Relatively mixing bimodules

2.5 Mixing operator bimodules

2.5.2 Relatively mixing bimodules

Given a dual normal operatorM-N-bimoduleW ⊂B(H)we say that a vectorw∈W is mixing relative toX×Y(or just mixing if X=K(L²M)andY=K(L²N)) if the mapM⊗_ConN^op3µ7→µ(w)∈W is continuous from theX-Y-topology to the weak^∗-topology. We note that this is equivalent to the mapM⊗_Con N^op3µ7→µ(w)being continuous on uniformly bounded subsets. We letW_X-Y−mixdenote the set of vectors that are mixing relative toX×Y(we denote this space byWmix in the case whenX=K(L²M)andY= K(L²N)). It is easy to see that this is a norm closed subspace ofW. We say thatW is mixing relative to X×YifWX-Y−mix=W.

The bimoduleWmixneed not be weak^∗-closed in general. However, it will always be closed in theM-N- topology, so that it is a strong operatorM-N-bimodule.

Proposition 2.5.1. Let M and N be a finite von Neumann algebras, and X⊂B(L²M)and Y⊂B(L²N) boundary pieces for M and N respectively. SupposeW ⊂B(H)is a dual normal operator M-N-bimodule, thenW_X-Y−mixis a strong operator M-N-bimodule.

Proof. First note that sinceXB(L²N,L²M)Yis an(M(X)∩M)-(M(Y)∩N)bimodule it follows thatW_X-Y−mix

is also an(M(X)∩M)-(M(Y)∩N)bimodule.

We now fix x∈M andv∈W_X-Y−mix, then by Kaplansky’s Density Theorem there exists a uniformly bounded net{x_i}_i⊂M(X)∩Msuch thatkx−x_ik₂→0.

If{µ_j}_j⊂M⊗_ConN^opis uniformly bounded such thatµ_j→0 in theX-Y-topology, then by Lemma 2.4.2, for eachϕ∈W∗we have

i→∞limsup

j

|ϕ(µ_j((x−x_i)v)|=0.

It then follows that limi→∞ϕ(µ_j(xv)) =0, and since the net{µ_j}_j⊂M⊗_ConN^opwas arbitrary we then have thatxv∈WX-Y−mix. ThusWX-Y−mixis a leftM-module, and a similar argument shows that it is also a right N-module.

We now fixv∈W and suppose{p_i}_i⊂Mis an increasing sequence of projections which converge to 1 and such thatp_iv∈W_X-Y−mixfor alli. Letµj∈M⊗_ConN^opbe uniformly bounded such thatµj→0 in the X-Y-topology. Another application of Lemma 2.4.2 shows that for eachϕ∈W∗we have

i→∞limsup

j

|ϕ(µ_j(p^⊥_i v))|=0,

and it then follows just as above that limj→∞ϕ(µ_j(v)) =0, so thatv∈WX−mix. By Theorem 2.1 in [Mag98]

it then follows thatWX-Y−mix is a strong leftM-module. The fact thatWX-Y−mixis a strong rightN-module follows similarly.

Corollary 2.5.2. Under the hypotheses of the previous lemma, multiplication in M⊗_ConN^op is separately continuous in theX-Y-topology.

Proof. If µ∈M⊗_ConN^op and {µ_i}_i ⊂M⊗_ConN^op is uniformly bounded such that µi→0 in theX-Y- topology, then asB(L²N,L²M)3S7→µ(S)is a normal map it then follows thatµ(µ_i(T))→0 ultraweakly for allT∈XB(L²N,L²M)Y, and hence,µ·µ_i→0 in theX-Y-topology.

We also have thatµ_i(T)→0 ultraweakly for eachT∈B(L²N,L²M)_X-Y−mix, and by the previous propo- sition B(L²M)_X−mix is anM⊗_ConM^op-module, which then shows that µ_i(µ(T))→0 ultraweakly for all T ∈XB(L²N,L²M)Y, henceµi·µ→0 in theX-Y-topology.

