Single elements of finite CSL algebras (Q2701577)

An element \(s\) of an algebra \({\mathcal A}\) is a single element of \({\mathcal A}\) if \(asb=0\), with \(a\) and \(b\) in \({\mathcal A}\), implies that either \(as=0\) or \(sb=0\). This paper seeks to analyze the structure of the single elements for an important class of algebras of operators on Hilbert space. Let \(H\) be a nonzero complex separable Hilbert space and let \({\mathcal D}\) be a finite distributive subspace lattice on \(H\) with the property that the vector sum \(K+L\) is closed for every \(K\) and \(L\) in \(\mathcal D\). (This holds, for example, if \(\dim H\) is finite.) Let \(\{K_i:i=1,\ldots,n\}\) be the nonzero join-irreducible elements of \(\mathcal D\), ordered so that \(K_i\subseteq K_j\) implies that \(i\leq j\). With \((K_i)_{-}=\bigvee \{L\in {\mathcal D}:K_i\not\subseteq L\}\), we now let \(H_i=K_i\ominus (K_i\cap(K_i)_{-})\). Then \(H\) has the decomposition \(H=\sum_{i=1}^{n}{H_i}\). A partial ordering \(\preceq\) can be defined on the set \(\{1,\ldots,n\}\) by \(i\preceq j\) if \(K_i\subseteq K_j\). The operator algebra \(\text{Alg} {\mathcal D}\) consists of all \(n\times n\) operator-entried matrices \((T_{i,j})\) where each \(T_{i,j}\) is an operator from \(H_j\) to \(H_i\) and where \(T_{i,j}=0\) whenever \(i\not\preceq j\). (One can think of the elements of \(\text{Alg} {\mathcal D}\) as \`\` block upper triangular matrices''.) At the same time, one can define the matrix incidence algebra \({\mathcal A}_n(\preceq)\) corresponding to the given partial ordering to be the set of all \(n\times n\) scalar matrices \((a_{i,j})\) such that \(a_{i,j}=0\) whenever \(i\not\preceq j\). The single elements of \({\mathcal A}_n(\preceq)\) were characterized by the authors in an earlier paper. NEWLINENEWLINENEWLINEThe main theorem of this paper asserts that the single elements of \(\text{Alg} {\mathcal D}\) are obtained by taking the Hadamard (entry-wise) products of single elements of \({\mathcal A}_n(\preceq)\) with the matrices (relative to the decomposition \(H=\sum_{i=1}^{n}{H_i}\)) of the rank-one operators \({e_j}^*\otimes f_i\) on \(H\), where \(e_j\in H_j\), \(f_i\in H_i\), and \({e_j}^*\otimes f_i (z)=\langle z, e_j\rangle f_i\) for all \(z\in H\). Moreover, the rank of the single element of \(\text{Alg} {\mathcal D}\) is equal to the rank of the corresponding single element of \({\mathcal A}_n(\preceq)\).