Abstract Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$-algebras with additional $C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

中文翻译：

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不可逆和可逆皮姆斯纳算子代数的分类

摘要 自 1930 年代由冯诺依曼 (von Neumann) 提出以来，算子代数已被用于阐明许多数学理论。自伴随和非自伴随算子代数的分类结果表明了这种方法，但自 1960 年代后期出现以来，人们一直在寻求两者之间的明确联系。我们通过揭示非自伴随算子代数和 $C^{*}$-代数与附加 $C^{*}$-代数结构的分类层次结构来连接这些看似独立的结果类型。我们的方法自然适用于由 $C^{*}$-correspondences 产生的代数，以解决文献中的自伴随和非自伴随同构问题。我们应用我们的策略来完全阐明这个新发现的由有向图产生的算子代数的层次结构。