给定一个C*-代数$A$，一个离散交换群$X$和一个定义对偶作用群$Γ\subset$ aut $A$的同态$Θ: X\to$ Out $A$，本文给出了Hilbert $\{A,Γ\}$的存在性和刻画，其中作用由$\hat{X}$给出.它们是在（抽象）C*-水平上陈述的，因此可以被认为是例如Jones [Mem.Am.Math.Soc. 28 Nr 237（1980）]或Sutherland [Publ.Res.Inst.Math.Sci. 16（1980）135]。Hilbert扩张存在当且仅当存在广义2-上圈。这些结果推广了[Commun.Math.Phys. 15（1969）173]中的结果，这些结果是在超选择理论的背景下形成的，其中假设代数$A$有平凡中心，即$Z=C1$ .特别是众所周知的“外部特征”的第二上同调$H^2(X,{\cal U}(Z),α_X)$可以重新制定：有一个双射的集合的所有$A$ -模同构类的希尔伯特扩张。最后，一个希尔伯特空间表示（由于萨瑟兰在冯诺依曼的情况下）被提及。希尔伯特扩张的C*-范数用这个表示的范数表示，它与超选择理论中出现的所谓正则表示有关。
英文摘要：
Given a C*-algebra $A$, a discrete abelian group $X$ and a homomorphism $Θ: X\to$ Out$A$ defining the dual action group $Γ\subset$ aut$A$, the paper contains results on existence and characterization of Hilbert $\{A,Γ\}$, where the action is given by $\hat{X}$. They are stated at the (abstract) C*-level and can therefore be considered as a refinement of the extension results given for von Neumann algebras for example by Jones [Mem.Am.Math.Soc. 28 Nr 237 (1980)] or Sutherland [Publ.Res.Inst.Math.Sci. 16 (1980) 135]. A Hilbert extension exists iff there is a generalized 2-cocycle. These results generalize those in [Commun.Math.Phys. 15 (1969) 173], which are formulated in the context of superselection theory, where it is assumed that the algebra $A$ has a trivial center, i.e. $Z=C1$. In particular the well-known ``outer characterization'' of the second cohomology $H^2(X,{\cal U}(Z),α_X)$ can be reformulated: there is a bijection to the set of all $A$-module isomorphy classes of Hilbert extensions. Finally, a Hilbert space representation (due to Sutherland in the von Neumann case) is mentioned. The C*-norm of the Hilbert extension is expressed in terms of the norm of this representation and it is linked to the so-called regular representation appearing in superselection theory.