We recall thatB(L²N,L²M)^M]N denotes the space of bounded linear functionalsϕ∈B(L²N,L²M)^∗such that for eachT∈B(L²N,L²M)the mapM×M3(a,b)7→ϕ(aT b)is separately normal in each variable. We let

B(L²N,L²M)^]_J=B(L²N,L²M)^M]N∩B(L²N,L²M)^M0]N0.

Note that by Theorem 4.2 in [Mag98] ifA is a unitalC^∗-algebra withM⊂A, then we haveϕ∈A^]if and only if the mapsL_ϕ:M→A^∗,R_ϕ:M→A^∗defined byL_ϕ(x)(T) =ϕ(x^∗T), andR_ϕ(x)(T) =ϕ(T x)are

continuous from the ultrastrong topology onMto the norm topology onA^∗. In particular, it follows that if ϕ∈A^], then for eachT∈A the mapM²3(x,y)7→ϕ(x^∗Ty)is jointly strong operator topology continuous on bounded sets.

Lemma 2.5.3. SupposeA is a unital normal M-C^∗-algebra. Ifϕ∈A^]is Hermitian with Jordan decompo- sitionϕ=ϕ+−ϕ−, thenϕ+,ϕ−∈A^].

Proof. Suppose first thatϕ∈A^]. Takex_i∈M₊increasing withx_i→1 strongly. For all T ∈A we then haveϕ(x^1/2_i T x^1/2_i )→ϕ(T). Take arbitrary subnets such thatψ+, andψ− are weak^∗limits ofA 3T 7→

ϕ+(x^1/2_i T x^1/2_i )andA 3T 7→ϕ−(x^1/2_i T x^1/2_i )respectively. Thenϕ=ψ+−ψ−andkψ+k+kψ−k ≤ kϕ+k+ kϕ₋k=kϕkso that we haveψ+=ϕ+andψ−=ϕ−by uniqueness of Jordan decomposition. As the subnets were arbitrary we then haveϕ+(x_i)→ϕ+(1), andϕ−(x_i)→ϕ−(1). It therefore follows thatϕ+|M,ϕ_−|M∈M∗

and henceϕ+,ϕ−∈A^].

Recall that we may view K(L²(M,τ))^\ as a subspace ofB(L²(M,τ))^\. Indeed, by the discussion in Section 2.4.1, we have natural operator space isomorphisms

K(L²(M,τ))^\∼=CBM−M(K(L²(M,τ)),B(L²(M,τ)))∼=CB^σ_M−M(B(L²(M,τ)))

andB(L²(M,τ))^\∼=CBM−M(B(L²(M,τ))). In the saem way, we may viewK(L²(M,τ))^\_J as a subspace of B(L²(M,τ))^\_J.

Lemma 2.5.4. Let(M,τ)be a tracial von Neumann algebra. Given a stateϕ∈B(L²(M,τ))^\(resp.B(L²(M,τ))^\_J) there exists a net of statesµ_i∈K(L²(M,τ))^\(resp.K(L²(M,τ))^\_J) such thatµ_i→ϕweak^∗andkµ_ik_\≤2kϕk_\ (resp.kµ_ik_Con≤4kϕk_Con).

Proof. Given a stateϕ∈B(L²(M,τ))^\, we may find a net of positive functionalsµ_i⁰∈K(L²M)^∗converging weak^∗ toϕ, withkµ_i⁰k ≤ kϕk. Sinceϕ|M is normal we have that µi|M →ϕ|M in the weak topology in M∗and hence by taking convex combinations we may assume thatkµ_i|M−ϕ|Mk →0. Ifa_i∈M₊is such that µi(x) =τ(xa_i)for x∈M, then setting p_i=1_[0,2kϕk\](a_i)and ˜µi(·) =µi(p_i·p_i)/τ(p_ia_i)we have that kµ˜_ik_\≤2kϕk_\. Moreover, sincekµ_i|M−ϕ_|Mk →0 we haveµ_i(p_i)→1 so that the Cauchy-Schwarz Inequality giveskµ_i−µ˜_ik →0. Hence, ˜µ_i→ϕin the weak^∗-topology.

