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Nest algebras are among the simplest examples of Commutative Subspace Lattice Algebra s. Indeed, they are formally defined as the algebra of Bounded Operator s leaving Invariant each Subspace contained in a Subspace Nest , that is, a set of subspaces which is Totally Ordered by Inclusion . Since the Orthogonal Projection s corresponding to the subspaces in a nest Commute , nests are commutative subspace lattices.

By way of an example, let us apply this definition to recover the finite-dimensional upper-triangular matrices. Let us work in the n- Dimension al Complex Vector Space \mathbb{C}^n, and let e_1,e_2,\dots,e_n be the Standard Basis . For j=0,1,2,\dots,n, let S_j be the j-dimensional Subspace of \mathbb{C}^n Span ned by the first j basis vectors e_1,\dots,e_j. Let

:N=\{ (0)=S_0, S_1, S_2, \dots, S_{n-1}, S_n=\mathbb{C}^n \};

then N is a subspace nest, and the corresponding nest algebra of n imes n complex matrices M leaving each subspace in N invariant -- that is, satisfying MS\subseteq S for each S in N -- is precisely the set of upper-triangular matrices.

If we omit one or more of the subspaces ''Sj'' from ''N'' then the corresponding nest algebra consists of block upper-triangular matrices.


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