The parabolic algebra \({{\cal A}_p}\) is the weakly closed operator algebra on \({L^2}(\mathbb{R})\) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions \({e^{i\lambda x}},\lambda \ge 0\). It is reflexive, with an invariant subspace lattice \({\rm{Lat}}{{\cal A}_p}\) which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of \({\rm{Lat}}{{\cal A}_p}\) is used to classify strongly irreducible isometric representations of the partial Weyl commutation relations. A formal generalisation of Arveson’s notion of a synthetic commutative subspace lattice is given for general subspace lattices, and it is shown that \({\rm{Lat}}{{\cal A}_p}\) is not synthetic relative to the \({H^\infty}(\mathbb{R})\) subalgebra of \({{\cal A}_p}\). Also, various new operator algebras, derived from isometric representations and from compact perturbations of \({{\cal A}_p}\), are defined and identified.

抛物线代数\({{\cal A}_p}\)是由右平移酉半群和酉半群生成的\({L^2}(\mathbb{R})\)上的弱闭算子代数乘以解析指数函数\({e^{i\lambda x}},\lambda \ge 0\)。它是自反的，具有不变的子空间格\({\rm{Lat}}{{\cal A}_p}\)，它与单位圆盘自然同胚（Katavolos 和 Power，1997）。\({\rm{Lat}}{{\cal A}_p}\)的结构用于对部分 Weyl 交换关系的强不可约等距表示进行分类。对于一般子空间格给出了 Arveson 合成交换子空间格概念的形式化推广，并且表明\({\rm{Lat}}{{\cal A}_p}\)相对于\不是综合的({H^\infty}(\mathbb{R})\) \({{\cal A}_p}\)的子代数。此外，定义和识别了从等距表示和\({{\cal A}_p}\)的紧凑扰动导出的各种新算子代数。