Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional Hilbert spaces.
Let $B\_1, \ldots, B\_k \in B(\mathcal{H}).$
Define $L: B(\mathcal{K})^k \rightarrow B(\mathcal{H}\otimes \mathcal{K})$ via the formula
$$L(X\_1,\ldots, X\_k) = \sum B\_i \otimes X\_i.$$


For each $\varepsilon > 0,$ does there exist a finite rank projection $P \in B(\mathcal{H})$ such that
$$\|L(X\_1,\ldots, X\_k )\| \leq (1+\varepsilon)\|(P \otimes I)L(X\_1,\ldots, X\_k )(P \otimes I)\|$$
for all $(X\_1,\ldots, X\_k ) \in B(\mathcal{K})^k?$



Approximating the norm of an operator-valued linear function with operator inputs via a matrix-valued linear function
No, there does not always exist such a finite rank projection.


Indeed, this implies that the linear space spanned by $B\_1,\dots,B\_k$ is **exact** as an operator space, and there are non-exact operator spaces. Actually one can check that the converse holds: if $B\_1,\dots,B\_n$ span an exact operator space, then such a $P$ exists (**Edit**: see the end of this answer for a proof). For a reference, I suggest Pisier's book operators spaces, or the original paper <http://arxiv.org/abs/math/9308204>.


To make this answer a bit more self-contained, recall that


* An operator space is (by definition) a subspace of $B(H)$ for a Hilbert space $H$.
* The completely bounded (cb) norm of a linear map $u \colon E \to F$ between two operator spaces is $\|u\|\_{cb}$, the supremum over all $n$, all $x\_1,\dots,x\_k \in E$, all Hilbert spaces $K$ and all $a\_1,\dots,a\_n \in B(K)$ of $\frac{ \|\sum\_i a\_i \otimes u(x\_i)\|}{ \|\sum\_i a\_i \otimes x\_i\|}$.
* The cb-distance between $E$ and $F$ is $d\_{cb}(E,F)$, the infimum over all linear bijections $u \colon E \to F$ of $\|u\|\_{cb} \|u^{-1}\|\_{cb}$.
* The exactness constant of a (finite dimensional) operator space $E$, denoted $ex(E)$, is the infimum, over all subspaces $F$ of a matrix algebra, of $d\_{cb}(E,F)$.


With this terminology, it is immediate that a family $B\_1,\dots,B\_k$ for which your question has a positive answer satisfies that $ex(span(B\_1,...,B\_k))=1$. Indeed, your inequality is exactly that the map $u \colon B\_i \mapsto P B\_i P$ satisfies $\|u^{-1}\|\_{cb} \leq 1+\varepsilon$, whereas the inequality $\|u\|\_{cb} \leq 1$ is always true.


To completely answer your question, it remains to give examples of finite dimensional operator spaces with $ex(E)>1$.


The simplest one is probably the one that is denoted by $\max(\ell^1\_k)$. It is the dual of $\ell^\infty\_k$ in the category of operator spaces. Explicitely, it is linearly spanned by $B\_1,\dots,B\_k \in B(H)$ and satisfies that 
$$ \| \sum X\_k \otimes B\_k\| = \sup\_{U\_1,\dots,U\_k \in B(\ell^2), \|U\_i\| \leq 1} \|\sum X\_k \otimes U\_k\|.$$
However, the existence of a family $B\_1,\dots,B\_k$ satisfying this equality is not obvious. There are only nonexplicit constructions based on the Hahn-Banach theorem. Look at Ruan's theorem in Pisier's book.


Another example is obtained using groups. Consider $G$ a group generated by a finite set $g\_1,\dots,g\_k$, and take $B\_i=\lambda(g\_i) \in B(\ell^2(G))$, the operator by left-translation by $g\_i$. Assume furthermore that the $g\_1$ is equal to the identity of $G$. Then $ex(span(B\_1,\dots,B\_k))=1$ if and only if the group $G$ is exact. And there are groups constructed by Gromov that are not exact, as proved by Ozawa in <http://arxiv.org/abs/math/0002185>.




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As requested in the comments, let me show that such a $P$ exists if $B\_1,\dots,B\_n$ span an operator space $E$ with exactness constant $ex(E)=1$. 


The tool is a lemma by Roger Smith (see Proposition 1.12 in Pisier's book) asserting that for a map $u$ from an operator space to $M\_n(\mathbf C)$ the cb-norm of $u$ is equal to the supremum (denoted $\|u\|\_n$ in general) of $\frac{ \|\sum\_i a\_i \otimes u(x\_i)\|}{ \|\sum\_i a\_i \otimes x\_i\|}$ over all $k$, all $x\_1,\dots,x\_k \in E$ and all $a\_1,\dots,a\_k \in M\_n(\mathbf C)$.


First, it is easy to see that for every finite subset $F$ of $B(K)^k$ and each $\varepsilon>0$, there exists a finite dimensional projection $P$ such that 
$$ \| L(X\_1,\dots,X\_k)\| \leq (1+\varepsilon) \| (P \otimes 1)L(X\_1,\dots,X\_k) (P \otimes 1)\| .$$


By the compactness of the unit ball of $M\_n(\mathbf C)$, for every integer $n$ there exists a projection $P$ such that the preceding holds for all $X\_1,\dots,X\_n \in M\_n(\mathbf C)$.


The problem is to have such a $P$ which does not depend on $n$, and this is where exactness is used: since $ex(E)=1$, there exists $n$, a subspace $F \subset M\_n(\mathbf C)$ and a map $u \colon E \to F$ such that $\|u\|\_{cb} \|u^{-1}\|\_{cb} \leq 1+\varepsilon$. Let $P$ be projection as above for this value of $n$. If $u\_P$ is the map sending $B\_i$ to $P B\_i P$, then I claim that $\|u\_P^{-1}\|\_{cb} \leq (1+\varepsilon)^2$, QED. Indeed: 
$$ \|u\_P^{-1}\|\_{cb} \leq \|u^{-1}\|\_{cb} \|u \circ u\_P^{-1}\|\_{cb} = \|u^{-1}\|\_{cb} \|u \circ u\_P^{-1}\|\_{n} \leq \|u^{-1}\|\_{cb} \|u\|\_n \|u\_P^{-1}\|\_{n} \leq (1+\varepsilon)^2.$$
The equality is Smith's lemma, and the last inequality comes from the choice of $P$.


