Extending traces on fixed point C∗ algebras under Xerox product type actions of compact Lie groups
Extending traces on fixed point C∗ algebras under Xerox product type actions of compact Lie groups
复制标题
紧李群 Xerox 产品类型作用下定点 C* 代数上的扩展迹
DOI：
10.1016/0022-1236(87)90080-2
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发表时间：
1987
影响因子：
1.7
通讯作者：
D. Handelman
中科院分区：
文献类型：
--
作者：
D. Handelman
设G表示一个紧连通李群，其极大环面T（例如维数为d）.令n：G-tM，，C是酉表示，并构成nJL UHF C* 代数A= 0 M，，C（n是固定的）;这是n个On矩阵环的副本的无限张量积。我们有一个行动tt：G 3 Aut（A），通过设置a（g）= 0 Ad n（g），BM Baker建议将其命名为Xerox产品类型action，因为表示7 c被反复复制。我们可以形成交叉积A x 1 G和不动点代数A”"（或简单地AC，如果有歧义的可能性很小）。通过对T的限制，我们还得到了d-环面T的Xerox作用（也称为CI）。显然AC是A '的一个单位子代数.主结果断言AC上的每个迹线都扩展到A '上的迹线。在证明过程中，我们还证明了A”的Grothendieck群K_1（AG）是环生成的，而KO（AT）（也是环）是K_1（A '）-模生成的. AG上的忠实纯迹空间是A '的纯迹空间的一个稠密开子集，并且与严格正d元组在Weyl群自然作用下的轨道空间（Rd）++/W同胚。是忠诚的如果B=@ FL，M，C是具有相应乘积类型作用的UHF代数（不一定是Xerox）/？= 0 Ad nrr则自然包含A '3'-+（A@ B）G 30 1@B在忠实的纯迹上诱导一个双射（如果c（是忠实的），且后者的集合在（A@ B 0 '，"@？
Let G denote a compact connected Lie group, with maximal torus T (of dimension d, say). Let n: G--t M,, C be a unitary representation, and form the nJL UHF C* algebra A= 0 M,, C (n is fixed); this is the infinite tensor product of copies of the n On matrix ring. We have an action tt: G 3 Aut (A), by setting a (g)= 0 Ad n (g), BM Baker suggested the name Xerox product type action, because the representation 7c is duplicated over and over. We may form the crossed product A x 1 G and the fixed point algebra A””(or simply AC if there is little likelihood of ambiguity). By restriction of c (to T, we also obtain a Xerox action of the d-torus T (also called CI). Clearly AC is a unital subalgebra of A ‘. The principal result asserts that every trace on AC extends to a trace on A ‘. In the course of the proof, we also show that the Grothendieck group of A”, K,(AG), is finitely generated as a ring, and that KO (AT)(which is also a ring) is finitely generated as a K,,(A’)-module.There are several consequences of these results. The space of faithful pure traces on AG is a dense open subset of the pure trace space of A’, and moreover is homeomorphic to (Rd)++/W, the orbit space of the strictly positive d-tuples under the natural action of the Weyl group, provided that o! is faithful. If B=@ FL, M,,,, C is a UHF algebra with corresponding product type action (not necessarily Xerox)/?= 0 Ad nrr then the natural inclusion A ‘3’-+(A@ B) G301@ B induces a bijection on faithful pure traces (if c (is faithful), and the set of the latter is dense in the pure trace space of (A@ B)‘,“@?