Morita equivalence of dual operator algebras

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Journal of Pure and Applied Algebra

journal homepage:www.elsevier.com/locate/jpaa

Morita equivalence of dual operator algebras

David P. Blecher

∗

_{, Upasana Kashyap}

Department of Mathematics, University of Houston, Houston, TX 77204-3008, United States

a r t i c l e i n f o

Article history:

Received 5 September 2007

Received in revised form 29 January 2008 Available online 8 May 2008

Communicated by C. Kassel MSC: Primary: 47L30 47L45 46L08 secondary: 16D90 47L25 a b s t r a c t

We consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel’sW∗

-algebraic Morita equivalence.

Published by Elsevier B.V.

1. Introduction and notation

By definition, an operator algebra is a subalgebra ofB

(

H

)

, the bounded linear operators on a Hilbert spaceH, which is closed in the norm topology. It is a dual algebra if it is closed in the weak* topology (also known as the

σ

-weak topology). In [11], the first author, Muhly, and Paulsen generalized Rieffel’s strong Morita equivalence ofC∗_{-algebras [24], to general}

operator algebras. At that time however, we were not clear about how to generalize Rieffel’s variant forW∗_{-algebras [23], to}

dual operator algebras. Recently, two approaches have been suggested for this, in [9,17,18], each of which reflects (different) important aspects of Rieffel’sW∗_{-algebraic Morita equivalence. For example, the notion introduced in [17,18] is equivalent}

to the very important notion of (weak*) ‘stable isomorphism’ [20]. The fact remains, however, that neither approach seems able to treat certain important examples, such as the second dual of a strong Morita equivalence. In the present paper we examine a framework, part of which was suggested at the end of [9], which does include all examples hitherto considered, and which represents a natural setting for the Morita equivalence of dual algebras. It is also one to which all the relevant parts of the earlier theory of strong Morita equivalence (from e.g. [11,10]) transfers in a very clean manner, indeed which may in some sense be summarized as ‘just changing the tensor product involved’ to one appropriate to the weak* topology. Since many of the ideas and proofs are extremely analogous to those from our papers on related topics, principally [11,

1,10] and to a lesser extent [3–5,9], we will be quite brief in many of the proofs. That is, we assume that the reader is a little familiar with these earlier ideas and proof techniques. We will often merely indicate the modifications to weak* topologies. For a more detailed exposition see the second authors Ph. D. thesis [21], along with many other related results.

In Section2, we develop some basic tensor product properties which we shall need. In Section3, we define our variant of Morita equivalence, and present some of its consequences. Section4is centered on the ‘weak linking algebra’, the key tool for dealing with most aspects of Morita equivalence, and in Section5we prove that ifMandNare weak* Morita equivalent dual operator algebras, then the von Neumann algebras generated byMandNare Morita equivalent in Rieffel’sW∗_-algebraic

sense.

∗ _{Corresponding author.}

E-mail addresses:dblecher@math.uh.edu(D.P. Blecher),upasana@math.uh.edu(U. Kashyap). 0022-4049/$ – see front matter Published by Elsevier B.V.

Turning to notation, ifE

,

Fare sets, thenEFwill mean the norm closure of the span of productszwforz∈E

,

w∈F. We reserve the lettersHandKfor Hilbert spaces. We will assume that the reader is familiar with basic notions from operator space theory, as may be found in any of the current texts on that subject, e.g. [15], and the application of this theory to operator algebras, as may be found in e.g. [8]. We study operator algebras from an operator space point of view. Thus an abstract operator algebraAis an operator space and a Banach algebra, for which there exists a Hilbert spaceHand a completely isometric homomorphism

π

:A→B

(

H

)

.

We will often abbreviate ‘weak*’ to ‘w∗

’. A dual operator algebra is an operator algebraMwhich is also a dual operator space. By well known duality principles, anyw∗_{-closed subalgebra ofB}

(

_H

)

_{, is a dual operator algebra. Conversely, it is known}

(see e.g. [8]), that for any dual operator algebraM, there exists a Hilbert spaceHand aw∗_{-continuous completely isometric}

homomorphism

π

:M→B

(

H

)

. In this case, the range

π(

M

)

is aw∗

-closed subalgebra ofB

(

H

)

, which we may identify with

Min every way. In this paperMandNare dual operator algebras which are unital, that is we assume they each possess an identity of norm 1. Nondual operator algebras in this paper, in contrast, will usually be approximately unital, that is, they possess a contractive approximate identity (cai). A normal representation ofMis aw∗-continuous unital completely contractive homomorphism

π

:M→B

(

H

)

.

For cardinals or sets I

,

J, we use the symbolMI,J

(

X

)

for the operator space of I × J matrices overX, whose ‘finite submatrices’ have uniformly bounded norm. We write KI,J

(

X

)

for the norm closure of these ‘finite submatrices’. Then

Cw

J

(

X

)

=MJ,1

(

X

),

RwJ

(

X

)

=M1,J

(

X

)

, andCJ

(

X

)

=KJ,1

(

X

)

andRJ

(

X

)

=K1,J

(

X

)

. We sometimes writeMI

(

X

)

forMI,I

(

X

)

.

A concrete left operator module over an operator algebraA, is a subspaceX⊂B

(

K

,

H

)

such that

π(

A

)

X⊂Xfor a completely contractive representation

π

:A→B

(

H

)

. An abstract operator A-module is an operator spaceXwhich is also anA-module, such thatXis completely isometrically isomorphic, via anA-module map, to a concrete operatorA-module. Similarly for right modules, or bimodules. Most of the interesting modules over operator algebras are operator modules, such as Hilbert

C∗_{-modules, and Hilbert modules (see next paragraph).}

A concrete dual operatorM–N-bimodule is aw∗_{-closed subspaceXofB}

(

_K

,

_H

)

_{such that}

θ(

_M

)

_X

π(

_N

)

_{⊂ X, where}

θ

_and

π

are normal representations ofMandNonHandKrespectively. An abstract dual operatorM–N-bimodule is defined to be a nondegenerate operatorM–N-bimoduleX, which is also a dual operator space, such that the module actions are separately weak* continuous. Such spaces can be represented completely isometrically as concrete dual operator bimodules, and in fact this can be done under even weaker hypotheses (see e.g. [8,9,14]). Similarly for one-sided modules (the caseMorNequals

C). We use standard notation for module mapping spaces, e.g.CB

(

X

,

N

)

N(resp.w∗CB

(

X

,

N

)

