aa r X i v : . [ m a t h . OA ] J a n LATTICES OF LOGMODULAR ALGEBRAS
B.V. RAJARAMA BHAT AND MANISH KUMAR
Abstract.
A subalgebra A of a C ∗ -algebra M is logmodular (resp. has factorization) if theset { a ∗ a ; a is invertible with a, a − ∈ A} is dense in (resp. equal to) the set of all positive andinvertible elements of M . There are large classes of well studied algebras, both in commutativeand non-commutative settings, which are known to be logmodular. In this paper, we showthat the lattice of projections in a von Neumann algebra M whose ranges are invariant undera logmodular algebra in M , is a commutative subspace lattice. Further, if M is a factor thenthis lattice is a nest. As a special case, it follows that all reflexive (in particular, completelydistributive CSL) logmodular subalgebras of type I factors are nest algebras, thus answering aquestion of Paulsen and Raghupathi [18]. We also discuss some sufficient criteria under whichan algebra having factorization is automatically reflexive and is a nest algebra. Introduction
The well-known Cholesky factorization theorem states that any positive and invertible n × n matrix can be written as U ∗ U for some invertible upper triangular n × n matrix U (so the inverse U − is also upper triangular). We then say that the algebra of upper triangular matrices has factorization in M n , the algebra of all n × n complex matrices. Using this or otherwise, one can showthat an algebra consisting of block upper-triangular matrices (with respect to some orthonormalbasis) also admits such factorization in M n . Is there any other algebra in M n with this property?Paulsen and Raghupathi [18] showed that any algebra in M n containing all diagonal matrices, hasfactorization in M n if and only if it is unitarily equivalent to an algebra of block upper triangularmatrices (they actually studied the notion called logmodular algebra, which is our interest in thispaper as well). The general characterization was settled by Juschenko [14] who showed that uptoa change of basis, all algebras with factorization in M n contain the diagonal algebra, thus showingthat algebras of block upper triangular matrices are all which have factorization.A natural question that arises is up to what extent these results generalize to infinite-dimensionalsetting. Let H be a complex and separable Hilbert space, and let B ( H ) denote the algebra of allbounded operators on H . Let E be a collection of closed subspaces of H totally ordered withrespect to inclusion (such a collection is called a nest ), and let Alg E (called nest algebra ) denotethe algebra of all operators in B ( H ) which leave all subspaces in E invariant. Note that an algebraof block upper triangular matrices is nothing but Alg E for some nest E in C n , and vice versa. So ageneralization of Cholesky factorization would be to ask whether Alg E has factorization in B ( H )for a nest E in H . Gohberg and Krein [9] appear to be the first who studied factorization along nestalgebras, mainly examining positive and invertible operators ‘close’ to identity operator. Arveson[3] considered factorization property of nest algebras arising out of nests of order type Z . It wasLarson [15] who investigated factorization property of arbitrary nest algebras, and he proved inparticular that if E is a countable complete nest in H , then Alg E has factorization in B ( H ). Againwhat can be said about the converse? That is, if a subalgebra A has factorization in B ( H ), arethey of the form Alg E for some countable complete nest E ? In this paper, we show that this isindeed the case if we also assume that A is reflexive.More generally, factorization property of subalgebras of arbitrary von Neumann algebras areconsidered. A classical result says that the Hardy algebra H ∞ ( T ) on the unit circle has factorizationin L ∞ ( T ) i.e. for any non-negative element f ∈ L ∞ ( T ) with 1 /f ∈ L ∞ ( T ), there is an element Mathematics Subject Classification.
Primary: 47L35. Secondary: 47L30, 46K50.
Key words and phrases.
