The centralizer algebra of the diagonal action of the group $GL_n(\mathbb C)$ in a mixed tensor space

We consider the walled Brauer algebra $Br_{k,l}(n)$ introduced by V. Turaev and K. Koike. We prove that this algebra is a subalgebra of the Brauer algebra and that it is isomorphic, for sufficiently large $n\in\mathbb N$, to the centralizer algebra of the diagonal action of the group $GL_n(\mathbb C)$ in a mixed tensor space. We also give a presentation of the algebra $Br_{k,l}(n)$ by generators and relations. For the generic parameter, the algebra is semisimple, and in this case we describe the Bratteli diagram for the family of algebras under consideration and give realizations of the irreducible representations. We also give a new, more natural, proof of the formulas for the characters of the walled Brauer algebras.