N) are the completely bounded (resp. andw∗_{-continuous) rightN-module mapsX→N. Important examples of left dual operator modules overM, are the}

(completely contractive) normal HilbertM-modules. By this, we mean a pair

(

H

, π)

, whereHis a (column) Hilbert space (see e.g. 1.2.23 in [8]), and

π

: M → B

(

H

)

is a normal representation. The module action is expressed through the equation

m·

ζ

=

π(

m

)ζ

. The morphisms between HilbertM-modules are the boundedM-module maps (‘intertwiners’). If M is a dual operator algebra, then the maximalW∗_-coverW∗

max

(

M

)

is a W∗-algebra containingM as a w∗-closed

subalgebra, and which is generated byMas aW∗_{-algebra, and which has the universal property: any normal representation}

π

: M → B

(

H

)

extends uniquely to a (unital) normal∗-representation

π

˜ : W∗

max

(

M

)

→ B

(

H

)

(see [13]). A normal

representation

π

: M → B

(

H

)

of a dual operator algebraM, or the associated spaceHviewed as anM-module, will be called normal universal, if any other normal representation is unitarily equivalent to the restriction of a ‘multiple’ of

π

to a reducing subspace (see [13]).

Lemma 1.1. A normal representation

π

: M → B

(

H

)

of a dual operator algebraMis normal universal iff its extension

π

˜ to

W∗

max

(

M

)

is one-to-one.

Proof. The (⇐) direction is stated in [13]. Thus any faithful normal representation ofW∗

max

(

M

)

restricts to a normal universal

representation

π

whose extension

π

˜ toW_max∗

(

M

)

is one-to-one. It is observed in [13] that any other normal universal representation

θ

is quasiequivalent to

π

. It follows that the extension

θ

˜toW∗

max

(

M

)

is quasiequivalent to

π

˜, and it is easy to

see from this that

_θ

˜_{is one-to-one. }

2. Some tensor products

We begin by recalling the definition of the Haagerup tensor product. SupposeXandYare two operator spaces. Define kuk_nforu∈Mn

(

X⊗Y

)

as:

kuk_n=inf {kakkbk : u=ab

,

a∈Mnp

(

X

),

b∈Mpn

(

Y

),

p∈N}

.

Hereabstands for then×nmatrix whosei

,

j-entry isPp

k=1aik⊗bkj. The algebraic tensor productX⊗Ywith this sequence of matrix norms is an operator space. The completion of this operator space in the above norm is called Haagerup tensor product, and is denoted byX⊗_hY. The completion of an operator space is an operator space, henceX⊗_hYis an operator space.

IfXandYare respectively right and left operatorA-modules, then the module Haagerup tensor productX⊗_hAYis defined to be the quotient ofX⊗_hYby the closure of the subspace spanned by terms of the formxay−xay, forx∈X,y∈Y,a∈A. LetXbe a right andYbe a left operatorA-module whereAis an operator algebra. We say that a bilinear map

ψ

:X×Y→W

linearizes balanced bilinear maps which are completely contractive (or completely bounded) in the sense of Christensen and Sinclair (see e.g. 1.5.4 in [8]).

IfXandYare two operator spaces, then the extended Haagerup tensor productX⊗ehYmay be defined to be the subspace of

(

X∗_⊗

hY∗

)

∗corresponding to the completely bounded bilinear maps fromX∗×Y∗ → Cwhich are separately weak∗ -continuous. IfXandYare dual operator spaces, with predualsX∗andY∗, then this coincides with the weak

∗

Haagerup tensor product defined earlier in [12], and indeedX⊗ehY=

(

X∗⊗hY∗

)

∗. The normal Haagerup tensor productX⊗σhYis the operator

space dual ofX∗⊗ehY∗. The canonical maps are complete isometries

X⊗hY→X⊗ehY→X⊗σhY

.

See [16] for more details.

Lemma 2.1. For any dual operator spacesXandY, Ball

(

X⊗_hY

)

isw∗_{-dense in Ball}

₍

_X⊗σh_Y

₎

_.

Proof. Letx∈Ball

(

X⊗σh_Y

)

_\Ball

(

_X⊗ hY

)

w∗

. By the geometric Hahn-Banach theorem, there exists a

φ

∈

(

X⊗σh_Y

)

∗, andt∈R,

such that Re

φ(

x

) >

t

>

Re

φ(

y

)

for ally∈Ball

(

X⊗_hY

)

. We view

φ

as a mapX⊗_hY→C. It follows that Re

φ(

x

) >

t

>

|

φ(

y

)

| for ally∈Ball

(

X⊗_hY

)

, which implies thatk

φ

k ≤t. Thus|Re

φ(

x

)

| ≤ k

φ

kkxk ≤t, which is a contradiction. 

Lemma 2.2. The normal Haagerup tensor product is associative. That is, ifX,Y,Zare dual operator spaces then

(

X⊗σh_Y

₎

_⊗σh_Z=

X⊗σh

(

_Y⊗σh_Z

)

_{as dual operator spaces.}

Proof. This follows by the definition of the normal Haagerup tensor product and using associativity of the extended

Haagerup tensor product (e.g. see [16]). 

We now turn to the module version of the normal Haagerup tensor product, and review some definitions and facts from [20]. LetX be a right dual operatorM-module andY be a left dual operatorM-module. Let

(

X⊗_hMY

)

∗

σ denote the

subspace of

(

X⊗_hY

)

∗

corresponding to the completely bounded bilinear maps from

ψ

:X×Y →Cwhich are separately weak∗_{-continuous andM-balanced (that is,}

ψ(

_xm

,

_y

)

₌

ψ(

_x

,

_my

)

_{). Define the module normal Haagerup tensor productX⊗}σh

M Y to be the operator space dual of

(

X⊗_hMY

)

∗

σ. Equivalently,X⊗σMhYis the quotient ofX⊗σhYby the weak

∗_{-closure of the}

subspace spanned by terms of the formxm⊗y−x⊗my, forx∈X,y∈Y,m∈M. The module normal Haagerup tensor product linearizes completely contractive, separately weak∗_{-continuous, balanced bilinear maps. That is, the universal property of}

⊗σh

M is: every completely contractive separately weak* continuous mapu :X×Y → Zsuch thatu

(

xm

,

y

)

= u

(

x

,

my

)

for

m ∈ M

,

x ∈ X

,

y ∈ Y, yields a weak* continuous complete contractionu˜ : X⊗σ_MhY → Z, whose composition with the canonical map⊗ : X×Y → X⊗σh

M Y, equalsu(see [20, Proposition 2.2]). The map⊗here is a completely contractive, separately weak∗_{-continuous, balanced bilinear map.}

Lemma 2.3. Let X1

,

X2

,

Y1

,

Y2be dual operator spaces. If u : X1 → Y1 andv : X2 → Y2 arew∗-continuous, completely

bounded, linear maps, then the mapu⊗vextends to a well definedw∗_{-continuous, linear, completely bounded map from}

X₁⊗σh_X

2→Y1⊗σhY2, withku⊗vkcb≤ kukcbkvkcb.

