Let $V$ be an $n$-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form, and let ${\rm O}(V)$ be the orthogonal group preserving this form. Then ${\rm O}(V)$ has a natural action on the tensor product $\bigotimes^k V$. It is well known that the Brauer algebra $\mathcal{B}_k(n)$ is isomorphic to ${\rm End}_{{\rm O}(V)}(\bigotimes^k V)$ when $n \geq 2k$. This is the analogue of Schur-Weyl duality between ${\rm GL}(V)$ and the symmetric group $\mathfrak{S}_k$.
Can such a construction still make sense when $V$ is infinite-dimensional? I believe that the original Schur-Weyl setting still holds good, since $\mathbb{C}[\mathfrak{S}_k]$ is still the commutant of $\mathbb{C}[{\rm GL}(V)]$. But in the case of ${\rm O}(V)$, what exactly happens to $\mathcal{B}_k(n)$ if we try to make the parameter $n$ infinite?