Logmodular algebra, factorization, nest, commutative subspace lattice, nest algebra,CSL algebra. h ∈ H ∞ ( T ) with 1 /h ∈ H ∞ ( T ) such that f = ¯ hh . Some other function algebras like weak ∗ -Dirichlet algebras introduced by Srinivasan and Wang [25] have factorization property. Taking cuefrom analytic function algebras, Arveson [2] introduced the theory of finite maximal subdiagonalalgebras as noncommutative variant and considered many results analogous to the classical Hardyspace theory, showing in particular that they have factorization property. Later several authorshave examined such algebras in different settings. For more about algebras with factorization see[2, 3, 15, 16, 19, 23, 17, 7, 9], and for some closely related properties see [20, 21, 1, 13, 17, 24] toname a few.An algebra with factorization property is a particular case of logmodular algebras . The notionof logmodularity was first introduced by Hoffman [11] for subalgebras of commutative C ∗ -algebras,whose main idea was to generalize some classical results of analytic function theory in the unit disc.Blecher and Labuschagne [4] extended this notion to subalgebras of non-commutative C ∗ -algebras.They studied completely contractive representations on such algebras and their extension proper-ties. Paulsen and Raghupathi [18] also studied representations of logmodular algebras and exploredconditions under which a contractive representation is automatically completely contractive. In[14], Juschenko gave a complete characterization of all logmodular subalgebras of M n . See [5]for a beautiful survey on logmodular algebras arising out of tracial subalgebras and their relationto finite subdiagonal algebras among others. They show how most results generalized in 1960’sfrom the Hardy space on the unit disc to more general function algebras generalize further to thenon-commutative situation, though more sophisticated proof techniques had to be developed forthe purpose. We list some additional references on logmodular algebras in [11, 12, 8, 4, 5, 18, 14].In this article, our aim is to understand the behaviour of lattice of subspaces (or projections)invariant under logmodular algebras, and use it to characterize reflexive logmodular algebras. Themain result answers a conjecture by [18] in the affirmative, which asks whether every completelydistributive CSL logmodular algebra of B ( H ) is a nest algebra. In fact, we show more generallythat any M -reflexive logmodular subalgebra of a factor M is a nest subalgebra. Moreover, weexplore some sufficient criteria under which an algebra with factorization is automatically reflexiveand is a nest algebra. In particular it is proved that a subalgebra of B ( H ) with factorization, whoselattice consists of finite dimensional atoms, is reflexive and so it is a nest algebra. Also we showthat any subalgebra with factorization in a finite dimensional von Neumann algebra must be anest subalgebra. Finally we give an example of a subalgebra in a von Neumann algebra (certainlyinfinite dimensional), which has factorization but it is not a nest subalgebra.2. Definitions, examples and main results
All Hilbert spaces considered in this paper are complex and separable. Throughout B ( H ) denotesthe algebra of all bounded operators on a Hilbert space H . By subspaces, projections and operatorson H , we mean closed subspaces, orthogonal projections and bounded operators respectively. Forany subspace F contained in a subspace E , we denote the orthogonal complement of F in E by E ⊖ F . We write E ⊥ for the orthogonal complement of E i.e. E ⊥ = H ⊖ E . The projectiononto a subspace E is denoted by P E . For any projection p , we write p ⊥ for the projection 1 − p ,where 1 is the identity operator on H . If { p i } i ∈ Λ is a collection of projections, then ∨ i ∈ Λ p i denotesthe projection onto the smallest subspace containing ranges of all p ′ i s , and ∧ i ∈ Λ p i denotes theprojection onto the intersection of ranges of all p ′ i s . For any operator x in B ( H ), ker x and R ( x )denote the kernel and range of x respectively. All algebras considered will be subalgebras of B ( H ),which are always assumed to be norm closed, and contain the identity operator which we shalldenote by 1. Unless said otherwise, convergence of any sequence of operators is taken in normtopology.We briefly recall some basic notions of von Neumann algebra theory. A von Neumann algebra is a self-adjoint subalgebra of B ( H ) containing 1 and closed in weak operator topology (WOT).A von Neumann algebra M is called factor if its center M ∩ M ′ is trivial. Here M ′ denotesthe commutant of M in B ( H ). Let p, q be two projections in M . Then p and q are said to be (Murray-von Neumann) equivalent , and denoted p ∼ q , if there