Crossed products by compact group actions with the Rokhlin property

Compact group actions on UHF algebras obtained by folding the interval

This dissertation focuses on finite group actions with the tracial Rokhlin property and the structure of the corresponding crossed products. It consists of two major parts. For the first part, we study several different aspects of finite group actions with certain versions of the Rokhlin property. We are able to give an explicit characterization of product-type actions with the tracial Rokhlin property or strict Rokhlin property. We also show that, in good circumstances, the actions with the tracial Rokhlin property are generic. In the second portion of this dissertation, we introduce the weak tracial Rokhlin property for actions on non-simple C*-algebras. The main results are as follows. Let A be a unital non-simple C*-algebra and α be an action of G on A with the weak tracial Rokhlin property. Assume that the crossed product C*: G, A, α) is simple. Suppose A has either of the following property: tracial rankk, stable rank one, real rank zero. Then C*: G, A, α) has the same property.

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove that if $\alpha \colon G \rightarrow \mathrm{Aut}(A)$ is an action of a finite group $G$ on a simple C*-algebra $A$ with tracial rank zero and $\alpha$ has the weak tracial Rokhlin property, then the crossed product $A \rtimes _{\alpha} G$ and the fixed point algebra $A^{\alpha}$ are simple with tracial rank zero. This extends a result of N. C. Phillips to the nonunital case. We use the machinery of Cuntz subequivalence to work in this nonunital setting.

For any finite groupG we construct examples of an AF algebraA and an action byG onA such that the fixed point algebra is not AF. The construction ofA is done by successive foldings and cuttings of the interval in a way originally suggested by Blackadar and, in a different context, by Connes in his talk in Oslo in 1978.

The fixed-point algebra of tensor-product actions of finite Abelian groups on UHF-algebras

Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth manifold $M$, we study the operation, which takes each operator $D$ on the ambient manifold to a certain operator on the submanifold. The latter operator is called the trace of $D$. More precisely, we study traces of operators, associated with actions of compact Lie groups on $M$. We show that traces of such operators are localized at special submanifolds in $X$ and study the structure of the traces on these submanifolds.

The theory of compact Lie transformation groups is a large and well researched area of mathematics, in which topological methods are widely used and which requires a separate survey. In this chapter we only touch upon certain problems of this theory. In §1 we shall survey the classical results of Gleason, Montgomery, Mostow, Palais, Samelson and other authors concerning the structure of of orbits of actions of compact Lie groups. We emphasize the differentiable aspects of the theory and a number of results can be formulated in a more general setting — for proper actions of not necessarily compact Lie groups. §2 is devoted to properties of differentiable invariants and almost-invariants of compact Lie groups. In §3 we consider complexifications of homogeneous spaces of compact Lie groups and factorizations of reductive algebraic groups. Surveys of many questions not considered here can be found in (Bourbaki 1982), (Bredon 1972), (Hsiang 1975), (Janich 1968), (Palais 1960), (Schultz 1984).

We study the topological variant of Rokhlin dimension for topological dynamical systems (X,{\alpha},Z^m) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C*-algebraic level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias. In particular, it implies under these conditions that the transformation group C*-algebra has finite nuclear dimension. Generalizing a result of Yonatan Gutman, we show that free Z^m-actions on finite dimensional spaces satisfy a strengthened version of the so-called marker property, which yields finite Rokhlin dimension for said actions.

We establish sharp upper bounds for the dimensions of compact groups which act effectively on finite-dimensional compact generalized quadrangles with four-dimensional point rows. These bounds are attained, or indeed approached, only for explicitly known actions of Lie groups on Moufang quadrangles.

We investigate the representation theory of the crossed–product C*–algebra associated with a compact group G acting on a locally compact space X when the stability subgroups vary discontinuously. Our main result applies when G has a principal stability subgroup or X is locally of finite G–orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation V of a stability subgroup is obtained by restricting V to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of V. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup; the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the C*–algebra of the motion group ℝn ⋊ SO(n) is a Fell algebra. This uses the classical branching theorem for the special orthogonal group SO(n) with respect to SO(n − 1). Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Introduction. 1. Hilbert's fifth problem and proper actions of Lie groups S. Illman. 2. Equivariant algebraic vector bundles over representations - a survey M. Masuda. 3. G-manifolds and G-vector bundles in algebraic, semialgebraic, and definable categories T. Kawakami. 4. Geometry of Finite Topological and equivariant finite topological spaces S. Kono, F. Ushitaki. 5. On the theory of homotopy representations. A survey I. Nagasaki. 6. Manifolds as fixed point sets of smooth compact Lie group actions K. Pawalowski. 7. Surgery and homotopy theory in study of the transformation groups Y. Kitada. 8. Kervaire's obstructions of free actions of finite cyclic groups on homotopy spheres Y. Kitada. 9. The Burnside ring revisited M. Morimoto. 10. Multiplicative stabilization and transformation groups S. Kwasik, R. Schultz. 11. Symmetries on manifolds, deformations and rational homotopy: a survey M. Raussen. 12. Rigidity of codimension one locally free actions of solvable Lie groups N. Tsuchiya, A. Yamakawa. 13. Smooth actions of non-compact semi-simple Lie groups F. Uchida, K. Mukoyama. 14. Hamiltonian group actions and equivariant indices T. Takakura. 15. Controlled methods in equivariant topology, a survey E. Pedersen. Index.

Inclusions of simple C⁎-algebras arising from compact group actions

Let (A, G, a) be a C*-dynamical system such that G is compact, and let d be a closed *-derivation in A commuting with a. If -Aap\^(5)Cker5 and Aa is contained in the center of A, then d is the generator of a strongly continuous one-parameter group of ^-automorphis ms of A commuting with a. § 1. Introduction. We shall consider the problem of whether a closed *-derivation in a C*algebra commuting with an action of a locally compact group is the generator of a strongly continuous one-parameter group of ^-automorphis ms of the C*algebra. S. Sakai [14] first proposed this question and proved that a non-zero closed derivation in C(T) commuting with translations by elements of T is a constant multiple of differentiati on. The same result for C0CR) has been obtained by C. J. K. Batty [4] and F. Goodman [8]. F. Goodman [9] and H. Nakazato [13] generalized it to every locally compact group and compact group, respectively. In [9] the case that a locally compact group acts on its homogeneous space has also been considered. In the present paper we answer this problem in the affirmative for every ergodic action of a compact group. We shall also give some related results on non-ergodic actions of compact groups.

Let a compact Lie group act ergodically on a unital C^* -algebra A . We consider several ways of using this structure to define metrics on the state space of A . These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak-* topology.

A unital C*-algebra A is said to have cancellation of projections if the semigroup D(A) of Murray–von Neumann equivalence classes of projections in matrices over A is cancellative. It has long been known that stable rank one implies cancellation for any A, and some partial converses have been established. In this paper it is proved that cancellation does not imply stable rank one for simple, stably finite C*-algebras. 2000 Mathematics Subject Classification 46L80 (primary), 46L85 (secondary).