Proof. This follows by considering preduals of the maps, and using the functoriality of the extended Haagerup tensor

product [16]. 

Corollary 2.4. LetNbe a dual algebra, letY1andY2be dual operator spaces which are leftN-modules, and letX1,X2be dual

operator spaces which are rightN-modules. Ifu:X1→X2andv:Y1 →Y2are completely bounded,w∗-continuous,N-module

maps, then the mapu⊗vextends to a well defined linear,w∗_{-continuous, completely bounded map fromX}

1⊗σNhY1→X2⊗σNhY2,

withku⊗vkcb≤ kukcbkvkcb.

Proof. Lemma 2.3gives aw∗_{-continuous, completely bounded, linear mapX}

1⊗σhY1→X2⊗σhY2takingx⊗ytou

(

x

)

⊗v

(

y

)

.

Composing this map with thew∗_{-continuous, quotient mapX}

2⊗σhY2→X2⊗σNhY2, we obtain aw

∗_{-continuous, completely}

bounded mapX1⊗σhY1→X2⊗NσhY2. It is easy to see that the kernel of the last map contains all terms of formxn⊗Ny−x⊗Nny, withn∈N

,

x∈X1

,

y∈Y1. This gives a mapX1⊗σNhY1→X2⊗σNhY2with the required properties. 

Lemma 2.5. If Xis a dual operatorM–N-bimodule and ifYis a dual operatorN–L-bimodule, thenX⊗σh

N Yis a dual operator

M–L-bimodule.

Proof. To show e.g. it is a left dual operatorM-module, use the canonical maps

M⊗_h

(

X⊗σhY

)

→M⊗σh

(

X⊗σhY

)

→

(

M⊗σhX

)

⊗σhY→X⊗σhY

.

Composing the mapM⊗σh

(

_X⊗σh_Y

)

_→X⊗σh_{Yabove with the canonical mapM×}

(

_X⊗σh_Y

)

_→M⊗σh

(

_X⊗σh_Y

)

_{, one sees the} action ofMonX⊗σh_{Yis separately weak* continuous (see also [20]). That}

₍

_a

1a2

)

z=a1

(

a2z

)

forai∈M

,

z∈X⊗σhY, follows from the weak* density ofX⊗Y, and since this relation is true ifzis finite rank. It follows from 3.3.1 in [8], thatX⊗σh_{Yis an} operatorM-module. By 3.8.8 in [8],X⊗σh

N Yis a dual operatorM-module.  There is clearly a canonical mapX⊗_hMY→X⊗σh

Corollary 2.6. For any dual operatorM-modulesXandY, the image of Ball

(

X⊗_hMY

)

isw∗_{-dense in Ball}

(

_X⊗σh M Y

)

.

Proof. Consider the canonicalw∗_{-continuous quotient mapq:X⊗}σh_Y→X⊗σh

M Y. Ifz∈X⊗σ h

M Ywithkzk

<

1, then there existsz0 ∈X⊗σhYwithkz0k

<

1 such thatq

(

z0

)

=z. ByLemma 2.1, there exists a net

(

zt

)

in Ball

(

X⊗hY

)

such thatzt

w∗ →z0. Thenq

(

zt

)

w∗ →q

(

z0

)

_=z. 

Lemma 2.7. For any dual operatorM-modulesXandY, andm

,

n ∈ N, we haveMmn

(

X⊗MσhY

)

∼= Cm

(

X

)

⊗σMhRn

(

Y

)

completely isometrically and weak* homeomorphically. This is also true withm

,

nreplaced by arbitrary cardinals or sets:MIJ

(

X⊗σMhY

)

∼=

Cw

I

(

X

)

⊗σMhRwJ

(

Y

)

.

Proof. We just prove the case thatm

,

n∈N, the other being similar (or can be deduced easily fromProposition 2.9). First we claim thatMmn

(

X⊗σhY

)

∼=Cm

(

X

)

⊗σhRn

(

Y

)

. Using facts from [16] and basic operator space duality, the predual of the latter space is Cm

(

X

)

∗⊗ehRn

(

Y

)

∗ ∼=

(

Rm⊗hX∗

)

⊗eh

(

Y∗⊗hCn

)

∼ =

(

Rm⊗ehX∗

)

⊗eh

(

Y∗⊗ehCn

)

∼ = Rm⊗eh

(

X∗⊗ehY∗

)

⊗ehCn ∼ = Rm⊗h

(

X∗⊗ehY∗

)

⊗hCn ∼ =

(

X∗⊗ehY∗

)

⊗_

(

Mmn

)

∗

.

We have used for example 1.5.14 in [8], 5.15 in [16], and associativity of the extended Haagerup tensor product [16]. Now

(

X∗⊗ehY∗

)

⊗_

(

Mmn

)

∗is the predual ofMmn

(

X⊗σhY

)

, by e.g. 1.6.2 in [8]. This gives the claim. If

θ

is the ensuing completely isometric isomorphismCm

(

X

)

⊗σhRn

(

Y

)

→Mmn

(

X⊗σhY

)

, it is easy to check that

θ

takes[x1x2

. . .

xm]T⊗ [y1y2

. . .

yn]to the matrix[xi⊗yj]. NowCm

(

X

)

⊗σMhRn

(

Y

)

=Cm

(

X

)

⊗σhRn

(

Y

)/

NwhereN= [xt⊗y−x⊗ty]−w

∗

withx∈Cm

(

X

),

y∈Rn

(

Y

),

t∈M. LetN0_{= [xt⊗y−x⊗ty]}−w∗_wherex∈X

,

_y∈Y

,

_{t∈M, then clearly}

θ(

_N

)

_=M

mn

(

N0

)

. Hence

Cm

(

X

)

⊗σhRn

(

Y

)/

N∼=Mmn

(

X⊗σhY

)/θ(

N

)

=Mmn

(

X⊗σhY

)/

Mnm

(

N

0

_),

which in turn equalsMmn

(

X⊗σhY

/

N0

)

=Mmn

(

X⊗σMhY

)

. 

