Corrigendum to: “A Morita type equivalence for dual operator algebras” [J. Pure Appl. Algebra 212 (5) (2008) 1060–1071]

Journal of Pure and Applied Algebra 212 (2008) 2581–2582

Contents lists available at ScienceDirect

Journal of Pure and Applied Algebra journal homepage: www.elsevier.com/locate/jpaa

Corrigendum

Corrigendum to: “A Morita type equivalence for dual operator algebras” [J. Pure Appl. Algebra 212 (5) (2008) 1060–1071] G.K. Eleftherakis University of Athens, Department of Mathematics, 15784 Panepistimioupolis, Athens, Greece

article

info

Article history: Received 18 January 2008 Received in revised form 25 January 2008 Available online 4 June 2008 Communicated by C. Kassel

In the introduction to [3, Section 2] we define on the algebraic tensor product U ⊗ H a seminorm



2 m m

X

X



Tj ⊗ xj = sup α(STj )xj





S∈Ball(Mn,1 (V )),n∈N j=1 j=1 FU (H)

and we claim that it satisfies the parallelogram identity. The argument given there is not correct, but the assertion is true. Here is a proof: P By [3, Lemma 2.1], there exist partial isometries {Vk , k ∈ I} ⊂ M such that Vk Vk∗ ⊥ Vm Vm∗ for k 6= m and IK0 = k ⊕Vk Vk∗ . ∗ ∗ t Let E = {i1 , . . . , im } be a finite subset of I. Define SE = (Vi1 , . . . , Vim ) and define a linear map θE by

θE : U ⊗ H → Hm : T ⊗ x → α(SE T )(x). By the definition of the seminorm k · kFU (H) we have kθE (ξ)kHm ≤ kξkFU (H) for all ξ ∈ U ⊗ H. Also we define a map

γE : Hm → FU (H) : (x1 , . . . , xm )t →

m X

Vik ⊗ xk .

k=1

This map is a contraction. Indeed if h = (x1 , . . . , xm )t ∈ Hm and  > 0 by [3, Proposition 2.2] there exist r ∈ N and

S ∈ Ball(Mr,1 (M ∗ )) such that

2 m

X

Vik ⊗ xk



2 m

X

−  < α(SVik )(xk )

k=1 k=1 FU (H) E XD X

 = α(Vi∗l S∗ SVik )(xk ), xl α(SVik )(xk ), α(SVil )(xl ) Hr = k ,l

k ,l

D

H

E

2

= α((Vi∗1 , . . . , Vi∗m )t S∗ S(Vi1 , . . . , Vim ))(x1 , . . . , xm )t , (x1 , . . . , xm )t ≤ (x1 , . . . , xm )t . Therefore kγE (h)kFU (H) ∗-homomorphism.

≤ khkHm . In the above proof we used that the restriction of α on the diagonal of A is a

DOI of original article: 10.1016/j.jpaa.2007.07.022. E-mail address: [email protected]. 0022-4049/$ – see front matter doi:10.1016/j.jpaa.2008.03.018

G.K. Eleftherakis / Journal of Pure and Applied Algebra 212 (2008) 2581–2582

2582

Since kVik ⊗ α(Vi∗k T )x − Vik Vi∗k T ⊗ xkFU (H) = 0 for all k we have

γE ◦ θE (T ⊗ x) =

m X k=1

Vik Vi∗k T ⊗ x = SE∗ SE T ⊗ x

in FU (H), for all T ∈ U, x ∈ H. P If ξ = ni=1 Ti ⊗ xi ∈ U ⊗ H, for an arbitrary finite subset E of I we have

n

X

∗ kξkFU (H) ≥ kθE (ξ)kH|E| ≥ kγE (θE (ξ))kFU (H) = SE SE Ti ⊗ xi .

i=1

(1)

FU (H)

If  > 0 there exist m ∈ N and S ∈ Ball(Mm,1 (V )) such that

n

X



kξkFU (H) −  < α(STi )(xi ) −

m 2 i=1 H

1

and there exists a finite subset E0 of I such that



n n

X

 X

α(STi )(xi )

− ≤

α(SSE∗ SE Ti )(xi )



m

2 i=1 i=1

Hm

H

n

X

SE∗ SE Ti ⊗ xi

≤

i=1

FU (H)

for all finite sets E such that I ⊃ E ⊃ E0 . Therefore

n

X

kξkFU (H) −  < SE∗ SE Ti ⊗ xi

i=1

(2)

FU (H)

for all finite sets E such that I ⊃ E ⊃ E0 . From (1) and (2) it follows that

kξkFU (H) = lim kθE (ξ)kH|E| E

for all ξ ∈ U ⊗ H. It is now easy to see that the seminorm k · kFU (H) on U ⊗ H satisfies the parallelogram identity: if ξ, η ∈ U ⊗ H then

kξ + ηk2FU (H) + kξ − ηk2FU (H) = lim(kθE (ξ + η)k2H|E| + kθE (ξ − η)k2H|E| ) E

= lim(2kθE (ξ)k2H|E| + 2kθE (η)k2H|E| ) = 2kξk2FU (H) + 2kηk2FU (H) . E

Note added in proof. After [3] was completed, the author with V.I. Paulsen in [2] developed the notion of the ‘normal module Haagerup tensor product’ of two dual operator modules. Subsequently, Blecher and Kashyap proved in [1] that, in the case where one of the modules comes from a Hilbert space, this tensor product is in fact a Hilbert space. One can show that this tensor product agrees with FU (H) and this would provide an indirect proof that FU (H) is a Hilbert space. In retrospect, use of the normal module Haagerup tensor product would shorten some of the proofs in [3]. Acknowledgment I wish to thank David Blecher for pointing out a gap in the argument leading to the fact that the norm of FU (H) is a Hilbert space norm. References [1] D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, preprint, arxiv:0709.0757. [2] G.K. Eleftherakis, V.I. Paulsen, Stably isomorphic dual operator algebras, Math. Ann. 341 (1) (2008) 99–112. [3] G.K. Eleftherakis, A Morita type equivalence for dual operator algebras, J. Pure Appl. Algebra 5 (2008) 1060–1071.

1 Choose y ∈ Ball(Hm ) such that kωk

DP E Pn α(STi )(xi ). Since limE ni=1 α(SSE∗ SE Ti )(xi ), y = hω, yi there exists E0 such that i=1

* + n

X X  − ≤ α(SSE∗ SE Ti )(xi ), y ≤

α(SSE∗ SE Ti )(xi )

2 i=1 i=1 Hm Hm

kωkHm

for all finite sets E such that I ⊃ E ⊃ E0 .

= hω, yi , where ω =