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Berger-Shaw theorem of self-commutators in semifinite von Neumann algebras
J. Math. Anal. Appl. 479 (2019) 718–732
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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Berger-Shaw theorem of self-commutators in semifinite von Neumann algebras ✩ Qihui Li a,∗ , Ze Li b , Wenhua Qian c , Liguang Wang d a b c d
School of Science, East China University of Science and Technology, Shanghai, 200237, PR China College of Science, Xi’an Polytechnic University, South Jinhua Road 19, 710048, Xi’an, PR China School of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, PR China School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, PR China
a r t i c l e
i n f o
Article history: Received 24 February 2018 Available online 17 June 2019 Submitted by H. Lin
a b s t r a c t In the current article, we extend Voiculescu’s version of Berger-Shaw Theorem to properly infinite semifinite von Neumann algebras acting on separable Hilbert spaces. © 2019 Elsevier Inc. All rights reserved.
Keywords: The Berger-Shaw theorem Modulus of quasitrangularity von Neumann algebras
1. Introduction In [3], Berger and Shaw gave an beautiful inequality for insuring that the self-commutators of hypernormal operators lie in the trace class. Recall that [A, B] = AB − BA for operators A and B on a Hilbert space H, and self-commutator of A is [A, A∗ ]. Later, Berger and Ben-Jacob extended Berger-Shaw inequality to all von Neumann algebra [2]. In 1980, Voiculescu give a generalization of Berger-Shaw Theorem [9]. The argument giving by Voiculescu is based on the modulus of quasitriangularity which is an analogue of Apostol’s modulus of quasitrangularity relative to a Schatten-von Neumann class [1]. Voiculescu’s result is as follows: Theorem 1. ([9]) Suppose T is an operator on a Hilbert space such that [T ∗ , T ] = T ∗ T − T T ∗ is the sum of a positive operator and a trace class operator. If X is a Hilbert-Schmidt operator and T + X is rationally m-cyclic, then ✩ The first author was partially supported by NSFC (Grant No. 11671133). The third author was partially supported by NSFC (Grant No. 11801050). The last author was partially supported by NSFC (Grant No. 11871303) and NSFC of Shandong Province (Gant No. ZR2019MA039). * Corresponding author. E-mail addresses: [email protected] (Q. Li), [email protected] (Z. Li), [email protected] (W. Qian), [email protected] (L. Wang).
https://doi.org/10.1016/j.jmaa.2019.06.047 0022-247X/© 2019 Elsevier Inc. All rights reserved.
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πtr [T ∗ , T ] ≤ m · Area (σ (T + X)) where σ (T + X) is the spectrum of T + X and we agree ∞ · 0 = 0. In this article, we will mainly extend Voiculescu’s version of Berger-Shaw Theorem (Theorem 1) into the setting of properly infinite semifinite von Neumann algebras acting on separable Hilbert spaces. Let M be a countably decomposable properly infinite von Neumann algebra with a faithful tracial weight τM . For estimating the bounded of τM ([T ∗ , T ]) as T ∈ M and [T ∗ , T ] ∈ M+ + L1 (M, τM ) ∩ M, we will first extend Voiculescu’s modulus of quasitriangularity qΦ (T1 , · · · , Tn ) of n-tuple (T1 , · · · , Tn ) in B (H) to M with respect to a unitarily invariant norm Φ. In B (H), the modulus of quasitriangularity with respect to a unitarily invariant norm Φ is closely related to the invariant KΦ defined in [8]. For more information about KΦ , we refer the reader to [8,10–13]. The invariant KΦ can be extended to semifinite von Neumann algebras. From the argument in [6] (or arXiv:1706.09566), we can conclude that the equation KΦ (T1 , · · · , Tn ) = 0 for self-adjoint operators {T1 , · · · , Tn } in M may not be able to yield that {T1 , · · · , Tn } is diagonalizable simultaneously modulo a norm-ideal as it did in B (H), but the inequality KΦ (T1 , · · · , Tn ) = 0 is sufficient to ensure that {T1 , · · · , Tn } is not diagonalizable simultaneously modulo a certain norm-ideal in M. In current paper, we will build up a relation between qΦ and KΦ in M which is similar to their relation in B (H). So for T ∈ M and [T ∗ , T ] ∈ M+ + L1 (M, τM ) ∩ M, we are possible to estimate the bound of τM ([T ∗ , T ]) by using KΦ (T ) and qΦ (T, T ∗ ). When M is a properly infinite semifinite von Neumann algebra acting on a separable Hilbert space, we can prove Voiculescu’s version of Berger-Shaw Theorem in which another bound estimation of τM ([T ∗ , T ]) for T ∈ M and [T ∗ , T ] ∈ M+ + L1 (M, τM ) ∩ M will be given. This paper is divided into four sections. In the second section, we will recall some concepts such as unitarily invariant norm and dual norm on M, polynomial (rational) multiplicity of T in M. Section 3 will be devoted to giving the concept of modulus of quasitrangularity qΦ in M and KΦ in M as well as several properties of qΦ and KΦ . In this section, we will give an estimation of τM ([T, X]) as [T, X] ∈ M+ + L1 (M, τM ) ∩ M. Section 4 will deal with our main result in this paper. We will extend Voiculescu’s version of Berger-Shaw Theorem to a properly infinite semifintie von Neumann algebra acting on a separable Hilbert space. 2. Preliminaries and notation 2.1. Norm and dual norm on F (M, τM ) Let M be a countably decomposable, properly infinite von Neumann algebra with a faithful normal tracial weight τM . Let PF (M, τM ) = P : P = P ∗ = P 2 ∈ M and τM (P ) < ∞ , F (M, τM ) = {AP B : P ∈ PF (M, τM ) and A, B ∈ M} , K (M, τM ) = · -norm closure of F (M, τM ) in M, K(M) = · -norm closed ideal generated by finite projections in M, be the sets of finite rank projections, finite rank operators, compact operators and norm closed ideal generated by finite projections respectively in (M, τM ). In general K (M, τM ) ⊂ K(M). If M is a countably decomposable semifinite factor, we have K (M, τM ) = K(M). We say that a closed densely defined operator T is affiliated with a von Neumann algebra M (and write T ηM) when U ∗ T U = T for each unitary operator U commuting with M. An operator T ηM (affiliated
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with M) is said to be τM -measurable if, for every ε > 0, there exists a projection P ∈ M such that P H ⊆ D (T ) (domain of T ) and τM P ⊥ ≤ ε, where P ⊥ = 1 − P . The set of all τM -measurable operators which is a *-algebra with sum and product being the respective closure of the will be denoted by M, algebraic operations. Definition 2.1.1. A norm ideal KΦ (M, τM ) of M is a two sided ideal of M equipped with a norm Φ : KΦ (M, τM ) → [0, ∞), which satisfies (1) KΦ (M, τ ) is a Banach space with respect to the norm Φ; (2) Φ (U XV ) = Φ (X) for all X ∈ KΦ (M, τM ) and unitary elements U, V in M, i.e. Φ is a unitarily invariant norm; (3) there exists λ > 0 such that Φ(x) ≥ λ x for x ∈ KΦ (M, τM ), i.e. Φ is ·-dominating; (4) F(M, τ ) ⊂ KΦ (M, τM ) ⊂ K (M, τM ). For the purpose of convenience, if x ∈ / KΦ (M, τM ), then we set Φ(x) = ∞. For 1 ≤ p < ∞, the mapping ·p : F (M, τM ) → [0, ∞) is defined by p
1/p
Ap = (τM (|A| ))
, for any A ∈ F (M, τM ) .
Note ·p is a unitarily invariant norm on F (M, τM ). We let Lp (M, τM ) be the completion of F (M, τM ) (see [7] for more details). When p = ∞, we let A = A for all with respect to the norm ·p in M ∞ ∞ A ∈ M and let L (M, τM ) = M. Let Jp = Lp (M, τM ) ∩ M. Then Jp is a two-sided ideal of M with respect to the norm Φp defined in the form Φp (x) = max{x , xp } as x ∈ Jp for 1 ≤ p < ∞. We also have ∗
(Lp (M, τM )) = Lq (M, τM ) (isometrically) where
1 p
+
1 q
= 1 and p = 1 as q = ∞. It yields that 1 ∗ L (M, τM ) = M.
In general, we can define the dual norm Φ∗ for any norm Φ on F(M, τ ), we refer the reader to [7] and [4] for more information about dual norms. Definition 2.1.2. Let Φ be a norm on F(M, τ ). For T ∈ F(M, τ ), define the dual norm Φ∗ of Φ by Φ∗ (T ) = sup {|τM (T X)| : X ∈ F(M, τ ), Φ (X) ≤ 1} . The following lemma can be drawn from the definition of dual norms.
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Lemma 2.1.3. ([14]) Let Φ be a norm on KΦ (M, τM ) and Φ∗ be the dual norm of Φ. Then for any S ∈ KΦ (M, τM ) and T ∈ KΦ∗ (M, τM ), we have |τM (ST )| ≤ Φ (S) Φ∗ (T ). 2.2. Multiplicity of an operator in M In this subsection, let M be a von Neumann algebra acting on a Hilbert space H with a faithful tracial weight τM . The following concepts can be found in [2]. Definition 2.2.1. Let P represent the set of polynomials. Suppose P ∈ M is a projection and T ∈ M. Denote m P H T := the · -closure of { i=1 pi (T ) v : pi ∈ P , v ∈ P H} in H. If P H T = H, P is said to be polynomially cyclic for T . Let n (T ) = inf {τM (P ) : P ∈ M and P is polynomials cyclic for T } . Then n (T ) is called the polynomial multiplicity of T . Definition 2.2.2. Let P ∈ M be a projection. Suppose T ∈ M and Q (σ (T )) is the set of rational functions with poles off σ (T ). Denote m P H T := the · -closure of { i=1 qi (T ) v : qi ∈ Q (σ (T )) , v ∈ P H} in H.