Corollary 2.8. For any dual operatorM-modulesXandY, andm

,

n∈ N, we have that Ball

(

Mmn

(

X⊗hMY

))

isw∗-dense in Ball

(

Mmn

(

X⊗σMhY

))

.

Proof. If

η

∈ Ball

(

Mmn

(

X⊗MσhY

))

, then byLemma 2.7,

η

corresponds to an element

η

0 _{∈ C}

m

(

X

)

⊗σMhRn

(

Y

)

. ByCorollary 2.6, there exists a net

(

ut

)

inCm

(

X

)

⊗hMRn

(

Y

)

such thatut

w∗ →

η

0

. By 3.4.11 in [8],utcorresponds tou0t∈Ball

(

Mmn

(

X⊗hMY

))

such thatu0

t w∗

→

η

. 

Proposition 2.9. The normal module Haagerup tensor product is associative. That is, ifMandNare dual operator algebras, ifXis a right dual operatorM-module, ifYis aM–N-dual operator bimodule, andZis a left dual operatorN-module, then

(

X⊗σh

M Y

)

⊗σNhZ is completely isometrically isomorphic toX⊗σ_Mh

(

Y⊗σ_NhZ

)

.

Proof. We defineX⊗σh

M Y⊗σNhZ to be the quotient ofX⊗σhY⊗σhZ by thew

∗_{-closure of the linear span of terms of the}

formxm⊗y⊗z−x⊗my⊗zandx⊗yn⊗z−x⊗y⊗nz, withx ∈ X

,

y ∈ Y

,

z ∈ Z

,

m ∈ M

,

n ∈ N. By extending the arguments of Proposition 2.2 in [20] to the threefold normal module Haagerup tensor product, one sees thatX⊗σh

M Y⊗σNhZ has the following universal property: ifWis a dual operator space andu:X×Y×Z →Wis a separatelyw∗_-continuous,

completely contractive, balanced, trilinear map, then there exists aw∗

-continuous and completely contractive, linear map ˜

u : X⊗σh

M Y⊗σNhZ → Wsuch thatu˜

(

x⊗My⊗Nz

)

= u

(

x

,

y

,

z

)

. We will prove that

(

X⊗MσhY

)

⊗σNhZhas the above universal property definingX⊗σh

M Y⊗σNhZ. Letu:X×Y×Z →Wbe a separatelyw

∗_{-continuous, completely contractive, balanced,}

trilinear map. For each fixedz ∈ Z, defineuz : X×Y → Wbyuz

(

x

,

y

)

= u

(

x

,

y

,

z

)

. This is a separatelyw∗-continuous, balanced, bilinear map, which is completely bounded. Hence we obtain aw∗_{-continuous completely bounded linear map}

u0 z:X⊗σMhY →Wsuch thatu 0 z

(

x⊗My

)

=uz

(

x

,

y

)

. Defineu0 :

(

X⊗σMhY

)

×Z→Wbyu 0

(

_a

,

_z

)

_{= u}0 z

(

a

)

, fora∈ X⊗σMhY. Then

u0

(

x⊗My

,

z

)

=u

(

x

,

y

,

z

)

, and it is routine to check thatu0is bilinear and balanced overN. We will show thatu0is completely contractive on

(

X⊗_hMY

)

×Z, and then the complete contractivity ofu0_{follows fromCorollary 2.8. Leta∈M}

nm

(

X⊗hMY

)

with kak

<

1 andz∈Mmn

(

Z

)

withkzk

<

1. We want to showku0n

(

a

,

z

)

k

<

1. It is well known that we can writea=xMywhere

x∈ Mnk

(

X

)

andy∈ Mkm

(

Y

)

for somek∈N, withkxk

<

1 andkyk

<

1. Henceku0n

(

a

,

z

)

k = kun

(

x

,

y

,

z

)

k ≤ kxkkykkzk

<

1, provingu0_{is completely contractive. By Proposition 2.2 in [20], we obtain aw}∗_{-continuous, completely contractive, linear}

mapu˜ :

(

X⊗σh

M Y

)

⊗σNhZ → W such thatu˜

((

x⊗My

)

⊗Nz

)

= u0

(

x⊗My

,

z

)

= u

(

x

,

y

,

z

)

. This shows that

(

X⊗σMhY

)

⊗σNhZhas the defining universal property ofX⊗σh

M Y⊗σNhZ. Therefore

(

X⊗MσhY

)

⊗σNhZis completely isometrically isomorphic andw

∗ -homeomorphic toX⊗σh M Y⊗σ h N Z. SimilarlyX⊗σ h M

(

Y⊗σ h N Z

)

=X⊗σ h M Y⊗σ h N Z. 

Lemma 2.10. IfXis a left dual operatorM-module thenM⊗σh

M Xis completely isometrically isomorphic toX.

3. Morita contexts

We now define two variants of Morita equivalence for unital dual operator algebras, the first being seemingly more general than the second. There are many equivalent variants of these definitions, some of which we shall see later.

Throughout this section, we fix a pair of unital dual operator algebras,MandN, and a pair of dual operator bimodulesX

andY;Xwill always be aM–N-bimodule andYwill always be anN–M-bimodule.

Definition 3.1. We say thatMis weak* Morita equivalent toN, with equivalence bimodulesXandY, ifM ∼= X⊗σh N Yas dual operatorM-bimodules (that is, via a completely isometricw∗_{-homeomorphism which is also aM-bimodule map), and}

similarly ifN∼=Y⊗σ_MhXas dual operatorN-bimodules. We call

(

M

,

N

,

X

,

Y

)

a weak* Morita context in this case. For the next definition, we suppose that we have separately weak∗

-continuous completely contractive bilinear maps

(

·

,

·

)

:X×Y→M, and[·

,

·] :Y×X→N, and we will work with the 6-tuple, or context,

(

M

,

N

,

X

,

Y

, (

·

,

·

),

[·

,

·]

)

.

Definition 3.2. We say thatMis weakly Morita equivalent toN, if there exists a 6-tuple as above, and existw∗_{-dense operator}

algebrasAandBinMandNrespectively, and there exists aw∗_{-dense operatorA–B-submoduleX}0_{inX, and aw}∗_-denseB–A

-submoduleY0

inY, such that the ‘subcontext’

(

A

,

B

,

X0

_,

_Y0

₎

, together with restrictions of the pairings

(

·

,

·

)

and[·

,

·], is a (strong) Morita context in the sense of [11, Definition 3.1]. In this case, we call

(

M

,

N

,

X

,

Y

)

(or more properly the 6-tuple above the definition), a weak Morita context.

