On irreducible operators in factor von Neumann algebras
aa r X i v : . [ m a t h . OA ] M a y On irreducible operators in factor von Neumann algebras
Junsheng Fang, Rui Shi, and Shilin Wen
Abstract.
Let M be a factor von Neumann algebra with separable predual and let T ∈ M .We call T an irreducible operator (relative to M ) if W ∗ ( T ) is an irreducible subfactor of M ,i.e., W ∗ ( T ) ′ ∩ M = C I . In this note, we show that the set of irreducible operators in M is adense G δ subset of M in the operator norm. This is a natural generalization of a theorem ofHalmos.
1. Introduction
In [ ], Halmos proved the following theorem. Let H be a separable (finite or infinite-dimensional) complex Hilbert space. Then the set of irreducible operators on H is a dense G δ subset of B ( H ) in the operator norm. Recall that an operator T ∈ B ( H ) is irreducible if T has no nontrivial reducing subspaces, i.e., if P is a projection such that P T = T P then P = 0or P = I . We refer to [ ] for a beautiful short proof. In this note, we generalize the abovetheorem to arbitrary factor von Neumann algebras with separable predual. Let M be a factorvon Neumann algebra with separable predual and let T ∈ M . We call T an irreducible operator(relative to M ) if W ∗ ( T ) is an irreducible subfactor of M , i.e., W ∗ ( T ) ′ ∩ M = C I . We showthat the set of irreducible operators in M is a dense G δ subset of M in the operator norm.Let A ∈ B ( H ) and B ∈ B ( K ), where H , K are Hilbert spaces. For every operator X ∈B ( K , H ), we define an operator τ A,B ( X ) = AX − XB.τ
A,B is called a Rosenblum operator [
5, 4 ]. Lemma ]) . If σ ( A ) ∩ σ ( B ) = ∅ , then AX = XB implies X = 0 . Another ingredient in the proof of our main result is related to the generator problem offactor von Neumann algebras with separable predual. Precisely, we need the following lemma.
Lemma . Let M be a factor von Neumann algebra with separable predual. Then thereexists a singly generated irreducible subfactor N in M . Proof.
It is well-known that if M is type I, II ∞ or III, then M is singly generated. So thelemma is clear in these cases. When M is a type II factor with separable predual, by [ ], thereexists a hyperfinite irreducible subfactor in M which is singly generated. (cid:3) Mathematics Subject Classification.
Primary 47C15.
Key words and phrases. factor von Neumann alegbras, irreducible operators.Junsheng Fang was partly supported by NSFC(Grant No.11431011) and a start up funding from HebeiNormal University.Rui Shi was partly supported by NSFC(Grant No.11401071) and the Fundamental Research Funds for theCentral Universities (Grant No.DUT18LK23).
2. Main result
Theorem . Let M be a factor von Neumann algebra with separable predual. Then theset of irreducible operators in M is a dense G δ subset of M in the operator norm. Proof.
Let T ∈ M and ǫ >
0. We need to show that there exists an operator S ∈ M suchthat k T − S k < ǫ and S is irreducible relative to M , i.e., if P ∈ M is a projection such that P S = SP , then P = 0 or P = I .Write T = A + iB , where both A and B in M are self-adjoint operators. By the spec-tral theorem for self-adjoint operators, there exist λ < λ < · · · < λ n ∈ σ ( A ) and projec-tions E , E , . . . , E n ∈ M such that P nj =1 E j = I and k A − P nj =1 λ j E j k < ǫ . Let A = P ni =1 λ i E i and B ij = E i BE j . By the spectral theorem for self-adjoint operators again, thereexist η i , η i , . . . , η im i ∈ σ ( B ii ) and projections F i , F i , . . . , F im i ∈ M such that P m i j =1 F ij = E i and k B ii − P m i j =1 η ij F ij k < ǫ . Let B ′ ii = P m i j =1 η ij F ij .Define T = A + iB , where B is self-adjoint, defined in the form B = E E · · · E n E B ′ B · · · B n E B B ′ · · · B n ... ... ... . . . ... E n B n B n · · · B ′ nn . Then T can be expressed in the form T = A + iB = E E · · · E n E λ · · · E λ · · · E n · · · λ n + E E · · · E n E B ′ B · · · B n E B B ′ · · · B n ... ... ... . . . ... E n B n B n · · · B ′ nn . Note that k T − T k = k ( A + iB ) − ( A + iB ) k ≤ k A − A k + k B − B k < ǫ. (2.1)For 1 ≤ i ≤ n , we can choose real numbers λ i , λ i , . . . , λ im i such that(1) the inequality k λ i E i − P m i j =1 λ ij F ij k < ǫ holds for every i ;(2) λ < λ < · · · < λ m < λ < · · · < λ m < · · · < λ n < · · · < λ nm n .Define A = P ni =1 P m i j =1 λ ij F ij . Then k A − A k < ǫ . Now we make a small self-adjointperturbation B of B such that each off-diagonal entry of B , with respect to