Derivations in $C^*$-Algebras Commuting with Compact Actions

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).

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.

Here, a space is perfectly normal if it is normal, and every open subspace is a countable union of closed subspaces. Transfer maps for actions of finite groups were defined by Floyd [10], and used in particular for studying orbit spaces for actions on contractible spaces. When G is a compact Lie group acting freely on X, then n :X/H~X/G is a fiber bundle with fiber G/H, and the transfer constructed by Becker and Gottlieb [1] applies in this case. So our goal here is to generalize both of these constructions. One obvious consequence of the above theorem is that

Compact group actions on UHF algebras obtained by folding the interval

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.

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

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.

Compact Lie group action with the continuous Rokhlin property

We construct examples of locally compact quantum groups coming from bicrossed product construction, including non-Kac ones, which can faithfully and ergodically act on connected classical (noncompact) smooth manifolds. However, none of these actions can be isometric in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009), leading to the conjecture that the result obtained by Goswami and Joardar (Rigidity of action of compact quantum groups on compact, connected manifolds, 2013. arXiv:1309.1294) about nonexistence of genuine quantum isometry of classical compact connected Riemannian manifolds may hold in the noncompact case as well.

We compare several notions of equivariant formality for complex K-theory with respect to a compact Lie group action, including surjectivity of the forgetful map and the weak equivariant formality of Harada–Landweber, and find all are equivalent under standard hypotheses. As a consequence, we present an expression for the equivariant K-theory of the isotropy action of $H$ on a homogeneous space $G/H$ in all the classical cases. The proofs involve mainly homological algebra and arguments with the Atiyah–Hirzebruch–Leray–Serre spectral sequence, but a more general result depends on a map of spectral sequences from Hodgkin’s Künneth spectral sequence in equivariant K-theory to that in Borel cohomology that seems not to have been otherwise defined. The hypotheses for the main structure result are analogous to a previously announced characterization of cohomological equivariant formality, first proved here, expanding on the results of Shiga and Takahashi.

Borsuk-Ulam type theorems for arbitrary compact Lie group actions are proven. The transfer plays a major role in this approach.

We compare algebraic objects related to a compact quantum group action on a unital C*-algebra in the sense of Podleś and Baum et al. and show that they differ by the kernel of the morphism describing the action. Then we address ways to remove the kernel without changing the Podleś algebraic core. A minimal such procedure is described. We end the paper with a natural example of an action of a reduced compact quantum group with non-trivial kernel.

Suppose that a compact quantum group $${\mathcal{Q}}$$ acts faithfully on a smooth, compact, connected manifold M, i.e. has a C* (co)-action α on C(M), such that the action α is isometric in the sense of [GOS09] for some Riemannian structure on M. We prove that $${\mathcal{Q}}$$ must be commutative as a C* algebra i.e. $${\mathcal{Q}\cong C(G)}$$ for some compact group G acting smoothly on M. In particular, the quantum isometry group of M (in the sense of [GOS09]) coincides with C(ISO(M)).

We present a systematic study of the structure of crossed products and fixed point algebras by compact group actions with the Rokhlin property on not necessarily unital C^* -algebras. Our main technical result is the existence of an approximate homomorphism from the algebra to its subalgebra of fixed points, which is a left inverse for the canonical inclusion. Upon combining this with results regarding local approximations, we show that a number of classes characterized by inductive limit decompositions with weakly semiprojective building blocks, are closed under formation of crossed products by such actions. Similarly, in the presence of the Rokhlin property, if the algebra has any of the following properties, then so do the crossed product and the fixed point algebra: being a Kirchberg algebra, being simple and having tracial rank zero or one, having real rank zero, having stable rank one, absorbing a strongly self-absorbing C^* -algebra, satisfying the Universal Coefficient Theorem (in the simple, nuclear case), and being weakly semiprojective. The ideal structure of crossed products and fixed point algebras by Rokhlin actions is also studied. The methods of this paper unify, under a single conceptual approach, the work of a number of authors, who used rather different techniques. Our methods yield new results even in the well-studied case of finite group actions with the Rokhlin property.

Compact group actions, spherical bessel functions, and invariant random variables