Remark Some authors use the term ‘weak Morita equivalence’ for a quite different notion, namely to mean that the

algebras have equivalent categories of Hilbert space representations.

Weak Morita equivalence, as we have just defined it, is really nothing more than the ‘weak∗_{-closure of’ a strong Morita}

equivalence in the sense of [11]. This definition includes all examples that have hitherto been considered in the literature: Examples:

(1) We shall see inCorollary 3.4that every weak Morita equivalence is an example of weak* Morita equivalence. (2) We shall see in Section4that every weak Morita equivalence arises as follows: LetA

,

Bbe subalgebras ofB

(

H

)

andB

(

K

)

respectively, for Hilbert spacesH

,

K, and letX⊂ B

(

K

,

H

),

Y⊂ B

(

H

,

K

)

, such that the associated subsetLofB

(

H⊕K

)

withA

,

B

,

X

,

Yas ‘corners’, is a subalgebra ofB

(

H⊕K

)

, for Hilbert spacesH

,

K. This is the same as specifying a list of obvious algebraic conditions, such asXY⊂A. Assume in addition thatApossesses a cai

(

et

)

with terms of the formxy, forx∈Ball

(

Rn

(

X

))

andy∈Ball

(

Cn

(

Y

))

, andBpossessing a cai with terms of a similar formyx(dictated by symmetry). Taking the weak* (that is,

σ

-weak) closure of all these spaces clearly yields a weak Morita equivalence ofAw∗andBw∗. (3) Every weak* Morita equivalence arises similarly to the setting in (2). The main difference is thatA,Bare unital, and

(

et

)

is not a cai, butet→1Aweak*, and similarly for the net inB.

(4) Von Neumann algebras which are Morita equivalent in Rieffel’sW∗_{-algebraic sense from [23], are clearly weakly Morita}

equivalent. We state this in the language of TROs. We recall that a TRO is a subspaceZ⊂B

(

K

,

H

)

withZZ∗_{Z⊂Z. Rieffel’s}

W∗-algebraic Morita equivalence ofW∗-algebrasMandNis essentially the same (see e.g. [8, Section 8.5] for more details) as having a weak* closed TRO (that is, a WTRO)Z, withZZ∗_{weak* dense inMandZ}∗_{Zweak* dense inN. Recall}

thatZ∗_{Zdenotes the norm closure of the span of productsz}∗_wforz

,

_{w ∈ Z. Here}

(

_ZZ∗

,

_Z∗_Z

,

_Z

,

_Z∗

)

_{is the weak* dense}

subcontext.

(5) More generally, the ‘tight Moritaw∗_{-equivalence’ of [9, Section 5], is easily seen to be a special case of weak Morita}

equivalence. In this case, the equivalence bimodulesXandYare ‘selfdual’. Indeed, this selfduality is the reason for the approach taken in [9, Section 5].

(6) The second duals of strongly Morita equivalent operator algebras are weakly Morita equivalent. Recall that ifAandB

are approximately unital operator algebras, thenA∗∗

andB∗∗

are unital dual operator algebras, by 2.5.6 in [8]. IfXis a non-degenerate operatorA–B-bimodule, thenX∗∗_{is a dual operatorA}∗∗_–B∗∗_{-bimodule in a canonical way. Let}

₍

_·

_,

_·

₎

_{be a}

bilinear map fromX×YtoAthat is balanced overBand is anA-bimodule map. Then notice that by 1.6.7 in [8], there is a unique separatelyw∗_{-continuous extension fromX}∗∗_×Y∗∗_toA∗∗_{, which we still call}

(

_·

,

_·

)

_{. Now the weak Morita}

equivalence follows easily from the Goldstine lemma.

(7) Any unital dual operator algebraMis weakly Morita equivalent toMI

(

M

)

, for any cardinalI. The weak* dense strong Morita subcontext in this case is

(

M

,

KI

(

M

),

RI

(

M

),

CI

(

M

))

, whereas the equivalence bimodulesXandYabove areRwI

(

M

)

andCw

I

(

M

)

respectively.

(8) TRO equivalent dual operator algebrasMandN, or more generally∆-equivalent algebras, in the sense of [17,18], are weakly Morita equivalent. IfM⊂B

(

H

)

andN⊂B

(

K

)

, then TRO equivalence means that there exists a TROZ⊂B

(

H

,

K

)

such thatM= [Z∗_NZ]w∗_{andN= [ZMZ}∗_]w∗

.

_{Eleftherakis shows that one may assume thatZis a WTRO and 1}

Nz=z1M=z forz∈Z. DefineXandYto be the weak* closures ofMZ∗NandNZMrespectively. DefineAandBto be, respectively,Z∗NZ

andZMZ∗_{. DefineX}0_andY0_{to be, respectively, the norm closures ofZ}∗_YZ∗_{andZXZ. SinceZis a TRO,Z}∗_{Zis aC}∗_-algebra,

and so it has a contractive approximate identity

(

et

)

whereet = Pnk(=t)1xktytkfor someytk ∈ Z, andxtk =

(

ytk

)

∗_{. It is easy}

to check that

(

et

)

is a cai forA, and a similar statement holds forB. Indeed it is clear that

(

A

,

B

,

X0

,

Y0

)

is a weak∗-dense strong Morita subcontext of

(

M

,

N

,

X

,

Y

)

. HenceMandNare weakly Morita equivalent. We remark that it is proved in [20] that, in our language,MandNare weak* Morita equivalent.

(9) Examples of weak and weak* Morita equivalence may also be easily built as at the end of [6, Section 6], from a weak* closed subalgebraAof a von Neumann algebraM, and a strictly positivef ∈M+satisfying a certain ‘approximation in

modulus’ condition. Then the weak linking algebra of such an example is Morita equivalent in the same sense toA(see Section4), but they are probably not always weak* stably isomorphic.

(10) A beautiful example from [19] (formerly part of [17]): two ‘similar’ separably acting nest algebras are clearly weakly Morita equivalent by the facts presented around [19, Theorem 3.5] (Davidson’s similarity theorem), indeed in this case the ‘subcontext’ (seeDefinition 3.2) equals the ‘context’, and the algebras are even strongly Morita equivalent. However, Eleftherakis shows they need not be ‘∆-equivalent’ (that is, weak* stably isomorphic [20]).