If P H T = H, P is called to be rationally cyclic for T . Let m (T ) = inf {τM (P ) : P ∈ M and P is rationally cyclic for T } . Then m (T ) is called the rational multiplicity of T . 3. The bound of τM ([T, X]) 3.1. Voiculescu’s modulus of quasitrangularity in M Now we are ready to give an analogue of Voiculescu’s modulus of quasitrangularity in [9]. In this whole section, let M be a countably decomposable, properly infinite von Neumann algebra with a faithful normal tracial weight τM . Definition 3.1.1. Let {T1 , · · · , Tn } be an n-tuple in M. The modulus of quasitriangularity qΦ (T1 , · · · , Tn ) of n-tuple {T1 , · · · , Tn } with respect to a unitarily invariant norm Φ(·) on KΦ (M, τM ) is the number defined in the following form qΦ (T1 , · · · , Tn ; M, τM ) =
lim inf
max {Φ ((1 − P ) Ti P )}
P ∈PF (M,τM ) 1≤i≤n
where the lim inf s is taken with respect to the natural order on PF (M, τM ), i.e. qΦ (T1 , · · · , Tn ; M, τM ) =
sup
inf
max {Φ ((1 − Q) Ti Q)}
P ∈PF(M,τM ) Q∈PF (M,τM ) 1≤i≤n Q≥P
Note P ∈ PF (M, τM ), then we know that Φ((1 − P ) T P ) < ∞ for every T ∈ M. The next lemma is very useful for the applications of qΦ (T1 , · · · , Tn ; M, τM ).
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∞
Lemma 3.1.2. There is a sequence of increasing projections {Pn }n=1 in PF (M, τM ) such that Pn → I (in strong operator topology), and qΦ (T1 , · · · , Tn ; M, τM ) = lim
max {Φ ((1 − Pn ) Ti Pn )} .
n→∞
1≤i≤n
Proof. Define νQ (T1 , · · · , Tn ; M, τM ) max {Φ ((1 − P ) Ti P )} for Q ∈ PF (M, τM ) . = inf P ∈PF(M,τM ),P >Q
1≤i≤n
It is clear that νQ ≤ νQ for Q ≤ Q . We can choose an increasing sequence {Qn } of projections in PF (M, τM ) such that Qn → I (in strong operator topology) and qΦ (T1 , · · · , Tn ; M, τM ) = lim vQn (T1 , · · · , Tn ; M, τM ) . n→∞
Suppose {Pn } is an increasing sequence of projections in PF (M, τM ) such that Pn > Qn ∨ Pn−1 , P1 = Q1 and 1 . n
max {Φ ((I − Pn ) Ti Pn )} ≤ νQn ∨Pn−1 (T1 , · · · , Tn ; M, τM ) +
1≤i≤n
It implies that vQn (T1 , · · · , Tn ; M, τM ) ≤ max {Φ ((I − Pn ) Ti Pn )} 1≤i≤n
≤ νQn ∨Pn−1 (T1 , · · · , Tn ; M, τM ) + ≤ qΦ (T1 , · · · , Tn ; M, τM ) +
1 n
1 . n
If we take limit on the both sides, the following inequality holds qΦ (T1 , · · · , Tn ; M, τM ) = lim vQn (T1 , · · · , Tn ; M, τM ) n→∞ max {Φ ((1 − Pn ) Ti Pn )} ≤ qΦ (T1 , · · · , Tn ; M, τM ) . ≤ lim n→∞
1≤i≤n
Therefore qΦ (T1 , · · · , Tn ; M, τM ) = lim
n→∞
max {Φ ((1 − Pn ) Ti Pn )} . 2
1≤i≤n
3.2. KΦ in M The following definition is an analogue of Voiculescu’s KΦ in B (H) in [8]. Definition 3.2.1. Let n ∈ N and T1 , · · · , Tn be an n-tuple of elements in M. We define KΦ (T1 , · · · , Tn ; M, τM ) =
lim inf
max (Φ (ATi − Ti A))
A∈F1+ (M,τM ) 1≤i≤n
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where the lim inf s are taken with respect to the natural orders on F1+ (M, τM ) in which F1+ (M, τM ) is the unit ball of positive operators in F (M, τM ). Using similar idea in the proof of Lemma 3.1.2, it is not hard to prove the following statement. So we omit its proof. Lemma 3.2.2. Given T1 , · · · , Tn ∈ M, we can find an increasing sequence of Am ∈ F1+ (M, τM ) with Am → I (in strong operator topology) such that lim max Φ ([Am , Ti ]) = KΦ (T1 , · · · , Tn ; M, τM ) .
m→∞ 1≤i≤n
Now we are ready to set up a relation between KΦ and qΦ . Proposition 3.2.3. Given T1 , · · · , Tn ∈ M, KΦ (T1 , · · · , Tn ; M, τM ) ≤ 2qΦ (T1 , · · · , Tn , T1∗ , · · · , Tn∗ ; M, τM ). Proof. From definition it is easy to see that KΦ (T1 , · · · , Tn ; M, τM ) ≤
lim inf
max (Φ (P Ti − Ti P ))
P ∈PF (M,τM ) 1≤i≤n
and for every i Φ (P Ti − Ti P ) ≤ Φ(P Ti (1 − P )) + Φ((P − I)Ti P ) = Φ(P Ti (1 − P )) + Φ((P Ti∗ (I − P ) ≤ 2 max{Φ(P Ti (1 − P ), Φ(P Ti∗ (1 − P )). Therefore KΦ (T1 , · · · , Tn ; M, τM ) ≤ 2qΦ (T1 , · · · , Tn , T1∗ , · · · , Tn∗ ; M, τM ).
2
3.3. The bound of τM ([T, X]) as [T, X] ∈ J1 + M+ Recall that J1 = L1 (M, τM ) ∩ M, J1 is a two-sided ideal of M with respect to the norm Φ1 defined in the form Φ1 (x) = max{x , x1 } as x ∈ J1 . The next proposition is an extension of Proposition 2.1 in [11]. For the completeness, we give its proof. Proposition 3.3.1. Let T1 , · · · , Tn ∈ M, M+ be the set of all positive element in M and let X1 , · · · , Xn ∈ KΦ∗ (M, τM ). If Y =
1≤j≤n
[Tj , Xj ] ∈ J1 + M+ ,
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then we have |τM (Y )| ≤ KΦ (T1 , · · · , Tn ; M, τM )
Φ∗ (Xj ) .
1≤j≤n
Proof. Let {Am }m∈N be a sequence in F1+ (M, τM ) with Am → I (in strong operator topology) such that lim max Φ ([Am , Ti ]) = KΦ (T1 , · · · , Tn ) .
m→∞ 1≤i≤n
Note Am [T, X] = [Am , T ] X + [T, Am X] and τM ([T, Am X]) = 0 for any T, X ∈ M. Hence by Lemma 2.1.3 |τM (Y )| = lim τM (Am Y ) m→∞ = lim τM ([Am , T1 ] X1 + [T1 , Am X1 ] + · · · + [Am , Tn ] Xn + [Tn , Am Xn ]) m→∞ n
n
= lim τM [Am , Ti ] Xi ≤ lim |τM ([Am , Ti ] Xi )| m→∞ m→∞ i=1 i=1
≤ KΦ (T1 , · · · , Tn ; M, τM ) Φ∗ (Xi ) . 2 1≤i≤n
Remark 3.3.2. By the above Proposition, for any T ∈ M, if there is an element X ∈ M such that [T, X] is a sum of a positive element in M and an element in J1 , then |τM ([T, X])| ≤ KΦ (T ; M, τM ) X . It implies that |τM ([T ∗ , T ])| ≤ K1 (T ; M, τM ) T ≤ 2q1 (T, T ∗ ; M, τM )||T || if [T ∗ , T ] ∈ J1 + M+ by Proposition 3.2.3. In the next section, we will give another bound estimation of τM ([T ∗ , T ]). 4. Voiculescu’s Berger-Shaw theorem in semifinite von Neumann algebras 4.1. Berger-Shaw theorem in semifinite factors In this subsection, suppose M is a countably decomposable, properly infinite von Neumann algebra with a faithful normal tracial weight τM . Before giving our main result, we need the following lemmas. Lemma 4.1.1. Let P ∈ PF (M, τM ) and T ∈ M. Then τM (P [T ∗ , T ] P ) ≤ (1 − P ) T P 2 . 2
Proof. Note T = P T P + P T (1 − P ) + (1 − P ) T P + (1 − P ) T (1 − P ) . Since τM ([P T ∗ P, P T P ]) = 0,
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τM (P [T ∗ , T ] P ) = τM ([P T ∗ P, P T P ]) + τM (P T ∗ (1 − P ) T P ) − τM (P T (1 − P ) T ∗ P 2
2
2
= (1 − P ) T P 2 − P T (1 − P )2 ≤ (1 − P ) T P 2 .
