The aim of this article is to describe a class of \(^*\)-algebras that allows to treat well-behaved algebras of unbounded operators independently of a representation. To this end, Archimedean ordered \(^*\)-algebras (\(^*\)-algebras whose real linear subspace of Hermitian elements are an Archimedean ordered vector space with rather weak compatibilities with the algebraic structure) are examined. The order induces a translation-invariant uniform metric which comes from a \(C^*\)-norm in the bounded case. It will then be shown that uniformly complete Archimedean ordered \(^*\)-algebras have good order properties (like existence of infima, suprema or absolute values) if and only if they have good algebraic properties (like existence of inverses or square roots). This suggests the definition of Su \(^*\)-algebras as uniformly complete Archimedean ordered \(^*\)-algebras which have all these equivalent properties. All methods used are completely elementary and do not require any representation theory and not even any assumptions of boundedness, so Su \(^*\)-algebras generalize some important properties of \(C^*\)-algebras to algebras of unbounded operators. Similarly, they generalize some properties of \(\varPhi \)-algebras (certain lattice-ordered commutative real algebras) to non-commutative ordered \(^*\)-algebras. As an example, Su \(^*\)-algebras of unbounded operators on a Hilbert space are constructed. They arise e.g. as \(^*\)-algebras of symmetries of a self-adjoint (not necessarily bounded) Hamiltonian operator of a quantum mechanical system.

本文的目的是描述一类\(^*\) -代数，它允许独立于表示处理无界算子的良好代数。为此，检查了阿基米德有序\(^*\) -代数（\(^*\) -代数，其 Hermitian 元素的实线性子空间是阿基米德有序向量空间，与代数结构的兼容性相当弱）。在有界情况下，该顺序会产生一个平移不变的统一度量，该度量来自\(C^*\) -norm。然后将证明一致完备的阿基米德有序\(^*\)- 代数具有良好的阶数性质（例如存在 infima、suprema 或绝对值）当且仅当它们具有良好的代数性质（例如存在倒数或平方根）。这表明Su \(^*\) -代数的定义是一致完备的阿基米德有序\(^*\) -代数，它们具有所有这些等价的性质。所有使用的方法都是完全初级的，不需要任何表示理论，甚至不需要任何有界假设，所以Su \(^*\) -algebras 将\(C^*\) -algebras 的一些重要性质推广到无界算子的代数. 类似地，他们概括了\(\varPhi \) 的一些性质-代数（某些格序可交换实代数）到非交换有序\(^*\) -代数。例如，构造了希尔伯特空间上无界算子的Su \(^*\) -代数。它们以\(^*\) - 量子力学系统的自伴随（不一定有界）哈密顿算符的对称代数形式出现。