  A subalgebra $\mathcal{A}$ of a $C^*$-algebra $\mathcal{M}$ is logmodular
(resp. has factorization) if the set $\{a^*a; a\text{ is invertible with
}a,a^{-1}\in\mathcal{A}\}$ is dense in (resp. equal to) the set of all positive
and invertible elements of $\mathcal{M}$. There are large classes of well
studied algebras, both in commutative and non-commutative settings, which are
known to be logmodular. In this paper, we show that the lattice of projections
in a von Neumann algebra $\mathcal{M}$ whose ranges are invariant under a
logmodular algebra in $\mathcal{M}$, is a commutative subspace lattice.
Further, if $\mathcal{M}$ is a factor then this lattice is a nest. As a special
case, it follows that all reflexive (in particular, completely distributive
CSL) logmodular subalgebras of type I factors are nest algebras, thus answering
a question of Paulsen and Raghupathi [Trans. Amer. Math. Soc., 363 (2011)
2627-2640]. We also discuss some sufficient criteria under which an algebra
having factorization is automatically reflexive and is a nest algebra.