2
Lemma 4.1.2. Assume that 1 ≤ p < ∞. Let T ∈ M with rational multiplicity m (T ). Then 1/p
qp (T ; M, τM ) ≤ (m (T ))
T .
Proof. Let {z1 , z2 , · · · } be a dense subset of C\σ (T ). If m (T ) = ∞, there is nothing to prove. So we assume m(T ) < ∞. By definition, for any ε > 0, we can find a projection P ∈ M such that τM (P ) < m (T ) + ε and P is rationally cyclic for T . Without loss of generality, we may assume τM (P ) = m (T ) + ε. Let Pm = ∨2m i=0 Ran
Ti P (T − z1 ) · · · (T − zm )
for m ∈ N
Ti Ti where T 0 = I and Ran (T −z1 )···(T P is the range projection of operator (T −z1 )···(T −zm ) −zm ) P . Since P is rationally cyclic for T , Pm ↑ I (in strong operator topology). Note for every i with 0 ≤ i ≤ 2m − 1, (1 − Pm ) T
Ti P (T − z1 ) · · · (T − zm )
= 0.
Hence Ran ((1 − Pm ) T Pm ) ≤ Ran (1 − Pm )
T 2m+1 P (T − z1 ) · · · (T − zm )
2m+1 and τM Ran (1 − Pm ) (T −zT1 )···(T −zm ) P ≤ τM (P ) = m (T ) + ε. It follows that p T 2m+1 p Ran (1 − P P (1 − P (1 − Pm ) T Pm p = ) ) T P m m m (T − z1 ) · · · (T − zm ) p p
≤ (m (T ) + ε) (1 − Pm ) T Pm . Consequently, 1/p
qp (T ; M, τM ) ≤ lim sup (I − Pm ) T Pm p ≤ (m (T ) + ε) n→∞
T .
Since ε > 0 is arbitrary, we have 1/p
qp (T ; M, τM ) ≤ (m (T ))
T .
2
We can also give a relation between τM ([T ∗ , T ]) and q2 (T ; M, τM ) in the following situation. Lemma 4.1.3. Let T ∈ M and [T ∗ , T ] be the sum of a positive operator and an element in J1 . Then τM ([T ∗ , T ]) ≤ (q2 (T ; M, τM )) . 2
Proof. By Lemma 3.1.2, there is a increasing sequence {Pn } of projections in PF (M, τM ) such that q2 (T ; M, τM ) = lim (1 − Pn ) T Pn 2 n→∞
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and Pn → I in strong operator topology. Since [T ∗ , T ] is the sum of a positive operator and an element in J1 , we have τM ([T ∗ , T ]) is well-defined and by Lemma 4.1.1 2
q22 (T ; M, τM ) = lim (1 − Pn ) T Pn 2 n→∞
≥ lim sup τM (Pn [T ∗ , T ] Pn ) = τM ([T ∗ , T ]) . n→∞
2
Lemma 4.1.4. Let T ∈ M and [T ∗ , T ] be the sum of a positive operator and an element in J1 . Assume X is an element in J2 such that T + X is rationally m (T + X)-cyclic. Then τM ([T ∗ , T ]) ≤ m (T + X) T + X . 2
Proof. If m (T + X) = ∞, there is nothing to prove. Suppose m (T + X) < ∞, since X ∈ J2 , we have (1 − Pn )XPn 2 ≤ (1 − Pn )X2 = (1 − Pn )XX ∗ (1 − Pn ) 2
2
→ 0 as Pn → I (in strong operator topology). Note (1 − Pn )(T + X)Pn 2 ≤ (1 − Pn )T Pn 2 + (1 − Pn )XPn 2 and (1 − Pn )T Pn 2 ≤ (1 − Pn )(T + X)Pn 2 + (1 − Pn )XPn 2 . Combining above inequalities and Lemma 3.1.2, it is not hard to see that q2 (T ; M, τM ) = q2 (T + X; M, τM ) . So by Lemma 4.1.3 and Lemma 4.1.2, we get τM ([T ∗ , T ]) = (q2 (T ; M, τM )) = (q2 (T + X; M, τM )) 2
2
2
≤ (m (T + X)) T + X . This completes the proof. 2 For giving our main result in this subsection, we need to recall Runge’s Theorem first. Theorem 4.1.5. (Runge’s Theorem) Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K. If S is a set containing at least one complex number from every bounded connected component of C\K, then there exists a sequence {rn }n∈N of rational functions which converges uniformly to f on K and such that all the poles of the functions {rn }n∈N are in S. The next theorem is an extension of Theorem 1 to the setting of semifinite factors. Theorem 4.1.6. Let N be a countably decomposable properly infinite factor acting on a Hilbert space H with a faithful tracial weight τN and T ∈ N . Suppose [T ∗ , T ] is a sum of a positive operator in N and an element in J1 . If X is in J2 such that T + X is rationally m (T + X)-cyclic, then
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τN ([T ∗ , T ]) ≤
727
1 m (T + X) Area (σ (T + X)) π
where we agree ∞ · 0 = 0 Proof. If m (T + X) = ∞ and Area (σ (T + X)) = 0, then there is nothing to prove. Now we only need to consider the following cases. Case 1. Suppose m (T + X) < ∞ and σ (T + X) = D (0, T + X) where D (0, T + X) is the closed 2 disc with center 0 and radius T + X. Then Area (σ (T + X)) = π T + X . So by Lemma 4.1.4, τN ([T ∗ , T ]) ≤ m (T + X) T + X
2
=
1 m (T + X) Area (σ (T + X)) . π
Case 2. Suppose m (T + X) < ∞ and σ (T + X) ⊂ D (0, T + X). For any ε > 0, we can find a projection P ∈ N such that P is rationally cyclic for T and τN (P ) < m (T + X) + ε. We may assume that τN (P ) = m (T + X) + ε. Now for ε > 0, we can find some closed disjoint disks D (a1 , r1 ) , · · · , D (an , rn ) with center ai and radius ri respectively in D (0, T + X) \σ (T + X) such that Area (D (0, T + X)) ≤ Area (σ (T + X)) +
n
k=1 Area (Dk )
+ ε.
It implies that 2
π T + X − π Define a normal tracial weight τ N =
n
2 k=1 rk
≤ Area (σ (T + X)) + ε.
on N . Hence τ N (P ) = 1. Since N is countably de , ρ with faithful tracial state ρ and composable, we can find a type II1 factor von Neumann algebra N ⊗ B (H0 ) and separable Hilbert space H0 such that N ∼ =N 1 m(T +X)+ε τN
τ N (A ⊗ B) = (ρ ⊗ tr) (A ⊗ B) = ρ (A) tr (B) where tr denotes the usual trace on B (H0 ). Let S ∈ B (H0 ) be the unilateral shift on H0 and Q be the rank one projection on the closed linear space generated by the unit cyclic vector for S. Define Sk = IN ⊗ ak IB(H0 ) + rk S for 1 ≤ k ≤ n. It follows that σ (Sk ) = D (ak , rk ) and ∗ 2 2 τ N ([Sk , Sk ]) = rk τ N (IN ⊗ Q) = rk
for 1 ≤ k ≤ n. Note IN ⊗ Q is cyclic projection for Sk , we also have Sk ≤ T + X and σ (T + X) , σ (S1 ) , · · · , σ (Sn ) are disjoint subsets of complex plane. Let A = (T + X) ⊕ S1 ⊕ · · · ⊕ Sn ∈ N ⊗ Mn+1 (C) , B = T ⊕ S1 ⊕ · · · ⊕ Sn ∈ N ⊗ Mn+1 (C) and
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⎛ ⎜ ⎜ P1 = P ⊗ ⎜ ⎜ ⎝
⎞
1 . 0
⎛ · · · , Pn+1 = IN
⎟ ⎜ ⎟ ⎜ ⎟ , P2 = IN ⊗ Q ⊗ ⎜ ⎟ ⎜ ⎠ ⎝
0 ..
⎛
..
. 0
⎟ ⎟ ⎟ in N ⊗ Mn+1 (C) ⎟ ⎠
0 ..
⎟ ⎟ ⎟, ⎟ ⎠
1
⎞
0
⎜ ⎜ ⊗Q⊗⎜ ⎜ ⎝
⎞
0
. 1
Since τ N ⊗ trn+1 (P1 ) = · · · = τ N ⊗ trn+1 (Pn+1 ) = 1 where trn+1 stands for the trace on Mn+1 (C), we are able to find a system of matrix units {Eij : 1 ≤ i, j ≤ n + 1} in the factor N ⊗ Mn+1 (C) such that ∗ ∗ ∗ Eii = Pi , Eij = Eji , Eij Eij = Pj and Eij Eij = Pi
as well as Eij Emk = 0 as j = m for 1 ≤ i, j ≤ n + 1. Define a projection ⎛ P =
1 ⎝ n+1
⎞ Eij ⎠ ∈ N ⊗ Mn+1 (C) .