exists a partial isometry v ∈ M such that v ∗ v = p and vv ∗ = q . We say p (cid:22) q if there is a projection q ∈ M such that q ≤ q ATTICES OF LOGMODULAR ALGEBRAS 3 and p ∼ q . Here ≤ denotes the usual order of self-adjoint operators, and < will denote the strictorder. A projection p ∈ M is called finite if the only projection q in M such that q ≤ p and q ∼ p is p . The von Neumann algebra M is called finite if 1 ∈ M is finite. Note that if p is a projectionin M , then p M p is a von Neumann algebra which is ∗ -isomorphic to a von Neumann subalgebraof B ( K ), where K is the range subspace of the projection p . See [6] for more details on these topics.We now define some notations relevant to our results. Fix a von Neumann algebra M , whichis always assumed to be acting on a separable Hilbert space. Let A be a norm closed subalgebra(not necessarily self-adjoint) of M . We denote by A ∗ the set { x ∈ M ; x ∗ ∈ A} , and by A − theset { x ∈ A ; x is invertible with x − ∈ A} . Let M − denote the set of all positive and invertibleelements of M . Note that all these notations make sense for any C ∗ -algebra. But our main focusin this paper is on von Neumann algebras.For M and A as above, let Lat M A denote the lattice of all projections in M whose ranges areinvariant under every operator in A i.e.Lat M A = { p ∈ M ; p = p = p ∗ and ap = pap ∀ a ∈ A} . If M = B ( H ), we denote Lat M A simply by Lat A . Note that if A is also considered as a subalgebraof B ( H ) (where M ⊆ B ( H )), then we have Lat M A = M ∩
Lat A . Also note that 0 , ∈ Lat M A and Lat M A is closed under the operations ∨ and ∧ of arbitrary collection as well as closed understrong operator topology (SOT).Dually, let E be a collection of projections in M (not necessarily a lattice), and let Alg M E (or Alg E when M = B ( H )) denote the algebra of all operators in M which leave range of everyprojection of E invariant i.e. Alg M E = { x ∈ M ; xp = pxp ∀ p ∈ E} . Again we note that Alg M A = M ∩
Alg E . Also it is clear that Alg M E is a unital subalgebra of M , which is closed in weak operator topology.Following [11, 4], we now consider the following definitions. Definition 2.1.
Let A be a subalgebra of a C ∗ -algebra M . Then A is called logmodular or haslogmodularity in M if the set { a ∗ a ; a ∈ A − } is norm dense in M − . The algebra A is said tohave factorization or strong logmodularity in M if { a ∗ a ; a ∈ A − } = M − .It is clear that any algebra having factorization is logmodular. Below we collect some knownand straightforward results about logmodular algebras whose proof is simple, so it is left to thereader. (See Proposition 4.6, [4]). Proposition 2.2.
Let φ : M → N be a ∗ -isomorphism between two C ∗ -algebras, and let A bea subalgebra of M . Then A has logmodularity (resp. factorization) in M if and only if φ ( A ) has logmodularity (resp. factorization) in N . In particular if U is an appropriate unitary, then U ∗ A U has logmodularity (resp. factorization) in U ∗ M U if and only if A has logmodularity (resp.factorization) in M . Proposition 2.3. (Proposition 4.1, [4] ) Let A be a subalgebra of a C ∗ -algebra M . We have thefollowing:(1) A has factorization in M if and only if A ∗ has factorization in M if and only if for everyinvertible element x ∈ M , there exist unitaries u, v ∈ M and invertible elements a, b ∈ A − such that x = ua = bv .(2) A is logmodular in M if and only if A ∗ is logmodular in M if and only if for each invertibleelement x ∈ M , there exist sequences { u n } , { v n } of unitaries in M and invertible elements { a n } , { b n } in A − such that x = lim n u n a n = lim n b n v n . There are plenty of such algebras known in literature. The following are examples of logmodularalgebras in commutative C ∗ -algebras. Example 2.4. (Function algebras) A classical result of Szeg¨o (Theorem 25.13, [6]) says that theHardy algebra H ∞ ( T ) has factorization in L ∞ ( T , µ ). Here T is the unit circle, µ is the one-dimensional Lebesgue measure on T and H ∞ ( T ) is the algebra of all essentially bounded functionson T whose negative Fourier coefficients are zero. B.V. RAJARAMA BHAT AND MANISH KUMAR
