We study ${{\text{w}}^{*}}$-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) ${{\text{w}}^{*}}$-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer, we derive that ${{\text{w}}^{*}}$-semicrossed products of factors (on a separableHilbert space) are reflexive. Furthermore, we show that ${{\text{w}}^{*}}$-semicrossed products of automorphic actions on maximal abelian self adjoint algebras are reflexive. In all cases we prove that the ${{\text{w}}^{*}}$-semicrossed products have the bicommutant property if and only if the ambient algebra of the dynamics does also.

Triangular algebras, and maximal triangular algebras in particular, have been objects of interest for over 50 years. Rich families of examples have been studied in the context of many w*- and C*-algebras, but there remains a dearth of concrete examples in $B({\cal H})$. In previous work, we described a family of maximal triangular algebras of finite multiplicity. Here, we investigate a related family of maximal triangular algebras with infinite multiplicity, and unearth a new asymptotic structure exhibited by these algebras.

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)$. The periodic ${{\text{C}}^{*}}$-algebras are characterized, and it is shown that ${{\text{C}}^{*}}\left( \mathbb{F}_{\theta }^{+} \right)\,\simeq \,\text{C}\left( \mathbb{T} \right)\,\otimes \,\mathfrak{U}$ where $\mathfrak{A}$ is a simple ${{\text{C}}^{*}}$-algebra.

We analyze the automorphism group for the norm closed quiver algebras 𝒯+(Q). We begin by focusing on two normal subgroups of the automorphism group which are characterized by their actions on the maximal ideal space of 𝒯+(Q). To further discuss arbitrary automorphisms we factor automorphism through subalgebras for which the automorphism group can be better understood. This allows us to classify a large number of noninner automorphisms. We suggest a candidate for the group of inner automorphisms.

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.

A left ideal on any C*-algebra is an example of an operator algebra with a right contractive approximate indentiy (r.c.a.i.). Indeed, left ideal in C*-algebras may be charcterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. shoulod be studied in terms of a certain let ideal of a C*-algebra. We study left ideals from the perspective of ‘Hamana theory’ and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approzimate indentity, including a Banach-Stone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.