1≤i,j≤n+1
= 1. Since σ (T + X) , σ (S1 ) , · · · , σ (Sn ) are disjoint subsets of complex plane, by So (τ N ⊗ trn+1 ) P 4.1.5, IN ⊕ 0 ⊕ · · · ⊕ 0, · · · , 0 ⊕ 0 ⊕ · · · ⊕ IN are in the norm closed algebra generated by the rational functions in operator A. Combining it with the facts that IN ⊗ Q is cyclic projection for Sk for 1 ≤ k ≤ n and P is rationally cyclic for T , it is not hard to see that P is rationally cyclic for A. By Lemma 4.1.4 n 2 ∗ ∗ ∗ (τ N ⊗ trn+1 ) ([B , B]) = τ N ([T , T ]) + N ([Si , Si ]) ≤ m(A) A i=1 τ 2 2 ≤ (τ N ⊗ trn+1 ) P A = T + X . Note τ N =
1 m(T +X)+ε τN
on N , then n 2 τN ([T ∗ , T ]) ≤ (m (T + X) + ε) T + X − i=1 ri2 ≤
1 (m (T + X) + ε) (Area (σ (T + X)) + ε) . π
Since ε is arbitrary, we have τN ([T ∗ , T ]) ≤
1 m (T + X) Area (σ (T + X)) . π
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Case 3. If m (T + X) = ∞ and Area (σ (T + X)) = 0. Let {Pm } be a increasing sequence of projections in N with τN (Pm ) < ∞ and Pm → IN (in strong operator topology) for m → ∞. We define a projection Pm = ∨q∈Q(σ(T +X)) Ran (q (T + X) Pm ) ∈ N Then Pm H is an invariant subspace of T +X and Pm is rationally cyclic for Pm (T + X) |Pm H with Pm → IN .
1 0 A B Note Pm = IN by the fact that m (T + X) = ∞. Therefore we can assume Pm = ,T = 0 0 C D V W ). Since Pm H is an invariant subspace of T + X, we have C = −Y . It follows that C is in Y Z J2 and then C ∗ C is in J1 by the fact that X is in J2 . We also notice that and X = (
Pm [T , T ] Pm = ∗
[A∗ , A] + C ∗ C − BB ∗ 0
0 0
where Pm [T ∗ , T ] Pm is the sum of a positive operator and an element in J1 since [T ∗ , T ] is in this form too. Therefore [A∗ , A] is the sum of a positive operator and an element in J1 by the fact that BB ∗ is positive and C ∗ C is in J1 . Note Pm (T + X) |Pm H = A + V and σ (A + V ) ⊆ σ (T + X). Then Area (σ (A + V )) = 0 and A + V is τN (Pm )-rational cyclic, so by case 2, we have τN ([A∗ , A]) ≤ 0. Therefore by the fact that C = −Y is in J2 , we have 2 2 τN Pm [T ∗ , T ] Pm ≤ τN (C ∗ C) = IN − Pm T Pm = IN − Pm X Pm 2
2 2 ≤ IN − Pm X Pm = τN X ∗ IN − Pm X 2 = τN IN − Pm XX ∗ IN − Pm < ∞.
2
It is obvious that τN IN − Pm XX ∗ IN − Pm → 0 and Pm [T ∗ , T ] Pm → [T ∗ , T ] in strong operator topology as m → ∞. It yields that τN ([T ∗ , T ]) ≤ 0 =
1 m (T + X) Area (σ (T + X)) . π
This is the end of the proof. 2 4.2. Berger-Shaw theorem in semifinite von Neumann algebras Next, we are going to use the theory of direct integral of separable Hilbert spaces and von Neumann algebras acting on separable Hilbert spaces (see [5] for more general knowledge of direct integral) to give the main result of this paper. For giving a clear discussion, we list some lemmas which will be needed later. Lemma 4.2.1. ([5]) Suppose M is a von Neumann algebra acting on a separable Hilbert space H and Z is the center of M. Then there is a direct integral decomposition of M relative to Z, i.e., there exists a locally compact complete separable metric (Borel measure) space (X, μ) such that
730
Q. Li et al. / J. Math. Anal. Appl. 479 (2019) 718–732
(1) H is (unitarily equivalent to) the direct integral of {Hs : s ∈ X} over (X, μ), where each Hs is a separable Hilbert space, s ∈ X; (2) M is (unitarily equivalent to) the direct integral of {Ms : s ∈ X} over (X, μ), where Ms is a factor in B(Hs ) almost everywhere. Also, if M is of type In (n could be infinite), II1 , II∞ or III, then the components Ms are, almost everywhere, of type In , II1 , II∞ or III respectively. Lemma 4.2.2. (([6]) Let M be a properly infinite von Neumann algebra acting on a separable Hilbert space H with a faithful, normal, semifinite tracial weight τM and let Z be the center of M. Suppose M = X ⊕Ms dμ and H = X ⊕Hs dμ are direct integral decompositions of M and H over (X, μ) relative to Z. Then there exist a μ-null set N and a family {ξn }n∈N of vectors in H such that (i) there is a family of faithful, normal, semifinite tracial weights τs on Ms for s ∈ X \ N satisfying, for every positive T ∈ M, τM (T ) =
τs (T (s))dμ; X
(ii) moreover, for all s ∈ X \ N ,
τs (T (s)) =
T (s)ξn (s), ξn (s) ,
for all T (s) ∈ Ms .
n∈N
Now we are ready to give our main result in this article. Theorem 4.2.3. Let M be a properly infinite semifinite von Neumann algebra acting on a separable Hilber space H with a faithful tracial weight τM . Suppose [T ∗ , T ] is the sum of a positive element in M and an element in J1 . If there is an element X ∈ J2 such that T + X is rationally m (T + X)-cyclic. Then τM ([T ∗ , T ]) ≤
m(T + X) Area(σ(T + X)). π
Proof. If m(T + X) = ∞, there is nothing to prove. Now we suppose m(T + X) < ∞. Let {xj } be a dense denumerable set spanning H. So for any ε, we can find a projection P in M such that τM (P ) ≤ m(T + X) + ε
(4.1)
and m
the · -closure of {
i=1 qi
(T + X) P xt : qi ∈ Q (σ (T + X)) , xt ∈ {xj }} = H.
Let M = X ⊕Ms dμ, H = X ⊕Hs dμ and τM = X ⊕τs dμ be direct integral decompositions of M, H and τM over (X, μ) relative to the center of M. We might assume that, for s ∈ X almost everywhere, Ms is a properly infinite factor with a faithful normal semifinite tracial weight τs . Assume [T ∗ , T ] = A + C in which 0 ≤ A ∈ M and C ∈ J1 . Hence by Proposition 14.1.8 in [5] and Lemma 4.2.2, we can find a μ-null set N0 such that for s ∈ X\N0 A(s) ≥ 0, C(s) ∈ J1,s and
Q. Li et al. / J. Math. Anal. Appl. 479 (2019) 718–732
∗
731
∗
(T ∗ T − T T ∗ ) (s) = (T (s)) T (s) − T (s) (T (s)) = A(s) + C(s) where J1,s = Ms ∩ L1 (Ms , τs ).
Suppose {λ1 , λ2 , · · · } is a dense subset of C\σ(T + X). Assume Bi ∈ B (H) such that Bi (T + X − λi ) = IH for i = 1, 2, · · · , then there is a μ-null set N1 such that Bi (s)(T (s) + X(s) − λi ) = IHs for every i and s ∈ X\ (N0 ∪ N1 ) . It follows that {λ1 , λ2 , · · · } ⊆ C\σ(T (s) + X(s)) for s ∈ X\ (N0 ∪ N1 ) . Hence σ(T (s) + X(s)) ⊆ σ(T + X) for s ∈ X\ (N0 ∪ N1 ) . We can also find μ-null set N2 such that for s ∈ X\ (N0 ∪ N1 ∪ N2 ) m
the · -closure of {
i=1 qi
(T (s) + X(s)) P (s)xt (s) : qi ∈ Q (σ (T + X)) , xt ∈ {xj }} = Hs .
By Theorem 4.1.6, m(T (s) + X(s)) ∗ Area(σ(T (s) + X(s))) τs ( (T (s)) , T (s) ) ≤ π τs (P (s)) ≤ Area(σ(T (s) + X(s))). π So by Lemma 4.2.2, (4.1) and (4.2), τM ([T ∗ , T ]) =
∗ τs ( (T (s)) , T (s) )dμ ≤
X
≤
Area(σ(T + X)) π
≤
τs (P (s)) Area(σ(T (s) + X(s)))dμ π
X
τs (P (s))dμ = τM (P )
Area(σ(T + X)) π
X
m(T + X) + ε Area(σ(T + X)). π
Since ε is arbitrary, we get τM ([T ∗ , T ]) ≤ This completes the proof. 2
m(T + X) Area(σ(T + X)). π
(4.2)
732
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References [1] Constantin Apostol, Quasitriangularity in Hilbert space, Indiana Univ. Math. J. 22 (1973) 817–825. [2] C.A. Berger, Marion Golazerman Ben-Jacob, Trace class self-commutators, Trans. Amer. Math. Soc. 227 (1983) 75–91. [3] C.A. Berger, B.I. Shaw, Selfcommutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc. 79 (1973) 1193–1199. [4] Junsheng Fang, Don hadwin, Unitarily invariant norms related to factors, Studia Math. 229 (2015) 13–44. [5] Richard Kadison, John Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II. Advanced Theory, Corrected reprint of the 1986 original, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. [6] Qihui Li, Junhao Shen, Rui Shi, A generalization of the Voiculescu theorem for normal operators in semifinite von Neumann algebras, arXiv:1706.09522. [7] Gilles Pisier, Quanhua Xu, Non-commutative Lp -spaces, in: Handbook of the Geometry of Banach Spaces, vol. 2, NorthHolland, Amsterdam, 2003, pp. 1459–1517. [8] Dan Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators, J. Operator Theory 2 (1979) 3–37. [9] Dan Voiculescu, A note on quasitriangularity and trace-class selfcommutators, Acta Sci. Math. 42 (1980) 195–199. [10] Dan Voiculescu, Some results on norm-ideal perturbations of Hilbert space operators. II, J. Operator Theory 5 (1981) 77–100. [11] Dan Voiculescu, On the existence of quasicentral approximate units relative to normed ideals. Part I, J. Funct. Anal. 91 (1990) 1–36. [12] Dan Voiculescu, Almost normal operators mod Hilbert-Schmidt and the K-theory of the Banach algebras EΛ (Ω), J. Noncommut. Geom. 8 (2014) 1123–1145. [13] Dan Voiculescu, Some C*-algebras which are coronas of non-C*-Banach algebras, J. Geom. Phys. 105 (2016) 123–129. [14] Keiichi Watanabe, Some results on non-commutative Banach function spaces II (Infinite cases), Hokkaido Math. J. 22 (1993) 349–364.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
Berger-Shaw theorem of self-commutators in semifinite von Neumann algebras ✩ Qihui Li a,∗ , Ze Li b , Wenhua Qian c , Liguang Wang d a b c d
School of Science, East China University of Science and Technology, Shanghai, 200237, PR China College of Science, Xi’an Polytechnic University, South Jinhua Road 19, 710048, Xi’an, PR China School of Mathematical Sciences, Chongqing Normal University, Chongqing, 400047, PR China School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, PR China
a r t i c l e
i n f o
Article history: Received 24 February 2018 Available online 17 June 2019 Submitted by H. Lin
a b s t r a c t In the current article, we extend Voiculescu’s version of Berger-Shaw Theorem to properly infinite semifinite von Neumann algebras acting on separable Hilbert spaces. © 2019 Elsevier Inc. All rights reserved.
