Cartan Subalgebras of $\mathfrak{gl}_\infty$

Let V be a vector space over a field K of characteristic zero and V_* be a space of linear functionals on V which separate the points of V. We consider V \otimes V_* as a Lie algebra of finite rank operators on V, and set mathfrak{gl} (V,V_*) := V \otimes V_*. We define a Cartan subalgebra of \mathfrak{gl} (V,V_*) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of mathfrak{gl} (V,V_*) under the assumption that K is algebraically closed. A subalgebra of mathfrak{gl} (V,V_*) is a Cartan subalgebra if and only if it equals bigoplus_j ( V_j otimes (V_j)_* ) oplus (V⁰ otimes V_*⁰) for some one-dimensional subspaces V_j \subseteq V and (V_j)_* subseteq V_* with (V_i)_* (V_j) = delta_ij K and such that the spaces V_*⁰ = \bigcap_j (V_j)^\bot \subseteq V_* and V⁰ = \bigcap_j \bigl( (V_j)_* \bigr)^\bot \subseteq V satisfy V_*⁰ (V⁰) = {0}. We then discuss explicit constructions of subspaces V_j and (V_j)_* as above. Our second main result claims that a Cartan subalgebra of mathfrak{gl} (V,V_*) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra mathfrak{h} which coincides with the maximal locally nilpotent mathfrak{h}-submodule of mathfrak{gl} (V,V_*), and such that the adjoint representation of mathfrak{h} is locally finite.