ERRATUM TO: SEEMINGLY INJECTIVE VON NEUMANN ALGEBRAS(Acta Mathematica Scientia,2021,41B(6): 2055–2085.)*

Department of Mathematics Texas A&M University,College Station,TX 77843,USA

E-mail: gilles.pisier@imj-prg.fr

In two instances,the author unfortunately confused the weak* separability and the separability of the predual for a von Neumann algebra.Therefore the Remark 1.3 and the proof of Proposition 7.1 must be replaced by what follows.

Remark 1.3IfMwith separable predual is seemingly injective (or remotely injective),we may takeH=ℓ2.Indeed,by the preceding remarku(M) has a separable predual,namelyB(H)*/ZwhereZis the preanihilator ofu(M).It follows thatu(M) is normed by a countable subset of the unit ball ofB(H)*.Therefore there is a separable Hilbert subspaceK⊂Hsuch that the compression Ψ :xPKx|Kis a normal unital positive isometric embedding ofu(M)inB(K).Repeating this argument forMn(u(M)) for alln≥1 (and augmentingKif necessary)we can obtain a separableK≃ℓ2such that the preceding embedding Ψ :u(M) →B(K) is also completely isometric.Then by the injectivity ofB(H) the embeddingu(M) ⊂B(H) factors aswithwcompletely contractive.Replacingvbyvwanduby Ψuthis shows that we may assume thatH=ℓ2.

Proof of Proposition 7.1Assume for contradiction that if either M=Rωor M=∏Mn/ω,the algebra M is seemingly injective.The goal is to reach a contradiction by showing that any QWEP von Neumann algebraMis then seemingly injective.Our first point is essentially unchanged: any QWEP (or “Connes embeddable”) finiteMwith separable predual embeds in M and hence is seemingly injective.WhenMis finite withM*non separable,this remains true ifMisσ-finite,or equivalently admits a faithful normal tracial state.Indeed,we may then viewMas the directed union of a family of finitely generated subalgebras {Mα}(with conditional expectationsPα:M→Mα).Since eachMαhas a separable predual,our first point (and Remark 7.2) implies thatMis seemingly injective.The rest of the proof leads to the announced result.