More generally, let m be a probability measure, and let A be a unital subalgebra of L ∞ ( m )satisfying the following: (i) R f gdm = R f dm R gdm for all f, g ∈ A , and ( ii ) if h ∈ L ( m ) with h ≥ R f hdm = R f dm for all f ∈ A , then h = 1 a.e. Let H ( m ) be the closure of A inthe Hilbert space L ( m ), and let H ∞ ( m ) = H ( m ) ∩ L ∞ ( m ). Then the proof of Theorem 4 in [12]says that H ∞ ( m ) has factorization in L ∞ ( m ). The algebra H ∞ ( m ) satisfies many other equivalentconditions analogues to classical Hardy space theory (see Theorem 3.1, [25] for details). Also see[11, 12, 25] for more concrete examples of such measures and algebras. Example 2.5. (Dirichlet algebras) A closed unital subalgebra A of a commutative C ∗ -algebra C ( X ) is called Dirichlet algebra if A + ¯ A is uniformly dense in C ( X ) (equivalently, Re A is uniformlydense in Re C ( X )), where Re A [Re C ( X )] denotes the set of real parts of the functions in A [ C ( X )].If A is a Dirichlet algebra, then since log |A − | ⊆ Re A , it is immediate that log |A − | is densein Re C ( X ); hence A is a logmodular subalgebra of C ( X ). But some Dirichlet algebras may nothave factorization. For example, consider the algebra A ( D ) of all continuous functions on theclosed unit disc D which is holomorphic on the open unit disc D . Then A ( D ) is a Dirichlet algebrawhen considered as the subalgebra of C ( T ), which is a consequence of Fej´er-Riesz Theorem onfactorization of positive trigonometric polynomials, but A ( D ) does not have factorization in C ( T ).On the other hand, H ∞ ( T ) is an example of an algebra which has factorization, but which is nota Dirichlet algebra. See [11] for details of these facts and more concrete examples of Dirichletalgebras.To see some examples and other properties of noncommutative algebras having factorization,we recall some notions to this end. Let M be a von Neumann algebra, and let E be a latticeof projections in M (i.e. E is closed under usual lattice operations ∨ and ∧ of finite family ofprojections). Then E is called complete if 0 , ∈ E , and ∨ i ∈ Λ p i and ∧ i ∈ Λ p i ∈ E for any arbitraryfamily { p i } i ∈ Λ in E . The lattice E is called commutative subspace lattice (CSL) if projections of E commute with one another. Moreover, E is called nest if E is ordered by usual operator orderingi.e. for any p, q ∈ E , either p ≤ q or q ≤ p holds true. We remark here that some authors assumea nest or a CSL to be always complete.A subalgebra A is called a nest subalgebra of M (or nest algebra when M = B ( H )) if A =Alg M E for a nest E in M . Further, a subalgebra A of M is called M - reflexive (or reflexive when M = B ( H )) if A = Alg M Lat M A . It is clear that any subalgebra of M of the form Alg M E forsome collection E of projections in M is always M -reflexive. In particular, a nest subalgebra of M is M -reflexive. It should be noted here that if M ⊆ B ( H ), then a subalgebra A of M can bereflexive in B ( H ), but it need not be M -reflexive.The following is a well-known result by Larson [15] regarding factorization property of nestalgebras. Theorem 2.6. (Theorem 4.7, [15] ) Let E be a complete nest in a separable Hilbert space H . Then Alg E has factorization in B ( H ) if and only if E is countable. The following are some examples of algebras having factorization in noncommutative von Neu-mann algebras.
Example 2.7. (Nest subalgebras) As already mentioned above, Alg E has factorization in B ( H )for any countable complete nest E in B ( H ). More generally, Pitts proved that if E is a completenest in a factor M , then Alg M E has factorization in M if and only if “certain” subnest E r of E iscountable (Theorem 6.4, [19]).Moreover, if E is a nest (not necessarily countable) in a finite von Neumann algebra M (notnecessarily a factor), then Alg M E has factorization in M (Corollary 5.11, [19]). Example 2.8. (Subdiagonal algebras) Let A be a subalgebra of a von Neumann algebra M , andlet φ : M → M be a faithful normal expectation (i.e. φ is positive, φ (1) = 1 and φ ◦ φ = φ ). Then A is called a subdiagonal algebra with respect to φ if it satisfies: ( i ) A + A ∗ is σ -weakly dense in M , ( ii ) φ ( ab ) = φ ( a ) φ ( b ) for all a, b ∈ A , and ( iii ) φ ( A ) ⊆ A ∩ A ∗ . Moreover, if the von Neumannalgebra M is finite with a distinguished trace τ , then the subdiagonal algebra A is called finite if τ ◦ φ = τ . Arveson proved that if A is a maximal (with respect to φ ) finite subdiagonal algebra of ATTICES OF LOGMODULAR ALGEBRAS 5 M , then A has factorization in M (Theorem 4.2.1, [2]). A nest subalgebra of a finite von Neumannalgebra is an example of maximal finite subdiagonal algebras (Corollary 3.1.2, [2]). See 4.10 foranother concrete example of a finite subdiagonal algebra.There are other subdiagonal algebras (not necessarily finite) as well, which are known to havefactorization. For example, all subdiagonal algebras arising out of periodic flow have factorization.See [23] for more details of these notions and Corollary 3.11 therein.We believe that some known facts about subdiagonal algebras can also be deduced from ourresult. One such is Theorem 5.1 in [17], which follows directly from Corollary 2.11. But we havenot explored other possible consequences in depth.Below we have some concrete examples of nest algebras which do not have factorization. Example 2.9.
