Real $K$-Theory for $C^*$-Algebras: Just the Facts

Jeff Boersema, Claude Schochet
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Abstract

This paper is intended to present the basic properties of $KO$-theory for real $C^*$-algebras and to explain its relationship with complex $K$-theory and with $KR$- theory. Whenever possible we will rely upon proofs in printed literature, particularly the work of Karoubi, Wood, Schr\"oder, and more recent work of Boersema and J. M. Rosenberg. In addition, we shall explain how $KO$-theory is related to the Ten-Fold Way in physics and point out how some deeper features of $KO$-theory for operator algebras may provide powerful new tools there. Commutative real $C^*$-algebras NOT of the form $C^R(X)$ will play a special role. We also will identify Atiyah's $KR^0(X, \tau ))$ in terms of $KO_0$ of an associated real $C^*$-algebra.
C^*$代数的实$K$理论:事实而已
本文旨在介绍实$C^*$格的$KO$理论的基本性质,并解释它与复$K$理论和$KR$理论的关系。在可能的情况下,我们将依靠印刷文献中的证明,特别是卡鲁比、伍德、施罗德的工作,以及波尔塞马和罗森伯格的最新工作。此外,我们还将解释 $KO$ 理论与物理学中的 "十重道 "是如何相关的,并指出算子代数的 $KO$ 理论的一些更深层次的特征是如何为物理学提供强大的新工具的。非$C^R(X)$形式的交换实$C^*$数组将发挥特殊作用。我们还将根据相关实$C^*$代数的$KO_0$来确定阿蒂亚的$KR^0(X, \tau ))$ 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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