the decomposition I = n X i =1 m i X j =1 F ij , N IRREDUCIBLE OPERATORS IN FACTOR VON NEUMANN ALGEBRAS 3 is nonzero. That is we can construct a self-adjoint operator B in M such that k B − B k < ǫ and F ij B F i ′ j ′ = 0 if i = i ′ or j = j ′ . Let T be defined in the form T = A + iB , for A = F F · · · F nm n F λ · · · F λ · · · F nm n · · · λ nm n and B = F F · · · F nm n F η ∗ · · · ∗ F ∗ η · · · ∗ ... ... ... . . . ... F nm n ∗ ∗ · · · η nm n , where each ∗ -entry is nonzero. By applying (2.1), it follows that k T − T k ≤ k T − T k + k T − T k < ǫ. (2.2)Since M is a separable factor, F ij M F ij is also a separable factor. By Lemma 1.2, we canfind positive elements X ij , Y ij ∈ F ij M F ij such that { X ij , Y ij } ′′ is an irreducible subfactor of F ij M F ij . Now we can choose δ > λ ij F ij + δX ij arepairwise disjoint, for 1 ≤ i ≤ n and 1 ≤ j ≤ m i .Let T be defined in the form T = A + iB , for A = F F · · · F nm n F λ + δX · · · F λ + δX · · · F nm n · · · λ nm n + δX nm n and B = F F · · · F nm n F η + δY ∗ · · · ∗ F ∗ η + δY · · · ∗ ... ... ... . . . ... F nm n ∗ ∗ · · · η nm n + δY nm n , where each ∗ -entry is the same as in T . Then, clearly, if δ > k T − T k < ǫ. (2.3)Hence, the inequalities (2.2) and (2.3) entail that k T − T k < ǫ .We assert that T is irreducible relative to M . Let P ∈ M be a projection commutingwith T . Then P A = A P and P B = B P . Write P = ( P ab ) ≤ a,b ≤ k with respect to thedecomposition I = P ni =1 P m i j =1 F ij , where k = P ni =1 m i . That P A = A P implies that( λ F + δX ) P = P ( λ F + δX ) . Since σ ( λ F + δX ) ∩ σ ( λ F + δX ) = ∅ , we have P = 0 by Lemma 1.1. Similarly,we have P ab = 0 for 1 ≤ a = b ≤ k . Thus P = P ka =1 P aa is diagonal with respect to thedecomposition I = P ni =1 P m i j =1 F ij . JUNSHENG FANG, RUI SHI, AND SHILIN WEN
By the construction that { X ij , Y ij } ′′ is an irreducible subfactor of F ij M F ij , it follows that P aa is either 0 or I F ij M F ij for each a . Since the off-diagonal entries of B are nonzero, an easycalculation shows that, if P = 0, then P aa = 0 for 1 ≤ a ≤ k . Therefore, P = 0 or P = I . Thisproves that T is irreducible relative to M .The remainder is to prove the set of irreducible operators relative to M is a G δ subset of M in the operator norm. The proof is similar to that provided by Halmos. For the sake ofcompleteness, we include the details. Let P be the set of all those selfadjoint operators P in M for which 0 ≤ P ≤ I . Let P be the subset of those elements of P that are not scalar multiplesof the identity. Since P is a weakly closed subset of the unit ball of M , it is weakly compact,and hence the weak topology for P is metrizable. Since the set of scalars is weakly closed, itfollows that P is weakly locally compact. Since the weak topology for P has a countable base,the same is true for P , and therefore P is weakly σ -compact. Let P , P , . . . be weakly compactsubsets of P such that ∪ ∞ n =1 P n = P .It is to be proved that the set of reducible operators relative to M , denoted by R ( M ), isan F σ set in the operator norm topology. Let ˆ P n be the set of all those operators T in M forwhich there exists a P ∈ P n such that T P = P T . Then ∪ ∞ n =1 ˆ P n = R ( M ).The proof can be completed by showing that each ˆ P n is closed in the operator norm. Supposethat T k ∈ ˆ P n and lim k →∞ T k = T in the operator norm. For each k , find a P k ∈ P n such that T k P k = P k T k . Since P n is weakly compact and metrizable, we may assume that P k is weaklyconvergent to P in P n . Then lim k →∞ T k P k = T P and lim k →∞ P k T k = P T in the weak operatortopology. Hence,
T P = P T and T ∈ ˆ P n . This implies that ˆ P n is closed in the operator normand R ( M ) is an F σ set in the operator norm. (cid:3) References [1] P.R. Halmos. Irreducible operators,
Michigan Math J. , 1968, 215–223.[2] S. Popa. On a problem of R. V. Kadison on maximal abelian ∗ -subalgebras in factors. Invent. Math. ,1981/82, no. 2, 269–281.[3] Heydar Radjavi and Peter Rosenthal. Shorter Notes: The Set of Irreducible Operators is Dense. Proceedingsof the American Mathematical Society. , No. 1 1969, p. 256[4] Heydar Radjavi and Peter Rosenthal. Invariant Subspaces (second edition). Dover Publications, Inc., Mine-ola, NY, 2003.[5] M.Rosenblum. On the operator equation BX − XA = Q . Duke Math. J. , 1956, 263–269. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
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