In the theory of strong Morita equivalence, and also in our paper, it is very important thatNhas some kind of ‘approximate identity’

(

fs

)

of the form

fs= ns X i=1 [ys_i

,

xs_i]

,

k[ys₁

, . . . ,

ys_n s]kk[x s 1

, . . . ,

x s ns] T_k

_<

₁

_,

_(3.1)

and similarly thatMhas some kind of ‘approximate identity’

(

et

)

of form

et= mt X i=1

(

xt_i

,

yt_i

),

k[xt₁

, . . . ,

xt_m t]kk[y t 1

, . . . ,

y t mt] T_k

_<

₁

_.

_(3.2) Herexs i

,

x t i ∈X

,

y s i

,

y t i∈Y. Indeed, by [11],x s i

,

x t i

,

y s i

,

y t imay be chosen inX 0 andY0

in the case of weak Morita equivalence. In what follows, we say, for example, that

(

·

,

·

)

is a bimodule map ifm

(

x

,

y

)

=

(

mx

,

y

)

and

(

x

,

y

)

m =

(

x

,

ym

)

for all

x∈X

,

y∈Y

,

m∈M.

Theorem 3.3.

(

M

,

N

,

X

,

Y

)

is a weak* Morita context iff the following conditions hold: there exists a separately weak∗_-continuous

completely contractiveM-bimodule map

(

·

,

·

)

: X×Y → Mwhich is balanced over N, and a separately weak∗_-continuous

completely contractive N-bimodule map[·

,

·] : Y ×X → N which is balanced overM, such that

(

x

,

y

)

x0 _{= x[y}

_,

_x0_{] and}

y0

(

_x

,

_y

)

_{= [y}0

,

_x]yforx

,

_x0 _{∈ X}

,

_y

,

_y0 _{∈ Y; and also there exist nets}

(

_f

s

)

inNand

(

et

)

inMof the form in(3.1)and(3.2)above, withfs→1Nandet→1Mweak*.

Proof. (⇐) Under these conditions, we first claim that if

π

:X⊗σh

N Y→Mis the canonical (w

∗_{-continuous)M–M-bimodule}

map induced by

(

·

,

·

)

, then

π(

u

)

x⊗Ny=u

(

x

,

y

)

for allx∈X

,

y∈Y, andu∈X⊗σNhY. To see this, fixx⊗Ny∈X⊗σNhY. Define

f

,

g:X⊗σh

N Y→X⊗σ h

N Y:f

(

u

)

=u

(

x

,

y

)

andg

(

u

)

=

π(

u

)

x⊗Nywhereu∈X⊗NσhY. We need to show thatf =g. SinceX⊗hNY isw∗_{-dense inX⊗}σh

N Y, andf

,

garew

∗_{-continuous, it is enough to check thatf =gonX⊗}

hNY. Foru=x0⊗Ny0, we have

u

(

x

,

y

)

=x0_⊗

Ny0

(

x

,

y

)

=x0⊗N[y0

,

x]y=x0[y0

,

x] ⊗Ny=

(

x0

,

y0

)

x⊗Ny=

π(

u

)

x⊗Ny

,

as desired in the claim.

To see thatM∼=X⊗σh

N Y, we shall show that

π

above is a complete isometry. SinceMis the weak* closure of the span of the range of

(

·

,

·

)

, it will follow from the Krein-Smulian theorem that

π

maps ontoM. Choose an approximate identity

(

et

)

forMof the form in(3.2). Define

ρ

t : M →X⊗σNhY:

ρ

t

(

m

)

=

Pnt

i=1mxti⊗Nyti. For[ujk] ∈ Mn

(

X⊗σNhY

)

, we have by the last paragraph that

ρ

t◦

π(

[ujk]

)

= "_n t X i=1

π(

ujk

)

xti⊗Nyti # = "_n t X i=1 ujk

(

xti

,

y t i

)

# = [ujket] w∗ →[ujk]

,

the convergence by [20, Lemma 2.3]. Since

ρ

tis completely contractive, we have

k[ujket]k = k

(ρ

t◦

π)(

[ujk]

)

k ≤ k

π(

[ujk]

)

k

.

As[ujk]is thew∗-limit of the net

(

[ujket]

)

t, by Alaoglu’s theorem we deduce thatk[ujk]k ≤ k

π(

[ujk]

)

k. Similarly,N∼=Y⊗σMhX. (⇒) The existence of the nets

(

fs

)

and

(

et

)

follows fromCorollary 2.6. Define

(

·

,

·

)

to be the composition of the canonical mapX×Y→X⊗σh

N Ywith the isomorphism of the latter space withM. Similarly one obtains[·

,

·], and these maps have all the desired properties except the relations

(

x

,

y

)

x0 _{= x[y}

,

_x0_]andy0

(

_x

,

_y

)

_{= [y}0

,

_{x]y. To obtain these we have to adjust}

(

_·

,

_·

)

by multiplying it by a certain unitary inM, as in the proof of [5, Proposition 1.3]. Indeed that proof transfers easily to our present setting, and in fact becomes slightly simpler, since in the latter proof the map calledTis weak* continuous in our case, andw∗_CB

M

(

M

)

∼=M. 

Corollary 3.4. Every weak Morita context is a weak* Morita context.

Proof. Let

(

M

,

N

,

X

,

Y

, (

·

,

·

),

[·

,

·]

)

be a weak Morita context with strong Morita subcontext

(

A

,

B

,

X0

_,

_Y0

₎

_{. If}

₍

_f

s

)

is a cai forBit is clear thatfs→1Nweak*. Indeed if a subnetfs_α→fin the weak∗-topology inN, thenbf=bfor allb∈B. By weak∗-density it follows thatbf = bfor allb ∈ N. Similarlyf b = b. Thusf = 1N. By Lemma 2.9 in [11] we may choose

(

fs

)

of the form

(3.1), and similarlyAhas a cai

(

et

)

of form in(3.2). That

(

x

,

y

)

x0 =x[y

,

x0]andy0

(

x

,

y

)

= [y0

,

x]yforx

,

x0∈X

,

y

,

y0∈Y, follows by weak* density, and from the fact that the analogous relations hold inX0_andY0_{. Similarly one sees that}

(

_·

,

_·

)

_and[·

,

_·]are

A key point for us, is that the condition involving(3.1)inTheorem 3.3becomes a powerful tool when expressed in terms of an ‘asymptotic factorization’ ofIY (resp.IY0) through spaces of the formCn