In the case whenϕ∈B(L²(M,τ))^\_Jwe may also assume thatkµ_i|JMJ−ϕ_|JMJk →0 and repeat the above argument by considering ˜µi|JMJto construct ˜˜µi∈K(L²(M,τ))^\_Jwithkµ˜˜_ik_Con≤4kϕk_Con.

Lemma 2.5.5. Let M and N be tracial von Neumann algebras. Ifϕ∈B(L²N,L²M)^]_J, then there is a uniformly bounded net{µ_i}_i⊂K(L²N,L²M)^\_Jsuch thatϕis the weak^∗limit of{µ_i}_i.

Proof. We first consider the case whenM=N. By considering the real and imaginary parts separately, it suffices to consider the case whenϕis Hermitian. Moreover, by Lemma 2.5.3 it then suffices to consider the case whenϕis positive. So supposeϕ∈B(L²M)^]_Jis a positive linear functional. For eachε>0 there exist K>0, and a projectionsp∈P(M), withτ(p),≥1−ε/2 such that|ϕ(pxp)| ≤Kkxk₁for allx∈M.

We may similarly find a projectionq∈P(JMJ)andL>0 such thatτ(q)≥1−εand|ϕ(pqyqp)| ≤Lkyk₁ for ally∈JMJ. Note that ifx∈Mhas polar decompositionx=v|x|, then by Cauchy-Schwarz we have

|ϕ(qpxpq)| ≤ϕ(pv|x|^1/2q|x|^1/2v^∗p)^1/2ϕ(p|x|^1/2q|x|^1/2p)^1/2

≤Kkv|x|v^∗k^1/2₁ k|x|k^1/2₁ =Kkxk₁.

Also, note that sinceϕ∈B(L²M)^]_J it follows that takingεtending to 0, the corresponding net qpϕpq converges weak^∗toϕ. Sinceqpϕpq∈B(L²M)^\_Jthe result then follows from Lemma 2.5.4.

To prove the general case we set ˜M=M⊕Nand realizeK(L²N,L²M) = (1_M⊕0)K(L²M)(0˜ ⊕1_N), and B(L²N,L²M) = (1_M⊕0)B(L²M)(0˜ ⊕1_N). If ϕ∈B(L²N,L²M)^M]N_J , then the mapϕ⁰defined byϕ⁰(T) = ϕ((1_M⊕0)T(0⊕1_N))defines an element ofB(L²M)˜ ^]_J, and ifµi∈K(L²M)˜ ^]_J are such thatµi→ϕ weak^∗, then the mapsµi|K(L²N,L²M)are inK(L²N,L²M)^]_Jand converge weak^∗toϕ.

Theorem 2.5.6. Let M and N be tracial von Neumann algebras. SupposeX⊂B(L²M)andY⊂B(L²N)are boundary pieces. Fix T∈B(L²N,L²M). The following conditions are equivalent:

1. T∈K^∞,1_X,Y(N,M).

2. T∈B(L²N,L²M)_X-Y−mix. 3. T∈KXB(L²N,L²M)KY

M−N JMJ−JNJ

.

4. ϕ(T) =0wheneverϕ∈B(L²N,L²M)^]_Jsuch thatϕ_|XB(L2N,L²M)Y=0.

Moreover, if M=N,X=Ywith M,JMJ⊂M(X), and if T ∈M(X), then the above conditions are also equivalent to either of the following conditions:

5. ϕ(T) =0for all statesϕ∈B(L²M)^]_Jsuch thatϕ_|X=0.

