $$ \lVert u\rVert_\vee =\lVert \pi.\sigma(u)\rVert $$ for $u\in E\odot F$.
Now if $V\subset B(H_1,K_1)$ and $W\subset B(H_2,K_2)$ are TROs, their canonical operator space structure comes from the embedding $B(H_j,K_j)\to B(H_j\oplus K_j)$ as lower left corner. You can check that the maps $\pi$ and $\sigma$ defined above are TRO homomrophisms in this case and that they satisfy the commutation relations from the definition of $\lVert\cdot\rVert_{\mathrm{tmax}}$.
Since $\lVert\cdot\rVert_{\mathrm{tmax}}$ involves the supremum over all such pairs of TRO homomorphisms, it dominates $\lVert\cdot\rVert_\vee$.