# Equivariant K-Theory and Freeness of Group Actions on C*-Algebras

Part of the Lecture Notes in Mathematics book series (LNM, volume 1274)

Part of the Lecture Notes in Mathematics book series (LNM, volume 1274)

Freeness of an action of a compact Lie group on a compact Hausdorff space is equivalent to a simple condition on the corresponding equivariant K-theory. This fact can be regarded as a theorem on actions on a commutative C*-algebra, namely the algebra of continuous complex-valued functions on the space. The successes of "noncommutative topology" suggest that one should try to generalize this result to actions on arbitrary C*-algebras. Lacking an appropriate definition of a free action on a C*-algebra, one is led instead to the study of actions satisfying conditions on equivariant K-theory - in the cases of spaces, simply freeness. The first third of this book is a detailed exposition of equivariant K-theory and KK-theory, assuming only a general knowledge of C*-algebras and some ordinary K-theory. It continues with the author's research on K-theoretic freeness of actions. It is shown that many properties of freeness generalize, while others do not, and that certain forms of K-theoretic freeness are related to other noncommutative measures of freeness, such as the Connes spectrum. The implications of K-theoretic freeness for actions on type I and AF algebras are also examined, and in these cases K-theoretic freeness is characterized analytically.

K-theory algebra group action lie group

- DOI https://doi.org/10.1007/BFb0078657
- Copyright Information Springer-Verlag Berlin Heidelberg 1987
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Springer Book Archive
- Print ISBN 978-3-540-18277-1
- Online ISBN 978-3-540-47868-3
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
- About this book