# AUTOMORPHIC ACTIONS OF COMPACT GROUPS ON OPERATOR ALGEBRAS

#### Abstract

This dissertation is in three essentially independent sections. The common unifying theme is the study of automorphic actions of compact groups on C*- and W*-algebras.^ (i) A "Mackey Machine" decomposition is obtained for the C*-algebras of transformation groups in which a compact Lie group acts on a locally compact space with one orbit type. This yields an explicit formula for the equivariant K-theory in this situation in terms of twisted K-theory of certain finite covering spaces of the orbit space.^ (ii) A study is made of ergodic actions of compact groups on von Neumann algebras. In particular, the notion of multiplier is developed for the dual of a not necessarily commutative compact group and is used to classify ergodic actions with factorial crossed products, in analogy to known results in the commutative case. It is also shown that the groups SU(2) and SO(3) admit no non-trivial multipliers; and further results are proved which seem to indicate that SU(2) admits essentially no non-classical ergodic actions, where classical actions are those derived from matrix algebra bundles on homogeneous spaces.^ (iii) Certain "product-type" actions of compact Lie groups on approximately finite-dimensional operator algebras are investigated. A sharpened form of Blattner's Theorem on outer actions is obtained for compact Lie groups; and formulas for equivariant K-theory and K-theory of fixed point algebras are determined in terms of localisations of character rings. Similar formulas are also established for certain canonical actions on Cuntz algebras. The link between K-theory and traces is explored, leading to a study of the normal (i.e. traceable primary) representations of the infinite symmetric group. In particular an asymptotic character formula is established, implying probability formulas for Young tableaux and explaining earlier character formulas of Thoma. ^

#### Subject Area

Mathematics

#### Recommended Citation

WASSERMANN, ANTONY JOHN, "AUTOMORPHIC ACTIONS OF COMPACT GROUPS ON OPERATOR ALGEBRAS" (1981). *Dissertations available from ProQuest*. AAI8127086.

https://repository.upenn.edu/dissertations/AAI8127086