Keywords: The Berger-Shaw theorem Modulus of quasitrangularity von Neumann algebras
1. Introduction In [3], Berger and Shaw gave an beautiful inequality for insuring that the self-commutators of hypernormal operators lie in the trace class. Recall that [A, B] = AB − BA for operators A and B on a Hilbert space H, and self-commutator of A is [A, A∗ ]. Later, Berger and Ben-Jacob extended Berger-Shaw inequality to all von Neumann algebra [2]. In 1980, Voiculescu give a generalization of Berger-Shaw Theorem [9]. The argument giving by Voiculescu is based on the modulus of quasitriangularity which is an analogue of Apostol’s modulus of quasitrangularity relative to a Schatten-von Neumann class [1]. Voiculescu’s result is as follows: Theorem 1. ([9]) Suppose T is an operator on a Hilbert space such that [T ∗ , T ] = T ∗ T − T T ∗ is the sum of a positive operator and a trace class operator. If X is a Hilbert-Schmidt operator and T + X is rationally m-cyclic, then ✩ The first author was partially supported by NSFC (Grant No. 11671133). The third author was partially supported by NSFC (Grant No. 11801050). The last author was partially supported by NSFC (Grant No. 11871303) and NSFC of Shandong Province (Gant No. ZR2019MA039). * Corresponding author. E-mail addresses: [email protected] (Q. Li), [email protected] (Z. Li), [email protected] (W. Qian), [email protected] (L. Wang).
https://doi.org/10.1016/j.jmaa.2019.06.047 0022-247X/© 2019 Elsevier Inc. All rights reserved.
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πtr [T ∗ , T ] ≤ m · Area (σ (T + X)) where σ (T + X) is the spectrum of T + X and we agree ∞ · 0 = 0. In this article, we will mainly extend Voiculescu’s version of Berger-Shaw Theorem (Theorem 1) into the setting of properly infinite semifinite von Neumann algebras acting on separable Hilbert spaces. Let M be a countably decomposable properly infinite von Neumann algebra with a faithful tracial weight τM . For estimating the bounded of τM ([T ∗ , T ]) as T ∈ M and [T ∗ , T ] ∈ M+ + L1 (M, τM ) ∩ M, we will first extend Voiculescu’s modulus of quasitriangularity qΦ (T1 , · · · , Tn ) of n-tuple (T1 , · · · , Tn ) in B (H) to M with respect to a unitarily invariant norm Φ. In B (H), the modulus of quasitriangularity with respect to a unitarily invariant norm Φ is closely related to the invariant KΦ defined in [8]. For more information about KΦ , we refer the reader to [8,10–13]. The invariant KΦ can be extended to semifinite von Neumann algebras. From the argument in [6] (or arXiv:1706.09566), we can conclude that the equation KΦ (T1 , · · · , Tn ) = 0 for self-adjoint operators {T1 , · · · , Tn } in M may not be able to yield that {T1 , · · · , Tn } is diagonalizable simultaneously modulo a norm-ideal as it did in B (H), but the inequality KΦ (T1 , · · · , Tn ) = 0 is sufficient to ensure that {T1 , · · · , Tn } is not diagonalizable simultaneously modulo a certain norm-ideal in M. In current paper, we will build up a relation between qΦ and KΦ in M which is similar to their relation in B (H). So for T ∈ M and [T ∗ , T ] ∈ M+ + L1 (M, τM ) ∩ M, we are possible to estimate the bound of τM ([T ∗ , T ]) by using KΦ (T ) and qΦ (T, T ∗ ). When M is a properly infinite semifinite von Neumann algebra acting on a separable Hilbert space, we can prove Voiculescu’s version of Berger-Shaw Theorem in which another bound estimation of τM ([T ∗ , T ]) for T ∈ M and [T ∗ , T ] ∈ M+ + L1 (M, τM ) ∩ M will be given. This paper is divided into four sections. In the second section, we will recall some concepts such as unitarily invariant norm and dual norm on M, polynomial (rational) multiplicity of T in M. Section 3 will be devoted to giving the concept of modulus of quasitrangularity qΦ in M and KΦ in M as well as several properties of qΦ and KΦ . In this section, we will give an estimation of τM ([T, X]) as [T, X] ∈ M+ + L1 (M, τM ) ∩ M. Section 4 will deal with our main result in this paper. We will extend Voiculescu’s version of Berger-Shaw Theorem to a properly infinite semifintie von Neumann algebra acting on a separable Hilbert space. 2. Preliminaries and notation 2.1. Norm and dual norm on F (M, τM ) Let M be a countably decomposable, properly infinite von Neumann algebra with a faithful normal tracial weight τM . Let PF (M, τM ) = P : P = P ∗ = P 2 ∈ M and τM (P ) < ∞ , F (M, τM ) = {AP B : P ∈ PF (M, τM ) and A, B ∈ M} , K (M, τM ) = · -norm closure of F (M, τM ) in M, K(M) = · -norm closed ideal generated by finite projections in M, be the sets of finite rank projections, finite rank operators, compact operators and norm closed ideal generated by finite projections respectively in (M, τM ). In general K (M, τM ) ⊂ K(M). If M is a countably decomposable semifinite factor, we have K (M, τM ) = K(M). We say that a closed densely defined operator T is affiliated with a von Neumann algebra M (and write T ηM) when U ∗ T U = T for each unitary operator U commuting with M. An operator T ηM (affiliated
Q. Li et al. / J. Math. Anal. Appl. 479 (2019) 718–732
720
with M) is said to be τM -measurable if, for every ε > 0, there exists a projection P ∈ M such that P H ⊆ D (T ) (domain of T ) and τM P ⊥ ≤ ε, where P ⊥ = 1 − P . The set of all τM -measurable operators which is a *-algebra with sum and product being the respective closure of the will be denoted by M, algebraic operations. Definition 2.1.1. A norm ideal KΦ (M, τM ) of M is a two sided ideal of M equipped with a norm Φ : KΦ (M, τM ) → [0, ∞), which satisfies (1) KΦ (M, τ ) is a Banach space with respect to the norm Φ; (2) Φ (U XV ) = Φ (X) for all X ∈ KΦ (M, τM ) and unitary elements U, V in M, i.e. Φ is a unitarily invariant norm; (3) there exists λ > 0 such that Φ(x) ≥ λ x for x ∈ KΦ (M, τM ), i.e. Φ is ·-dominating; (4) F(M, τ ) ⊂ KΦ (M, τM ) ⊂ K (M, τM ). For the purpose of convenience, if x ∈ / KΦ (M, τM ), then we set Φ(x) = ∞. For 1 ≤ p < ∞, the mapping ·p : F (M, τM ) → [0, ∞) is defined by p
1/p
Ap = (τM (|A| ))
, for any A ∈ F (M, τM ) .
Note ·p is a unitarily invariant norm on F (M, τM ). We let Lp (M, τM ) be the completion of F (M, τM ) (see [7] for more details). When p = ∞, we let A = A for all with respect to the norm ·p in M ∞ ∞ A ∈ M and let L (M, τM ) = M. Let Jp = Lp (M, τM ) ∩ M. Then Jp is a two-sided ideal of M with respect to the norm Φp defined in the form Φp (x) = max{x , xp } as x ∈ Jp for 1 ≤ p < ∞. We also have ∗
(Lp (M, τM )) = Lq (M, τM ) (isometrically) where
1 p
+
1 q
= 1 and p = 1 as q = ∞. It yields that 1 ∗ L (M, τM ) = M.
