紧群的DR-对偶理论在具有非平凡中心的C* 代数-代数的自同态范畴中的应用
An application of the DR-duality theory for compact groups to endomorphism categories of C*-algebras with nontrivial center
摘要：
在Rev.Math.Phys.4（1997）785中，我们研究了不动点代数A具有非平凡中心Z且满足A' \cap F=Z的Hilbert-C ~* 系统{F，G}. A的所有规范自同态的相应范畴包含特征的相互同构的Doplicher/Roberts型子范畴，这些子范畴与在相应的G-不变Hilbert Z-模内选择杰出的G-不变代数希尔伯特空间有关。本文给出了相应反问题的解。更精确地说，假设中心为Z的C* 代数-代数A的自同态范畴T包含DR-型的某个子范畴，则构造A的Hilbert扩张{F，G}，使得T同构于A的所有标准自同态的范畴w.r.t. {F，G}和A' \cap F=Z。进一步证明了容许子范畴之间存在自然等价关系，并证明了两个容许子范畴产生A-模同构Hilbert扩张当且仅当它们等价.解的关键步骤是将标准DR-理论应用于指定的子范畴。
英文摘要：
In Rev. Math. Phys. 4 (1997) 785 we study Hilbert-C* systems {F,G} where the fixed point algebra A has nontrivial center Z and where A'\cap F=Z is satisfied. The corresponding category of all canonical endomorphisms of A contains characteristic mutually isomorphic subcategories of the Doplicher/Roberts-type which are connected with the choice of distinguished G-invariant algebraic Hilbert spaces within the corresponding G-invariant Hilbert Z-modules. We present in this paper the solution of the corresponding inverse problem. More precisely, assuming that the given endomorphism category T of a C*-algebra A with center Z contains a certain subcategory of the DR-type, a Hilbert extension {F,G} of A is constructed such that T is isomorphic to the category of all canonical endomorphisms of A w.r.t. {F,G} and A'\cap F=Z. Furthermore, there is a natural equivalence relation between admissible subcategories and it is shown that two admissible subcategories yield A-module isomorphic Hilbert extensions iff they are equivalent. The essential step of the solution is the application of the standard DR-theory to the assigned subcategory.
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