Embeddings of reduced free products of operator algebras

Abstract

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 embeddings of full amalgamated free product C*–algebras

“…(In contrast, it is known [2] that in the more stringent situation of reduced amalgamated free products, the map analogous to λ is always injective. )…”

HNN extensions of von Neumann algebras

“…To this question, we have a satisfactory answer as simple application of Blanchard and Dykema's work [1]. Namely, if all given conditional expectations have the faithful GNS representations, then there is such an embedding in the amalgamated free product level, B 0 → B, and hence it is plain to see that A 0 is embedded into A in the above-mentioned way.…”

“…(ii) For each ∈ P D , the vector s becomes the canonical implementing one in P s of the state D • −1 s • E s , a consequence from (1).…”

Lipschitzness of *-homomorphisms between C*-metric algebras

“…be the reduced free-product C * -algebra with the faithful state σ [34,36,3]. We define a * -filtration {A n } on (A, σ) by setting A n to be the linear span of all products A ı1 n1 · · · · · A ı k n k with each ı j ∈ {1, 2}, with ı j = ı j+1 for 1 ≤ j ≤ k − 1, and with k j=1 n j ≤ n. We let (π ı , H ı , ξ ı ) denote the faithful GNS representation of (A ı , σ ı ) for ı ∈ {1, 2}, and we let (π, H, ξ) denote the faithful GNS representation of (A, σ).…”