In general, we can define the dual norm Φ∗ for any norm Φ on F(M, τ ), we refer the reader to [7] and [4] for more information about dual norms. Definition 2.1.2. Let Φ be a norm on F(M, τ ). For T ∈ F(M, τ ), define the dual norm Φ∗ of Φ by Φ∗ (T ) = sup {|τM (T X)| : X ∈ F(M, τ ), Φ (X) ≤ 1} . The following lemma can be drawn from the definition of dual norms.
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Lemma 2.1.3. ([14]) Let Φ be a norm on KΦ (M, τM ) and Φ∗ be the dual norm of Φ. Then for any S ∈ KΦ (M, τM ) and T ∈ KΦ∗ (M, τM ), we have |τM (ST )| ≤ Φ (S) Φ∗ (T ). 2.2. Multiplicity of an operator in M In this subsection, let M be a von Neumann algebra acting on a Hilbert space H with a faithful tracial weight τM . The following concepts can be found in [2]. Definition 2.2.1. Let P represent the set of polynomials. Suppose P ∈ M is a projection and T ∈ M. Denote m P H T := the · -closure of { i=1 pi (T ) v : pi ∈ P , v ∈ P H} in H. If P H T = H, P is said to be polynomially cyclic for T . Let n (T ) = inf {τM (P ) : P ∈ M and P is polynomials cyclic for T } . Then n (T ) is called the polynomial multiplicity of T . Definition 2.2.2. Let P ∈ M be a projection. Suppose T ∈ M and Q (σ (T )) is the set of rational functions with poles off σ (T ). Denote m P H T := the · -closure of { i=1 qi (T ) v : qi ∈ Q (σ (T )) , v ∈ P H} in H.
If P H T = H, P is called to be rationally cyclic for T . Let m (T ) = inf {τM (P ) : P ∈ M and P is rationally cyclic for T } . Then m (T ) is called the rational multiplicity of T . 3. The bound of τM ([T, X]) 3.1. Voiculescu’s modulus of quasitrangularity in M Now we are ready to give an analogue of Voiculescu’s modulus of quasitrangularity in [9]. In this whole section, let M be a countably decomposable, properly infinite von Neumann algebra with a faithful normal tracial weight τM . Definition 3.1.1. Let {T1 , · · · , Tn } be an n-tuple in M. The modulus of quasitriangularity qΦ (T1 , · · · , Tn ) of n-tuple {T1 , · · · , Tn } with respect to a unitarily invariant norm Φ(·) on KΦ (M, τM ) is the number defined in the following form qΦ (T1 , · · · , Tn ; M, τM ) =
lim inf
max {Φ ((1 − P ) Ti P )}
P ∈PF (M,τM ) 1≤i≤n
where the lim inf s is taken with respect to the natural order on PF (M, τM ), i.e. qΦ (T1 , · · · , Tn ; M, τM ) =
sup
inf
max {Φ ((1 − Q) Ti Q)}
P ∈PF(M,τM ) Q∈PF (M,τM ) 1≤i≤n Q≥P
Note P ∈ PF (M, τM ), then we know that Φ((1 − P ) T P ) < ∞ for every T ∈ M. The next lemma is very useful for the applications of qΦ (T1 , · · · , Tn ; M, τM ).
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∞
Lemma 3.1.2. There is a sequence of increasing projections {Pn }n=1 in PF (M, τM ) such that Pn → I (in strong operator topology), and qΦ (T1 , · · · , Tn ; M, τM ) = lim
max {Φ ((1 − Pn ) Ti Pn )} .
n→∞
1≤i≤n
Proof. Define νQ (T1 , · · · , Tn ; M, τM ) max {Φ ((1 − P ) Ti P )} for Q ∈ PF (M, τM ) . = inf P ∈PF(M,τM ),P >Q
1≤i≤n
It is clear that νQ ≤ νQ for Q ≤ Q . We can choose an increasing sequence {Qn } of projections in PF (M, τM ) such that Qn → I (in strong operator topology) and qΦ (T1 , · · · , Tn ; M, τM ) = lim vQn (T1 , · · · , Tn ; M, τM ) . n→∞
Suppose {Pn } is an increasing sequence of projections in PF (M, τM ) such that Pn > Qn ∨ Pn−1 , P1 = Q1 and 1 . n
max {Φ ((I − Pn ) Ti Pn )} ≤ νQn ∨Pn−1 (T1 , · · · , Tn ; M, τM ) +
1≤i≤n
It implies that vQn (T1 , · · · , Tn ; M, τM ) ≤ max {Φ ((I − Pn ) Ti Pn )} 1≤i≤n
≤ νQn ∨Pn−1 (T1 , · · · , Tn ; M, τM ) + ≤ qΦ (T1 , · · · , Tn ; M, τM ) +
1 n
1 . n
If we take limit on the both sides, the following inequality holds qΦ (T1 , · · · , Tn ; M, τM ) = lim vQn (T1 , · · · , Tn ; M, τM ) n→∞ max {Φ ((1 − Pn ) Ti Pn )} ≤ qΦ (T1 , · · · , Tn ; M, τM ) . ≤ lim n→∞
1≤i≤n
Therefore qΦ (T1 , · · · , Tn ; M, τM ) = lim
n→∞
max {Φ ((1 − Pn ) Ti Pn )} . 2
1≤i≤n
3.2. KΦ in M The following definition is an analogue of Voiculescu’s KΦ in B (H) in [8]. Definition 3.2.1. Let n ∈ N and T1 , · · · , Tn be an n-tuple of elements in M. We define KΦ (T1 , · · · , Tn ; M, τM ) =
lim inf
max (Φ (ATi − Ti A))
A∈F1+ (M,τM ) 1≤i≤n
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where the lim inf s are taken with respect to the natural orders on F1+ (M, τM ) in which F1+ (M, τM ) is the unit ball of positive operators in F (M, τM ). Using similar idea in the proof of Lemma 3.1.2, it is not hard to prove the following statement. So we omit its proof. Lemma 3.2.2. Given T1 , · · · , Tn ∈ M, we can find an increasing sequence of Am ∈ F1+ (M, τM ) with Am → I (in strong operator topology) such that lim max Φ ([Am , Ti ]) = KΦ (T1 , · · · , Tn ; M, τM ) .
m→∞ 1≤i≤n
Now we are ready to set up a relation between KΦ and qΦ . Proposition 3.2.3. Given T1 , · · · , Tn ∈ M, KΦ (T1 , · · · , Tn ; M, τM ) ≤ 2qΦ (T1 , · · · , Tn , T1∗ , · · · , Tn∗ ; M, τM ). Proof. From definition it is easy to see that KΦ (T1 , · · · , Tn ; M, τM ) ≤
lim inf
max (Φ (P Ti − Ti P ))
P ∈PF (M,τM ) 1≤i≤n
and for every i Φ (P Ti − Ti P ) ≤ Φ(P Ti (1 − P )) + Φ((P − I)Ti P ) = Φ(P Ti (1 − P )) + Φ((P Ti∗ (I − P ) ≤ 2 max{Φ(P Ti (1 − P ), Φ(P Ti∗ (1 − P )). Therefore KΦ (T1 , · · · , Tn ; M, τM ) ≤ 2qΦ (T1 , · · · , Tn , T1∗ , · · · , Tn∗ ; M, τM ).
2
3.3. The bound of τM ([T, X]) as [T, X] ∈ J1 + M+ Recall that J1 = L1 (M, τM ) ∩ M, J1 is a two-sided ideal of M with respect to the norm Φ1 defined in the form Φ1 (x) = max{x , x1 } as x ∈ J1 . The next proposition is an extension of Proposition 2.1 in [11]. For the completeness, we give its proof. Proposition 3.3.1. Let T1 , · · · , Tn ∈ M, M+ be the set of all positive element in M and let X1 , · · · , Xn ∈ KΦ∗ (M, τM ). If Y =
1≤j≤n
[Tj , Xj ] ∈ J1 + M+ ,
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then we have |τM (Y )| ≤ KΦ (T1 , · · · , Tn ; M, τM )
Φ∗ (Xj ) .
1≤j≤n
Proof. Let {Am }m∈N be a sequence in F1+ (M, τM ) with Am → I (in strong operator topology) such that lim max Φ ([Am , Ti ]) = KΦ (T1 , · · · , Tn ) .
m→∞ 1≤i≤n
Note Am [T, X] = [Am , T ] X + [T, Am X] and τM ([T, Am X]) = 0 for any T, X ∈ M. Hence by Lemma 2.1.3 |τM (Y )| = lim τM (Am Y ) m→∞ = lim τM ([Am , T1 ] X1 + [T1 , Am X1 ] + · · · + [Am , Tn ] Xn + [Tn , Am Xn ]) m→∞ n
n
= lim τM [Am , Ti ] Xi ≤ lim |τM ([Am , Ti ] Xi )| m→∞ m→∞ i=1 i=1
≤ KΦ (T1 , · · · , Tn ; M, τM ) Φ∗ (Xi ) . 2 1≤i≤n
Remark 3.3.2. By the above Proposition, for any T ∈ M, if there is an element X ∈ M such that [T, X] is a sum of a positive element in M and an element in J1 , then |τM ([T, X])| ≤ KΦ (T ; M, τM ) X . It implies that |τM ([T ∗ , T ])| ≤ K1 (T ; M, τM ) T ≤ 2q1 (T, T ∗ ; M, τM )||T || if [T ∗ , T ] ∈ J1 + M+ by Proposition 3.2.3. In the next section, we will give another bound estimation of τM ([T ∗ , T ]). 4. Voiculescu’s Berger-Shaw theorem in semifinite von Neumann algebras 4.1. Berger-Shaw theorem in semifinite factors In this subsection, suppose M is a countably decomposable, properly infinite von Neumann algebra with a faithful normal tracial weight τM . Before giving our main result, we need the following lemmas. Lemma 4.1.1. Let P ∈ PF (M, τM ) and T ∈ M. Then τM (P [T ∗ , T ] P ) ≤ (1 − P ) T P 2 . 2
Proof. Note T = P T P + P T (1 − P ) + (1 − P ) T P + (1 − P ) T (1 − P ) . Since τM ([P T ∗ P, P T P ]) = 0,
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τM (P [T ∗ , T ] P ) = τM ([P T ∗ P, P T P ]) + τM (P T ∗ (1 − P ) T P ) − τM (P T (1 − P ) T ∗ P 2
2
2
= (1 − P ) T P 2 − P T (1 − P )2 ≤ (1 − P ) T P 2 .
