We study restriction and extension properties for states on ${{\text{C}}^{*}}$-algebras with an eye towards hyperrigidity of operator systems. We use these ideas to provide supporting evidence for Arveson’s hyperrigidity conjecture. Prompted by various characterizations of hyperrigidity in terms of states, we examine unperforated pairs of self-adjoint subspaces in a ${{\text{C}}^{*}}$-algebra. The configuration of the subspaces forming an unperforated pair is in some sense compatible with the order structure of the ambient ${{\text{C}}^{*}}$-algebra. We prove that commuting pairs are unperforated and obtain consequences for hyperrigidity. Finally, by exploiting recent advances in the tensor theory of operator systems, we show how the weak expectation property can serve as a flexible relaxation of the notion of unperforated pairs.

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

We apply the methods developed in our previous work to study holomorphic functions of slow growth on coverings of pseudoconvex domains in Stein manifolds. In particular, we extend and strengthen certain results of Gromov, Henkin and Shubin on holomorphic $L^{2}$ functions on coverings of pseudoconvex manifolds in the case of coverings of Stein manifolds.

It is shown that given a uniform algebra $A$ and a peak point $x_0$ for $A$, every function in $A$ can be expressed as a linear combination of two functions in $A$ that peak at $x_0$.