Maximal Coactions

Abstract

A coaction δ of a locally compact group G on a C * -algebra A is maximal if a certain natural map from A × δ G ×δ G onto A ⊗ K(L 2 (G)) is an isomorphism. All dual coactions on full crossed products by group actions are maximal; a discrete coaction is maximal if and only if A is the full cross-sectional algebra of the corresponding Fell bundle. For every nondegenerate coaction of G on A, there is a maximal coaction of G on an extension of A such that the quotient map induces an isomorphism of the crossed products.

Operator Algebras and Their Applications

“…It follows from Proposition 4.8 that (1) implies (2) and (3) implies (1), and a careful examination of the construction of the maximalization in [EKQ04] (particularly Lemma 3.6 and the proof of Theorem 3.3 in that paper) shows that Proof. We must show that for all a, b 2…”

Categorical Landstad duality for actions

“…We adopt, more or less, the conventions of [3], [2], and [10] for actions and coactions of locally compact Hausdorff groups on C * -algebras. An action of a locally compact group G on a C * -algebra A is a homomorphism α from G to the automorphism group Aut A such that s → α s (a) is continuous for each a ∈ A.…”

“…A normalization of (A, δ) is a surjection η : (A, δ) → (B, ε) such that (B, ε) is normal and η × G : A × δ G → B × ε G is an isomorphism, and η is called a normalizing map. Maximalizations and normalizations always exist (see [2,Theorem 3.3] for maximalizations and [10, Proposition 2.3] for normalizations). The requirement that maximalizing and normalizing maps be surjective is redundant, as Lemma 3.2 below shows.…”

“…The second author proved a dual version of this in [9], giving a characterization of crossed products by coactions in terms of the existence of suitably equivariant actions. In [6], the authors applied the recently-developed theory of maximal coactions (see [2]) to give a version of Landstad's characterization for full, rather than reduced, crossed products by actions.…”

Coverings of k-graphs

“…For C * -coactions we adopt the conventions of [6,7,21,22]. A coaction of a group G on a C * -algebra A is an injective nondegenerate homomorphism δ of A into the spatial tensor product A ⊗ C * (G) satisfying the coaction identity (id ⊗ δ G )δ = (δ ⊗ id)δ, where δ G is the comultiplication on C * (G).…”