2
Lemma 4.1.2. Assume that 1 ≤ p < ∞. Let T ∈ M with rational multiplicity m (T ). Then 1/p
qp (T ; M, τM ) ≤ (m (T ))
T .
Proof. Let {z1 , z2 , · · · } be a dense subset of C\σ (T ). If m (T ) = ∞, there is nothing to prove. So we assume m(T ) < ∞. By definition, for any ε > 0, we can find a projection P ∈ M such that τM (P ) < m (T ) + ε and P is rationally cyclic for T . Without loss of generality, we may assume τM (P ) = m (T ) + ε. Let Pm = ∨2m i=0 Ran
Ti P (T − z1 ) · · · (T − zm )
for m ∈ N
Ti Ti where T 0 = I and Ran (T −z1 )···(T P is the range projection of operator (T −z1 )···(T −zm ) −zm ) P . Since P is rationally cyclic for T , Pm ↑ I (in strong operator topology). Note for every i with 0 ≤ i ≤ 2m − 1, (1 − Pm ) T
Ti P (T − z1 ) · · · (T − zm )
= 0.
Hence Ran ((1 − Pm ) T Pm ) ≤ Ran (1 − Pm )
T 2m+1 P (T − z1 ) · · · (T − zm )
2m+1 and τM Ran (1 − Pm ) (T −zT1 )···(T −zm ) P ≤ τM (P ) = m (T ) + ε. It follows that p T 2m+1 p Ran (1 − P P (1 − P (1 − Pm ) T Pm p = ) ) T P m m m (T − z1 ) · · · (T − zm ) p p
≤ (m (T ) + ε) (1 − Pm ) T Pm . Consequently, 1/p
qp (T ; M, τM ) ≤ lim sup (I − Pm ) T Pm p ≤ (m (T ) + ε) n→∞
T .
Since ε > 0 is arbitrary, we have 1/p
qp (T ; M, τM ) ≤ (m (T ))
T .
2
We can also give a relation between τM ([T ∗ , T ]) and q2 (T ; M, τM ) in the following situation. Lemma 4.1.3. Let T ∈ M and [T ∗ , T ] be the sum of a positive operator and an element in J1 . Then τM ([T ∗ , T ]) ≤ (q2 (T ; M, τM )) . 2
Proof. By Lemma 3.1.2, there is a increasing sequence {Pn } of projections in PF (M, τM ) such that q2 (T ; M, τM ) = lim (1 − Pn ) T Pn 2 n→∞
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and Pn → I in strong operator topology. Since [T ∗ , T ] is the sum of a positive operator and an element in J1 , we have τM ([T ∗ , T ]) is well-defined and by Lemma 4.1.1 2
q22 (T ; M, τM ) = lim (1 − Pn ) T Pn 2 n→∞
≥ lim sup τM (Pn [T ∗ , T ] Pn ) = τM ([T ∗ , T ]) . n→∞
2
Lemma 4.1.4. Let T ∈ M and [T ∗ , T ] be the sum of a positive operator and an element in J1 . Assume X is an element in J2 such that T + X is rationally m (T + X)-cyclic. Then τM ([T ∗ , T ]) ≤ m (T + X) T + X . 2
Proof. If m (T + X) = ∞, there is nothing to prove. Suppose m (T + X) < ∞, since X ∈ J2 , we have (1 − Pn )XPn 2 ≤ (1 − Pn )X2 = (1 − Pn )XX ∗ (1 − Pn ) 2
2
→ 0 as Pn → I (in strong operator topology). Note (1 − Pn )(T + X)Pn 2 ≤ (1 − Pn )T Pn 2 + (1 − Pn )XPn 2 and (1 − Pn )T Pn 2 ≤ (1 − Pn )(T + X)Pn 2 + (1 − Pn )XPn 2 . Combining above inequalities and Lemma 3.1.2, it is not hard to see that q2 (T ; M, τM ) = q2 (T + X; M, τM ) . So by Lemma 4.1.3 and Lemma 4.1.2, we get τM ([T ∗ , T ]) = (q2 (T ; M, τM )) = (q2 (T + X; M, τM )) 2
2
2
≤ (m (T + X)) T + X . This completes the proof. 2 For giving our main result in this subsection, we need to recall Runge’s Theorem first. Theorem 4.1.5. (Runge’s Theorem) Denoting by C the set of complex numbers, let K be a compact subset of C and let f be a function which is holomorphic on an open set containing K. If S is a set containing at least one complex number from every bounded connected component of C\K, then there exists a sequence {rn }n∈N of rational functions which converges uniformly to f on K and such that all the poles of the functions {rn }n∈N are in S. The next theorem is an extension of Theorem 1 to the setting of semifinite factors. Theorem 4.1.6. Let N be a countably decomposable properly infinite factor acting on a Hilbert space H with a faithful tracial weight τN and T ∈ N . Suppose [T ∗ , T ] is a sum of a positive operator in N and an element in J1 . If X is in J2 such that T + X is rationally m (T + X)-cyclic, then
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τN ([T ∗ , T ]) ≤
727
1 m (T + X) Area (σ (T + X)) π
where we agree ∞ · 0 = 0 Proof. If m (T + X) = ∞ and Area (σ (T + X)) = 0, then there is nothing to prove. Now we only need to consider the following cases. Case 1. Suppose m (T + X) < ∞ and σ (T + X) = D (0, T + X) where D (0, T + X) is the closed 2 disc with center 0 and radius T + X. Then Area (σ (T + X)) = π T + X . So by Lemma 4.1.4, τN ([T ∗ , T ]) ≤ m (T + X) T + X
2
=
1 m (T + X) Area (σ (T + X)) . π
Case 2. Suppose m (T + X) < ∞ and σ (T + X) ⊂ D (0, T + X). For any ε > 0, we can find a projection P ∈ N such that P is rationally cyclic for T and τN (P ) < m (T + X) + ε. We may assume that τN (P ) = m (T + X) + ε. Now for ε > 0, we can find some closed disjoint disks D (a1 , r1 ) , · · · , D (an , rn ) with center ai and radius ri respectively in D (0, T + X) \σ (T + X) such that Area (D (0, T + X)) ≤ Area (σ (T + X)) +
n
k=1 Area (Dk )
+ ε.
It implies that 2
π T + X − π Define a normal tracial weight τ N =
n
2 k=1 rk
≤ Area (σ (T + X)) + ε.
on N . Hence τ N (P ) = 1. Since N is countably de , ρ with faithful tracial state ρ and composable, we can find a type II1 factor von Neumann algebra N ⊗ B (H0 ) and separable Hilbert space H0 such that N ∼ =N 1 m(T +X)+ε τN
τ N (A ⊗ B) = (ρ ⊗ tr) (A ⊗ B) = ρ (A) tr (B) where tr denotes the usual trace on B (H0 ). Let S ∈ B (H0 ) be the unilateral shift on H0 and Q be the rank one projection on the closed linear space generated by the unit cyclic vector for S. Define Sk = IN ⊗ ak IB(H0 ) + rk S for 1 ≤ k ≤ n. It follows that σ (Sk ) = D (ak , rk ) and ∗ 2 2 τ N ([Sk , Sk ]) = rk τ N (IN ⊗ Q) = rk
for 1 ≤ k ≤ n. Note IN ⊗ Q is cyclic projection for Sk , we also have Sk ≤ T + X and σ (T + X) , σ (S1 ) , · · · , σ (Sn ) are disjoint subsets of complex plane. Let A = (T + X) ⊕ S1 ⊕ · · · ⊕ Sn ∈ N ⊗ Mn+1 (C) , B = T ⊕ S1 ⊕ · · · ⊕ Sn ∈ N ⊗ Mn+1 (C) and
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728
⎛ ⎜ ⎜ P1 = P ⊗ ⎜ ⎜ ⎝
⎞
1 . 0
⎛ · · · , Pn+1 = IN
⎟ ⎜ ⎟ ⎜ ⎟ , P2 = IN ⊗ Q ⊗ ⎜ ⎟ ⎜ ⎠ ⎝
0 ..
⎛
..
. 0
⎟ ⎟ ⎟ in N ⊗ Mn+1 (C) ⎟ ⎠
0 ..
