We introduce a notion of ‘hereditarily antisymmetric’ operator algebras and prove a structure theorem for them in finite dimensions. We also characterize those operator algebras in finite dimensions which can be made upper triangular and prove matrix analogs of the theorems of Dilworth and Mirsky for finite posets. Some partial results are obtained in the infinite dimensional case.

We consider the ${{w}^{*}}$-closed operator algebra ${{\mathcal{A}}_{+}}$ generated by the image of the semigroup $S{{L}_{2}}\left( {{\mathbb{R}}_{+}} \right)$ under a unitary representation $\rho$ of $S{{L}_{2}}\left( \mathbb{R} \right)$ on the Hilbert space ${{L}^{2}}\left( \mathbb{R} \right)$. We show that ${{\mathcal{A}}_{+}}$ is a reflexive operator algebra and ${{\mathcal{A}}_{+}}=\text{Alg }\mathcal{D}$ where $\mathcal{D}$ is a double triangle subspace lattice. Surprisingly, ${{\mathcal{A}}_{+}}$ is also generated as a ${{w}^{*}}$-closed algebra by the image under $\rho$ of a strict subsemigroup of $S{{L}_{2}}\left( {{\mathbb{R}}_{+}} \right)$.

We prove that two quiver operator algebras can be isometrically isomorphic only if the quivers (=directed graphs) are isomorphic. We also show how the graph can be recovered from certain representations of the algebra.

AF C*-algebras contain natural AF masas which, here, we call standard diagonals. Standard diagonals are unique, in the sense that two standard diagonals in an AF C*-algebra are conjugate by an approximately inner automorphism. We show that this uniqueness fails for non-selfadjoint AF operator algebras. Precisely, we construct two standard diagonals in a particular non-selfadjoint AF operator algebra which are not conjugate by an approximately inner automorphism of the non-selfadjoint algebra.