Real C*-Algebras, United K-Theory, and the Kunneth Formula
Real C*-Algebras, United K-Theory, and the Kunneth Formula
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实 C* 代数、联合 K 理论和 Kunneth 公式
DOI：
10.1023/a:1020671031447
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
Jeffrey L. Boersema
中科院分区：
文献类型：
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作者：
Jeffrey L. Boersema
我们为实 C* 代数定义了联合 K 理论，推广了 Bousfield 的拓扑联合 K 理论。联合 K 理论融合了三个函子——实 K 理论、复 K 理论和自共轭 K 理论——以及它们之间的自然变换。联合K理论相对于普通K理论的优势在于它的同调代数性质，它允许我们构造一个Kunneth型、非分裂、短精确序列，其中项是两个实数C*-代数A和B的张量积的联合K理论，只要A的复化属于自举范畴，该理论就成立。由于联合 K 理论包含普通 K 理论，因此我们的序列提供了一种计算两个实 C* 代数张量积的 K 理论的方法。
作为一个应用，我们计算两个实 Cuntz 代数的张量积的统一 K 理论。与复杂情况不同的是，张量积 O_{k+1} 有时 O_{l+1} 的同构类不仅仅由 k 和 l 的最大公约数决定。因此，我们有非同构、简单、纯无限、实 C* 代数的例子，其复化是同构的。
We define united K-theory for real C*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors -- real K-theory, complex K-theory, and self-conjugate K-theory -- and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Kunneth-type, non-splitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C*-algebras A and B which holds as long as the complexification of A is in the bootstrap category. Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C*-algebras.
As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product O_{k+1} otimes O_{l+1} is not determined solely by the greatest common divisor of k and l. Hence we have examples of non-isomorphic, simple, purely infinite, real C*-algebras whose complexifications are isomorphic.