(

M

)

(resp.Cn

(

A

)

in the case of a weak Morita equivalence). Indeed, define

ϕ

s

(

y

)

to be the column[

(

xsj

,

y

)

]jinCns

(

M

)

, foryinY, and define

ψ

s

(

[aj]

)

=

P

jysjajfor[aj]in

Cns

(

M

)

. Then

ψ

s

(ϕ

s

(

y

))

=fsy→yweak* ify∈Y(or in norm ify∈Y0, in the case of a weak Morita equivalence, in which case we can replaceCns

(

M

)

byCns

(

A

)

). Similarly,(3.1)may be expressed in terms of an ‘asymptotic factorization’ ofIXthrough spaces of the formRn

(

M

)

(orIX0 throughRn

(

A

)

in the ‘weak Morita’ case), Similarly, the condition involving(3.2)may be expressed in terms of an ‘asymptotic factorization’ ofIYthrough spaces of the formRn

(

N

)

(orIY0throughRn

(

B

)

in the ‘weak Morita’ case), or ofIXthroughCn

(

N

)

.

Henceforth in this section, let

(

M

,

N

,

X

,

Y

, (

·

,

·

),

[·

,

·]

)

be as inTheorem 3.3. We will also refer to this 6-tuple as the weak* Morita context.

Theorem 3.5. Weak* Morita equivalent dual operator algebras have equivalent categories of dual operator modules.

Proof. Write MR for the category of left dual operator modules over M. The morphisms here are the w∗-continuous completely boundedM-module maps. IfZ ∈

NRand ifF

(

Z

)

= X⊗σ h

N Z, thenF

(

Z

)

is a left dual operatorM-module by

Lemma 2.5. That is,F

(

Z

)

∈_MR. Further, ifT∈ w∗_CB

N

(

Z

,

W

)

, forZ

,

W ∈NR, and ifF

(

T

)

is defined to beI⊗NT :F

(

Z

)

→ F

(

W

)

, then by the functoriality of the normal module Haagerup tensor product we haveF

(

T

)

∈ w∗_CB

M

(

F

(

Z

),

F

(

W

))

, and k_F

(

T

)

k_cb≤ kTk_cb. ThusF is a contractive functor fromNRtoMR. Similarly, we obtain a contractive functorGfromMRto NR. Namely,G

(

W

)

=Y⊗σh

M W, forW ∈ MR, andG

(

T

)

= I⊗MTforT ∈ w∗CBM

(

W

,

Z

)

withW

,

Z ∈ MR. Similarly, it is easy to check that these functors are completely contractive; for example,T 7→F

(

T

)

is a completely contractive map on each spacew∗_CB

N

(

Z

,

W

)

of morphisms. It is more complicated to show that the mapT→F

(

T

)

is weak* continuous: a proof of this emerged during a conversation with Jon Kraus; we will present this elsewhere. If we composeF andG, we find that for

Z∈_NRwe haveG

(

F

(

Z

))

∈_NR. ByProposition 2.9andLemma 2.10, we have G

(

F

(

Z

))

∼=Y⊗_Mσh

(

X⊗σ_NhZ

)

∼=

(

Y⊗σ_MhX

)

⊗σ_NhZ∼=N⊗σ_NhZ∼=Z

.

where the isomorphisms are completely isometric. The rest of the proof follows as in Theorem 3.9 in [11]. 

Remark We imagine that the ideas of [5] show that the converse of the last theorem is true, and hope to pursue this in the future.

We shall adopt the convention from algebra of writing maps on the side opposite the one on which ring acts on the module. For example a leftA-module map onXwill be written on the right and a rightA-module map will be written on the left. The pairings and actions arising in the weak* Morita context give rise to eight maps:

RN:N→CBM

(

X

,

X

),

xRN

(

b

)

=x·b LN:N→CB

(

Y

,

Y

)

M

,

LN

(

b

)

y=b·y RM:M→CBN

(

Y

,

Y

),

yRM

(

a

)

=y·a LM:M→CB

(

X

,

X

)

N

,

LM

(

a

)

x=a·x RM_:Y→CB M

(

X

,

M

),

xRM

(

y

)

=

(

x

,

y

)

LN_:Y→CB

₍

_X

_,

_N

₎

N

,

LN

(

y

)

x= [y

,

x] RN:X→CBN

(

Y

,

N

),

yRM

(

x

)

= [y

,

x] LM:X→CB

(

Y

,

M

)

M

,

LM

(

x

)

y=

(

x

,

y

).

The first four maps are completely contractive since module actions are completely contractive. Also the mapsLNand

LMare homomorphisms andRNandRMare anti-homomorphisms. Similar proofs to the analogous results in [11] show that

RM

,

LN,RN, andLMare completely contractive.

Theorem 3.6. If

(

M

,

N

,

X

,

Y

, (

·

,

·

),

[·

,

·]

)

is a weak* Morita context, then each of the mapsRM_,RN_,LM_andLN_{is a weak* continuous} complete isometry. The range ofRM_isw∗_CB

M

(

X

,

M

)

, with similar assertions holding forRN,LMandLN. The mapLN(resp.RN) is a

w∗-continuous completely isometric isomorphism (resp. anti-isomorphism) onto thew∗-closed left (resp. right) idealw∗CB

(

Y

)

M (resp. w∗_CB

M

(

X

)

). Similar results hold forLMandRM.

Proof. Most of this can be proved directly, as in [11, Theorem 4.1]. Instead we will deduce it from the functoriality (Theorem 3.5). For example, because of the equivalence of categories via the (completely contractive) functorF =Y⊗σh

M −, we have completely isometrically:

M∼= w∗CBM

(

M

)

∼= w∗CBN

(

F

(

M

))

∼= w∗CBN

(

Y

),

and the composition of these maps is easily seen to beRM. ThusRMis a complete isometry. Similar proofs work for the other seven maps. To see thatLNisw∗-continuous, for example, let

(

bt

)

be a bounded net inNconverging in thew∗-topology of

Ntob∈N. ThenLN

(

bt

)

is a bounded net inCB

(

Y

)

M. As the module action is separatelyw∗-continuous, it is easy to see that

LN

(

bt

)

converges toLN

(

b

)

in thew∗-topology. ThusLNis aw∗-continuous isometry withw∗-closed range, by the Krein-Smulian theorem. To see that its range is a left ideal simply use the weak∗_{-density of the span of terms[y}

,

_{x]inN, and the equation}

Remarks (1) Note that in the case of weak Morita equivalence,CBA

(

X0

)

is an operator algebra ([11], Theorem 4.9). It is not true in general thatCBM

(

X

)

is an operator algebra, as we show in [21]. Nonetheless, the above shows thatw∗CBM

(

X

)

is a dual operator algebra (∼=N).

