A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C*-Algebras
A Duality Theorem for Ergodic Actions of Compact Quantum Groups on C*-Algebras
复制标题
C*-代数上紧量子群遍历作用的对偶定理
DOI：
10.1007/s00220-007-0371-7
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发表时间：
2006
影响因子：
2.4
通讯作者：
J. Roberts
中科院分区：
文献类型：
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作者：
C. Pinzari;J. Roberts
紧量子群G在单位C*-代数上的遍历作用的谱函子是拟量的，即两个谱子空间的张量积等距包含在张量积表示的谱子空间中，且包含映射满足自然性质.证明了从Rep(G)到Hilbert空间范畴的任一拟量*-函子都是G在单位C*-代数上的遍历作用的谱函子。作为应用，我们将单位C*-代数上的遍历G-作用与Rep(G)包含在抽象张量C*-范畴$${\mathcal{T}}$$中联系起来。如果包含来自G的量子子群K，则相应的G-系就是商空间K\G。如果G是一个群且$${数学{T}$$具有置换对称性，则G-系就是商空间K\G。如果一个张量C*-范畴有一个Hecke对称性，使得一个对象为d维ρ，且具有μ-行列式1，则在一个以$$(i，r)$为谱子空间的单位C*-代数上，存在U(D)的遍历作用.讨论了$$S_\MU(2)$$的特例。
The spectral functor of an ergodic action of a compact quantum group G on a unital C*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor *-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C*-algebra.As an application, we associate an ergodic G-action on a unital C*-algebra to an inclusion of Rep(G) into an abstract tensor C*-category $${\mathcal{T}}$$ .If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and $${\mathcal{T}}$$ has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G.If a tensor C*-category has a Hecke symmetry making an object ρ of dimension d and μ-determinant 1, then there is an ergodic action of SμU(d) on a unital C*-algebra having the $$(\iota,\rho^r)$$ as its spectral subspaces. The special case of $$S_\mu U(2)$$ is discussed.