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Local and global definitions of pure infiniteness for a C Ã -algebra A are compared, and equivalence between them is obtained if the primitive ideal space of A is Hausdorff and of finite dimension, if A has real rank zero, or if A is approximately divisible. Sufficient criteria are given for local pure infiniteness of tensor products. They yield that exact simple tensorially non-prime C Ã -algebras are purely infinite if they have no semi-finite lower semi-continuous trace. One obtains that A is isomorphic to A#O N if A is (1-)purely infinite, separable, stable, nuclear and PrimðAÞ is a Hausdorff space (not necessarily of finite dimension).

Given reduced amalgamated free products of C * -algebras (A, φ) = * ι∈I (A ι , φ ι ) and (D, ψ) = * ι∈I (D ι , ψ ι ), an embedding A ֒→ D is shown to exist assuming there are conditionalexpectation-preserving embeddings A ι ֒→ D ι . This result is extended to show the existence of the reduced amalgamated free product of certain classes of unital completely positive maps. Finally, analogues of the above mentioned results are proved for amagamated free products of von Neumann algebras.Note that the homomorphism κ exists if and only if the GNS representation π ψ : D → L(L 2 (D, ψ)) of ψ is faithful when restricted to the subalgebra of D generated by ι∈I κ ι (A ι ) As observed in [16, 1.3], the answer to Question 1 is "yes" if the state ψ on D is assumed to be faithful, (and a similar result holds in the amalgamated case). However, in general the answer is "no", as was shown by the elementary example [16, 1.4], (see also the erratum to [16]). K.D. was partially supported as an invited researcher funded by CNRS of France and by NSF Grant No. DMS 0070558.

On appelle pré-sous-groupe d'un unitaire multiplicatif V agissant sur un espace hilbertien de dimension finie H une droite vectorielle L de H telle que V (L ⊗ L) = L ⊗ L . Nous montrons que les pré-sous-groupes sont en nombre fini, donnons un équivalent du théorème de Lagrange et généralisons à ce cadre la construction du 'bi-produit croisé'. De plus, nous établissons des bijections entre pré-sous-groupes et sous-algèbres coïdéales de l'algèbre de Hopf associée à V , et donc avec les facteurs intermédiaires des inclusions de facteurs associées (cf. [11]). Enfin, nous montrons que les pré-sous-groupes classifient les sous-objets de (H, V ) .

Given two unital continuous C * -bundles, A and B, over the same compact Hausdorff base space X, we study the continuity properties of their different amalgamated free products over C(X).