(2) In factw∗_CB

₍

_Y

₎

M(resp.w∗CBM

(

X

)

) equals the one-sided operator space multiplier algebra (see [8, Chapter 4])M`

(

Y

)

(resp.Mr

(

X

)

). See [7].

Theorem 3.7. If MandNare weak* Morita equivalent dual operator algebras, then their centers are completely isometrically isomorphic via aw∗_{-homeomorphism.}

Proof. ByTheorem 3.6there is aw∗

-continuous complete isometryRM:M→w∗CBN

(

Y

)

. The restriction ofRMtoZ

(

M

)

maps intow∗_CB

(

_Y

)

M∼=N, and so we have defined aw∗-continuous completely isometric homomorphism

θ

:Z

(

M

)

→N. One easily sees that

θ(

a

)(

y

)

=ya, fora∈Z

(

M

)

. It is also easy to see that this implies that

θ

maps intoZ

(

N

)

, and to argue, by symmetry, that

θ

must be an isomorphism. 

Lemma 3.8. In the case of weak Morita equivalence, ifZis a left dual operatorM-module, then the canonical map fromY0_⊗

hAZ intoY⊗σh

M Zis completely isometric, and it maps the ball onto aw

∗_{-dense set in Ball}

(

_Y⊗σh M Z

)

.

Proof. The canonical map here is completely contractive, let us call it

θ

. On the other hand, let

(

fs

)

be as in(3.1), and let

ϕ

s

, ψ

sbe as defined just belowCorollary 3.4, with

ψ

s

(ϕ

s

(

y

))

=fsy→y. Then foru∈Mn

(

Y0⊗AZ

)

, we have k

θ

_n

(

u

)

k ≥ k

(ϕ

_s⊗I

)

n

(θ

n

(

u

))

k = k

(ϕ

s⊗I

)

n

(

u

)

k ≥ kfsuk

.

Taking a limit overs, givesk

θ

_n

(

u

)

k ≥ kuk. Letu ∈ Ball

(

Y⊗σh

M Z

)

. ByCorollary 2.6, there exists a net

(

ut

)

in the image of Ball

(

Y⊗hMZ

)

such thatut w∗

→u. We may assume that eachutis of the formwz, forw∈Ball

(

Rn

(

Y

)),

z∈Ball

(

Cn

(

Z

))

. We rewrite(3.1)and the lines below it, namely write eachfsin the form[y

,

x](in suggestive notation), fory∈Ball

(

Rm

(

Y0

))

andx∈Ball

(

Cm

(

X0

))

. Notewzis the weak* limit of termsfswz, andfswz=yv, wherevis a column withkth entry

P

j[xk

,

wj]zj. It is easy to check thatkvk ≤1. 

Proposition 3.9. Weak* Morita equivalence is an equivalence relation.

Proof. This follows the usual lines, for example the transitivity follows from associativity of the tensor products and

Lemma 2.10. 

Remark Concerning transitivity of weak Morita equivalence, it is convenient (and probably necessary) to consider

Definition 3.2as defining an equivalence between pairs

(

M

,

A

)

and

(

N

,

B

)

, as opposed to just betweenMandN. That is we also consider the weak∗_{-dense operator subalgebras. Then it is fairly routine to see that weak Morita equivalence is an}

equivalence relation [21].

Theorem 3.10. Weak* Morita equivalent dual operator algebras have equivalent categories of normal Hilbert modules. Moreover, the equivalence preserves the subcategory of modules corresponding to completely isometric normal representations.

Proof. IfHis a normal HilbertM-module, letK = Y⊗σh M H

c_{. By the discussion just belowCorollary 3.4, combined with}

Corollary 2.4, there are nets of maps

ϕ

s:K→Cns

(

M

)

⊗σ

h M Hc∼=Cns

(

H c

)

_{, and maps}

ψ

s:Cns

(

H c

)

_{→K, with}

ψ

s

(ϕ

s

(

z

))

=fsz→z weak* for allz∈K. Here

(

fs

)

is as in(3.1). LetΛbe the directed set indexings, and letUbe an ultrafilter with the property that lim_U zs=limΛzsfor scalarszs, whenever the latter limit exists. LetHUbe the ultraproduct of the spacesCns

(

H

c

₎

_{, which} is a column Hilbert space, as is well known and easy to see. DefineT:K→HUbyT

(

x

)

=

(ϕ

s

(

x

))

s, forx∈K. This is a complete contraction. To see that it is an isometry, note that for anyx∈K

, ρ

∈Ball

(

K∗

)

, we have

|

ρ(

x

)

| =lim

U |

ρ(ψ

s

(ϕ

s

(

x

)))

| ≤limU k

ϕ

s

(

x

)

k = kT

(

x

)

k

.

Similarly,Tis a complete isometry, as we leave to the reader to check. ThusKis a (column) Hilbert space. ThatK = Y⊗σh

M Hc is a normal HilbertN-module now follows fromLemma 2.5. Finally, suppose that Mis a weak* closed subalgebra ofB

(

H

)

, we will show that the induced representation

ρ

ofNonKis completely isometric. Certainly this map is completely contractive. Let[bpq] ∈Md

(

N

)

,[ykl] ∈Ball

(

Mm

(

Y

)),

[

ζ

rs] ∈Ball

(

Mg

(

Hc

)),

[xij] ∈Ball

(

Mn

(

X

))

. We have k[ykl⊗

ζ

rs]k ≤1 with respect to the operator space projective tensor product matrix norm, and hence also with respect to ⊗σh

M. Thus,

k[

ρ(

bpq

)

]k ≥ k[bpqykl⊗

ζ

rs]k ≥ k[

(

xij

,

bpqykl

)ζ

rs]k

.

Taking the supremum over all such[

ζ

_rs], gives

k[

ρ(

bpq

)

]k ≥sup{k[

(

xij

,

bpqykl

)

]k : [xij] ∈Ball

(

Mn

(

X

))

} = k[bpqykl]k

,

by Theorem 3.6. Taking the supremum over all such[ykl] ∈ Ball

(

Mm

(

Y

))

gives k[

ρ(

bpq

)

]k ≥ k[bpq]k, by Theorem 3.6 again.