Exactness of locally compact groups

Brodzki, Jacek; Cave, Chris; Li, Kang

doi:10.48550/arxiv.1603.01829

Cited by 4 publications

(4 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…see [Oza06] for a general discussion), the class of discrete exact groups is known to be identical to the class of all discrete groups which can act amenably on some compact Hausdorff space X (we refer to [ADR00] for a quite complete exposition of amenable actions). An analogous result for general second countable locally compact groups has been announced very recently by Brodzki, Cave and Li in [BCL16]. This implies a new proof that exactness passes to closed subgroups, since the restriction of an amenable action to a closed subgroup is amenable.…”

Section: The Double Dual Crossed Productmentioning

confidence: 54%

Crossed products and the Mackey–Rieffel–Green machine

Echterhoff

2017

Oberwolfach Seminars

View full text Add to dashboard Cite

We give an introduction into the ideal structure and representation theory of crossed products by actions of locally compact groups on C * -algebras. In particular, we discuss the Mackey-Rieffel-Green theory of induced representations of crossed products and groups. Although we do not give complete proofs of all results, we try at least to explain the main ideas. For a much more detailed exposition of many of the results presented here we refer to the beautiful book [Wil07] by Dana Williams.

show abstract

Section: The Double Dual Crossed Productmentioning

confidence: 54%

Crossed products and the Mackey–Rieffel–Green machine

Echterhoff

2017

Oberwolfach Seminars

View full text Add to dashboard Cite

show abstract

“…Moreover, it is straightforward to check that E restricts to a map C c pG, Jq Ñ C c pG, Iq, whence it takes J ¸µ, r C G onto I ¸µ,C G (it has closed range as it is an idempotent), and acts as the identity on I ¸µ, r C G. It follows from a diagram chase that if a P J ¸µ, r C G goes to zero under the quotient map to C ¸µ G, then Epaq " a, and thus that a P I ¸µ,C G. Hence the left hand vertical column is also exact. To complete the proof, note that we now have that the left two columns in diagram (5) above are exact, while the rows are all exact by definition. It follows from a diagram chase that the map A¸π G Ñ A¸r π G is an isomorphism, and thus that for any a P C c pG, Aq we have…”

Section: Conventionsmentioning

confidence: 96%

The Minimal Exact Crossed Product

2018

View full text Add to dashboard Cite

Given a locally compact group G, we study the smallest exact crossed-product functor pA, G, αq Þ Ñ A ¸E G on the category of G-C ˚-dynamical systems. As an outcome, we show that the smallest exact crossed-product functor is automatically Morita compatible, and hence coincides with the functor ¸E as introduced by Baum, Guentner, and Willett in their reformulation of the Baum-Connes conjecture (see [2]). We show that the corresponding group algebra C E pGq always coincides with the reduced group algebra, thus showing that the new formulation of the Baum-Connes conjecture coincides with the classical one in the case of trivial coefficients.

show abstract

“…We are now ready for Proof. By [BCL16], being exact is equivalent to the condition that there exists a compact amenable G-space X. Following the arguments given by Higson in [Hig00, Lemma 3.5 and Lemma 3.6] we may as well assume that X is a metrisable convex space and G acts by affine transformations.…”

Section: The Class Compmentioning

confidence: 99%

Bivariant KK-Theory and the Baum–Connes conjecure

Echterhoff

2017

K-Theory for Group C*-Algebras and Semigroup C*-Algebras

View full text Add to dashboard Cite

This is a survey on Kasparov's bivariant KK-theory in connection with the Baum-Connes conjecture on the K-theory of crossed products A ⋊r G by actions of a locally compact group G on a C*-algebra A. In particular we shall discuss Kasparov's Dirac dual-Dirac method as well as the permanence properties of the conjecture and the "Going-Down principle" for the left hand side of the conjecture, which often allows to reduce K-theory computations for A ⋊r G to computations for crossed products by compact subgroups of G. We give several applications for this principle including a discussion of a method developed by Cuntz, Li and the author in [CEL13] for explicit computations of the K-theory groups of crossed products for certain group actions on totally disconnected spaces. This provides an important tool for the computation of K-theory groups of semi-group C*-algebras.

show abstract

Exactness of locally compact groups

Cited by 4 publications

References 25 publications

Crossed products and the Mackey–Rieffel–Green machine

Crossed products and the Mackey–Rieffel–Green machine

The Minimal Exact Crossed Product

Bivariant KK-Theory and the Baum–Connes conjecure

Contact Info

Product

Resources

About