6. Whenever we have a net of statesµ_i∈K(L²M)^\_Jsuch that{µ_i}_iis uniformly bounded in M⊗_ConM^op and such thatµi(S)→0for all S∈X, then we haveµi(T)→0.

Proof. The equivalence between (3) and (1) follows from Proposition 2.3.1. An application of the Hahn- Banach theorem gives the equivalence between (3) and (4). By Proposition 2.5.1 we have thatB(L²N,L²M)_X-Y−mix is a strongM-Nbimodule, and it is easy to see that it is also a strongM⁰-N⁰bimodule. SinceB(L²N,L²M)_X-Y−mix

containsXB(L²N,L²M)Yit then follows from Corollary 2.3.2 thatK^∞,1_X,Y(N,M) =X B(L²N,L²M)Y

k·k_∞,1

⊂ B(L²N,L²M)_X-Y−mix, which shows that (1) =⇒ (2).

To see that (2) =⇒ (4) we suppose that condition (2) is satisfied and take ϕ ∈B(L²M)^]_J such that ϕ_|XB(L2N,L²M)Y=0. By Lemma 2.5.5 there exists a uniformly bounded net{µ_i}_i⊂M⊗_ConM^opsuch that µi→ϕ weak^∗. Sinceϕ_|XB(L2N,L²M)Y=0 we have that µi(S)→0 for all S∈XB(L²N,L²M)Yand hence ϕ(T) =limi→∞µi(T) =0.

We now suppose thatM=N,X=YwithM,JMJ⊂M(X), andT∈M(X). Clearly we have (4) =⇒ (5).

SupposeT∈M(X)satisfies (5) and letϕ∈B(L²M)^]_Jbe such thatϕ|X=0. Taking the real and imaginary part ofϕ separately we may assume thatϕ is Hermitian. We then consider the Jordan decompositionϕ|M(X)= ϕ₊−ϕ−. By Lemma 2.5.3 we have thatϕ₊, andϕ−each restrict to a normal state on both M andJMJ.

SinceXis an ideal inM(X)it follows that(ϕ±)_|X=0. If we extendϕ+andϕ−to arbitrary positive linear functionalsψ+andψ−onB(L²M)respectively that satisfykψ+k=kϕ+kandkψ−k=kϕ−k, then we have ψ±∈B(L²M)^]_J and hence by (5) we haveψ±(T) =0 showing that ϕ(T) =ϕ+(T)−ϕ−(T) =ψ+(T)− ψ−(T) =0.

Finally, the equivalence between (5) and (6) follows from Lemma 2.5.4.

Remark 2.5.7. Given a normal operator M-N andJMJ-JNJ bimodule X, we note that theM-N closure operation forXandJMJ-JNJclosure operation forXcommute. Indeed, this follows from Proposition 2.2.3 (2).

Remark 2.5.8. From the previous theorem it follows thatB(L²N,L²M)_X-Y−mix defines the same subspace whether we viewB(L²N,L²M)as anM-Nbimodule, or as anJMJ-JNJbimodule.

Analogous to Proposition 2.3.8, we also have the following version for pairs of boundary pieces.

Proposition 2.5.9. Let M and N be tracial von Neumann algebras. SupposeX⊂B(L²M)andY⊂B(L²N) are boundary pieces. If T∈B(L²N,L²M), then the following are equivalent:

1. T∈K_X,Y(N,M).

2. T^∗T∈KY(N)and T T^∗∈KX(M).

3. T^∗T∈K^∞,1_Y (N)and T T^∗∈K^∞,1_X (M).

Proof. The implications (1)=⇒(2)=⇒(3) are clear. If (3) holds, then by Proposition 2.3.8 we have T^∗T ∈KY(N) and T T^∗ ∈KX(M). Considering the polar decomposition T =V|T| we then have T =

|T^∗|^1/2V|T|^1/2∈KX,Y(N,M).