⎟ ⎟ ⎟, ⎟ ⎠
1
⎞
0
⎜ ⎜ ⊗Q⊗⎜ ⎜ ⎝
⎞
0
. 1
Since τ N ⊗ trn+1 (P1 ) = · · · = τ N ⊗ trn+1 (Pn+1 ) = 1 where trn+1 stands for the trace on Mn+1 (C), we are able to find a system of matrix units {Eij : 1 ≤ i, j ≤ n + 1} in the factor N ⊗ Mn+1 (C) such that ∗ ∗ ∗ Eii = Pi , Eij = Eji , Eij Eij = Pj and Eij Eij = Pi
as well as Eij Emk = 0 as j = m for 1 ≤ i, j ≤ n + 1. Define a projection ⎛ P =
1 ⎝ n+1
⎞ Eij ⎠ ∈ N ⊗ Mn+1 (C) .
1≤i,j≤n+1
= 1. Since σ (T + X) , σ (S1 ) , · · · , σ (Sn ) are disjoint subsets of complex plane, by So (τ N ⊗ trn+1 ) P 4.1.5, IN ⊕ 0 ⊕ · · · ⊕ 0, · · · , 0 ⊕ 0 ⊕ · · · ⊕ IN are in the norm closed algebra generated by the rational functions in operator A. Combining it with the facts that IN ⊗ Q is cyclic projection for Sk for 1 ≤ k ≤ n and P is rationally cyclic for T , it is not hard to see that P is rationally cyclic for A. By Lemma 4.1.4 n 2 ∗ ∗ ∗ (τ N ⊗ trn+1 ) ([B , B]) = τ N ([T , T ]) + N ([Si , Si ]) ≤ m(A) A i=1 τ 2 2 ≤ (τ N ⊗ trn+1 ) P A = T + X . Note τ N =
1 m(T +X)+ε τN
on N , then n 2 τN ([T ∗ , T ]) ≤ (m (T + X) + ε) T + X − i=1 ri2 ≤
1 (m (T + X) + ε) (Area (σ (T + X)) + ε) . π
Since ε is arbitrary, we have τN ([T ∗ , T ]) ≤
1 m (T + X) Area (σ (T + X)) . π
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Case 3. If m (T + X) = ∞ and Area (σ (T + X)) = 0. Let {Pm } be a increasing sequence of projections in N with τN (Pm ) < ∞ and Pm → IN (in strong operator topology) for m → ∞. We define a projection Pm = ∨q∈Q(σ(T +X)) Ran (q (T + X) Pm ) ∈ N Then Pm H is an invariant subspace of T +X and Pm is rationally cyclic for Pm (T + X) |Pm H with Pm → IN .
1 0 A B Note Pm = IN by the fact that m (T + X) = ∞. Therefore we can assume Pm = ,T = 0 0 C D V W ). Since Pm H is an invariant subspace of T + X, we have C = −Y . It follows that C is in Y Z J2 and then C ∗ C is in J1 by the fact that X is in J2 . We also notice that and X = (
Pm [T , T ] Pm = ∗
[A∗ , A] + C ∗ C − BB ∗ 0
0 0
where Pm [T ∗ , T ] Pm is the sum of a positive operator and an element in J1 since [T ∗ , T ] is in this form too. Therefore [A∗ , A] is the sum of a positive operator and an element in J1 by the fact that BB ∗ is positive and C ∗ C is in J1 . Note Pm (T + X) |Pm H = A + V and σ (A + V ) ⊆ σ (T + X). Then Area (σ (A + V )) = 0 and A + V is τN (Pm )-rational cyclic, so by case 2, we have τN ([A∗ , A]) ≤ 0. Therefore by the fact that C = −Y is in J2 , we have 2 2 τN Pm [T ∗ , T ] Pm ≤ τN (C ∗ C) = IN − Pm T Pm = IN − Pm X Pm 2
2 2 ≤ IN − Pm X Pm = τN X ∗ IN − Pm X 2 = τN IN − Pm XX ∗ IN − Pm < ∞.
2
It is obvious that τN IN − Pm XX ∗ IN − Pm → 0 and Pm [T ∗ , T ] Pm → [T ∗ , T ] in strong operator topology as m → ∞. It yields that τN ([T ∗ , T ]) ≤ 0 =
1 m (T + X) Area (σ (T + X)) . π
This is the end of the proof. 2 4.2. Berger-Shaw theorem in semifinite von Neumann algebras Next, we are going to use the theory of direct integral of separable Hilbert spaces and von Neumann algebras acting on separable Hilbert spaces (see [5] for more general knowledge of direct integral) to give the main result of this paper. For giving a clear discussion, we list some lemmas which will be needed later. Lemma 4.2.1. ([5]) Suppose M is a von Neumann algebra acting on a separable Hilbert space H and Z is the center of M. Then there is a direct integral decomposition of M relative to Z, i.e., there exists a locally compact complete separable metric (Borel measure) space (X, μ) such that
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(1) H is (unitarily equivalent to) the direct integral of {Hs : s ∈ X} over (X, μ), where each Hs is a separable Hilbert space, s ∈ X; (2) M is (unitarily equivalent to) the direct integral of {Ms : s ∈ X} over (X, μ), where Ms is a factor in B(Hs ) almost everywhere. Also, if M is of type In (n could be infinite), II1 , II∞ or III, then the components Ms are, almost everywhere, of type In , II1 , II∞ or III respectively. Lemma 4.2.2. (([6]) Let M be a properly infinite von Neumann algebra acting on a separable Hilbert space H with a faithful, normal, semifinite tracial weight τM and let Z be the center of M. Suppose M = X ⊕Ms dμ and H = X ⊕Hs dμ are direct integral decompositions of M and H over (X, μ) relative to Z. Then there exist a μ-null set N and a family {ξn }n∈N of vectors in H such that (i) there is a family of faithful, normal, semifinite tracial weights τs on Ms for s ∈ X \ N satisfying, for every positive T ∈ M, τM (T ) =
τs (T (s))dμ; X
(ii) moreover, for all s ∈ X \ N ,
τs (T (s)) =
T (s)ξn (s), ξn (s) ,
for all T (s) ∈ Ms .
n∈N
Now we are ready to give our main result in this article. Theorem 4.2.3. Let M be a properly infinite semifinite von Neumann algebra acting on a separable Hilber space H with a faithful tracial weight τM . Suppose [T ∗ , T ] is the sum of a positive element in M and an element in J1 . If there is an element X ∈ J2 such that T + X is rationally m (T + X)-cyclic. Then τM ([T ∗ , T ]) ≤
m(T + X) Area(σ(T + X)). π
Proof. If m(T + X) = ∞, there is nothing to prove. Now we suppose m(T + X) < ∞. Let {xj } be a dense denumerable set spanning H. So for any ε, we can find a projection P in M such that τM (P ) ≤ m(T + X) + ε
(4.1)
and m
the · -closure of {
i=1 qi
(T + X) P xt : qi ∈ Q (σ (T + X)) , xt ∈ {xj }} = H.
Let M = X ⊕Ms dμ, H = X ⊕Hs dμ and τM = X ⊕τs dμ be direct integral decompositions of M, H and τM over (X, μ) relative to the center of M. We might assume that, for s ∈ X almost everywhere, Ms is a properly infinite factor with a faithful normal semifinite tracial weight τs . Assume [T ∗ , T ] = A + C in which 0 ≤ A ∈ M and C ∈ J1 . Hence by Proposition 14.1.8 in [5] and Lemma 4.2.2, we can find a μ-null set N0 such that for s ∈ X\N0 A(s) ≥ 0, C(s) ∈ J1,s and
Q. Li et al. / J. Math. Anal. Appl. 479 (2019) 718–732
∗
731
∗
(T ∗ T − T T ∗ ) (s) = (T (s)) T (s) − T (s) (T (s)) = A(s) + C(s) where J1,s = Ms ∩ L1 (Ms , τs ).
Suppose {λ1 , λ2 , · · · } is a dense subset of C\σ(T + X). Assume Bi ∈ B (H) such that Bi (T + X − λi ) = IH for i = 1, 2, · · · , then there is a μ-null set N1 such that Bi (s)(T (s) + X(s) − λi ) = IHs for every i and s ∈ X\ (N0 ∪ N1 ) . It follows that {λ1 , λ2 , · · · } ⊆ C\σ(T (s) + X(s)) for s ∈ X\ (N0 ∪ N1 ) . Hence σ(T (s) + X(s)) ⊆ σ(T + X) for s ∈ X\ (N0 ∪ N1 ) . We can also find μ-null set N2 such that for s ∈ X\ (N0 ∪ N1 ∪ N2 ) m
the · -closure of {
i=1 qi
(T (s) + X(s)) P (s)xt (s) : qi ∈ Q (σ (T + X)) , xt ∈ {xj }} = Hs .
By Theorem 4.1.6, m(T (s) + X(s)) ∗ Area(σ(T (s) + X(s))) τs ( (T (s)) , T (s) ) ≤ π τs (P (s)) ≤ Area(σ(T (s) + X(s))). π So by Lemma 4.2.2, (4.1) and (4.2), τM ([T ∗ , T ]) =
∗ τs ( (T (s)) , T (s) )dμ ≤
X
≤
Area(σ(T + X)) π
≤
τs (P (s)) Area(σ(T (s) + X(s)))dμ π
X
τs (P (s))dμ = τM (P )
Area(σ(T + X)) π
X
m(T + X) + ε Area(σ(T + X)). π
Since ε is arbitrary, we get τM ([T ∗ , T ]) ≤ This completes the proof. 2
m(T + X) Area(σ(T + X)). π
(4.2)
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