We prove that a certain bialgebroid introduced recently by Kadison is isomorphic to a bialgebroid introduced earlier by Connes and Moscovici. At the level of total algebras, the isomorphism is a consequence of the general fact that an L-R-smash product over a Hopf algebra is isomorphic to a diagonal crossed product.

Mackey’s imprimitivity theorem characterises the unitary representations of a locally compact group G which have been induced from representations of a closed subgroupK; Rieffel’s influential reformulation says that the group C-algebra C(K) is Morita equivalent to the crossed product C0(G/K) × G [14]. There have since been many important generalisations of this theorem, especially by Rieffel [1...

By a partial dynamical system we mean a pair (X,α) where X is compact and α : ∆ → X is a continuous, not necessarily injective, mapping such that ∆ is clopen, additionally we assume that α(∆) is open. Such systems arise naturally when dealing with commutative C∗-dynamical system (A, δ). We associate with every pair (X,α), or (A, δ), a covariance C∗-algebra C∗(X,α) = C∗(A, δ) which agrees with a...

Let (G,Σ) be an ordered abelian group with Haar measure μ, let (A,G, α) be a dynamical system and let A oα Σ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism Aoα Σ ∼s ( A⊗K(G,Σ, μ) ) oα⊗Ad ρ G, where K(G,Σ, μ) denotes the compact operators in the CSL algebra AlgL(G,Σ, μ) and ρ denotes the right regular representation of G. We also show that there exis...

We consider four operads which are different varieties of the cacti operad. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting loops. We show that both types of cacti are a semi–direct products as operads of their normalized versions with a re–scaling operad based on R>0 which are homotopic to direct produc...

A subgroup of an amenable group is amenable. The C*-algebra version of this fact is false. This was first proved by M.-D. Choi [9] who proved that the non-nuclear C*-algebra C*(Z2 * Z3) is a subalgebra of the nuclear Cuntz algebra €2. A. Connes provided another example, based on a crossed product construction. More recently J. Spielberg [23] showed that these examples were essentially the same....

Pseudo-Galois extensions are shown to be depth two extensions. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups or involutive Hopf algebras and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observ...