REPRESENTATIONS AND AUTOMORPHISMS OF THE IRRATIONAL ROTATION ALGEBRA
REPRESENTATIONS AND AUTOMORPHISMS OF THE IRRATIONAL ROTATION ALGEBRA
复制标题
无理旋转代数的表示和自同构
DOI：
10.2140/pjm.1984.111.257
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发表时间：
1984
影响因子：
0.6
通讯作者：
B. Brenken
中科院分区：
文献类型：
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作者：
B. Brenken
给定一个无理数，AA是由两个酉算子U和V生成的唯一的C*-代数，满足扭曲对易关系UV=exp(2πia)vu。我们研究了AA的可分表示，当限制到由V生成的交换C*代数时，AA的可分表示具有一致重数m.这些表示按它们的重数来分类，即圆上的拟不变t-Borel测度(W.r.t.按角度里拉旋转)和么正一余周期。可分离因子表示存在于这类中，在这种情况下，度量是遍历的。因子表示在由Gb/生成的C*代数上具有一致重数Mf，且如果m，m‘是相对素的，则表示是不可约的。利用我们构造的SL(2，Z)的作用作为AA9的*-自同构，我们得到了AA的纯态分离族，其对应的不可约表示提供了明确的例子，其中m和m‘作为任意给定的非零相对素数对出现。
Given an irrational number «, Aa is the unique C*-algebra generated by two unitary operators, U and V, satisfying the twisted commutation relation UV= exp(2 πia)VU. We investigate separable representations of Aa which, when restricted to the abelian C* algebra generated by V, are of uniform multiplicity m. These representations are classified by their multiplicity, a quasi-invarian t Borel measure on the circle (w.r.t. rotation by the angle lira) and a unitary one cocycle. Separable factor representations lie in this class, the measure being ergodic in this case. A factor representation is of uniform multiplicity mf on the C* algebra generated by £/, and if m, m' are relatively prime, the representation is irreducible. By use of an action of SL(2, Z) as *-automorphisms of Aa9 that we construct, we arrive at a separating family of pure states of Aa whose corresponding irreducible representations provide explicit examples with m and m' occurring as any given pair of nonzero relatively prime numbers.