Let E be the nest { p t ; t ∈ [0 , } of projections on L ([0 , p t denotes theprojection onto L ([0 , t ]), considered as subspace of L ([0 , E is complete and uncountable;hence Alg E does not have factorization in B ( L ([0 , F = { p i ; i ∈ Q } be the nest of projections on ℓ ( Q ), where p i denotes the projection onto the subspacespan { e j ; j ≤ i } for the canonical basis { e i ; i ∈ Q } of ℓ ( Q ). Although F is a countable nest, it iseasy to verify that its completion is not countable (actually indexed by R ⊔ Q ) and hence Alg F does not have factorization in B ( ℓ ( Q )). At this point, we do not know whether these algebras arelogmodular.We now state the main result of this paper. This gives us the understanding of behaviour ofthe lattices of logmodular algebras. Theorem 2.10.
Let A be a logmodular algebra in a von Neumann algebra M . Then Lat M A is acommutative subspace lattice. Moreover if M is a factor, then Lat M A is a nest. We postpone the proof of Theorem 2.10 to the next section, and instead look at some of its con-sequences first. Note that since any algebra having factorization is also logmodular, the followingcorollary is immediate.
Corollary 2.11.
Let an algebra A have factorization in a von Neumann algebra M . Then Lat M A is a commutative subspace lattice. Moreover if M is a factor, then Lat M A is a nest. Remark 2.12. If M is an arbitrary von Neumann algebra which is not a factor, and A is asubalgebra of M , then we can never expect Lat M A to be a nest irrespective of whether A islogmodular or has factorization. In fact if P Z denotes the lattice of all projections in the center Z of M , then it is always true that P Z ⊆ Lat M A . So Lat M A can never be a nest if the center Z is non-trivial.Now let M be a factor, and let A be an M -reflexive subalgebra of M . If A is logmodular in M , then Lat M A is a nest by Theorem 2.10. But since A = Alg M Lat M A , it follows that A is anest subalgebra of M .We now answer an open question posed by Paulsen and Raghupathi (see pg. 2630, [18]) usingabove observation. They conjectured that every completely distributive CSL logmodular algebrain B ( H ) is a nest algebra. Here a (completely distributive) CSL algebra means an algebra of theform Alg E , where E is a (completely distributive) commutative subspace lattice (see [7] for moreon completely distributive CSL algebras). Note that all nests are completely distributive. Sinceany CSL algebra is a special case of reflexive algebras, we have thus answered their question inaffirmative. We record it below. Corollary 2.13. An M -reflexive logmodular algebra in a factor M is a nest subalgebra of M . Inparticular, all reflexive (hence completely distributive CSL) logmodular algebras in B ( H ) are nestalgebras. If an algebra A has factorization in B ( H ), then Alg Lat A also has factorization in B ( H ) as A is contained in Alg Lat A . Since Lat A is a complete nest, it then follows from Theorem 2.6 thatLat A is a countable nest. In particular, if A = Alg E for a lattice E of projections in H , then E isa countable nest because E ⊆
Lat A . Thus we get the following corollary, which is a strengtheningof Theorem 2.6 of Larson. B.V. RAJARAMA BHAT AND MANISH KUMAR
Corollary 2.14.
Let E be a complete lattice of projections on a separable Hilbert space H . Then Alg E has factorization in B ( H ) if and only if E is a countable nest. Proof of the main result
This section is devoted to the proof of our main result (Theorem 2.10). We first discuss somegeneral ingredients required for this. A simple observation that we shall be using throughout thearticle is the following remark. Recall that p ⊥ denotes the projection 1 − p for any projection p . Remark 3.1.
For any subalgebra A of a von Neumann algebra M , p ∈ Lat M A ⇐⇒ ap = pap ∀ a ∈ A ⇐⇒ pa ∗ = pa ∗ p ∀ a ∈ A ⇐⇒ a ∗ p ⊥ = p ⊥ a ∗ p ⊥ ∀ a ∈ A ⇐⇒ p ⊥ ∈ Lat M A ∗ .The following proposition says that logmodularity and factorization are preserved under com-pression of algebras by appropriates projections. Here p A p denotes the subspace { pap ; a ∈ A} forany projection p and an algebra A . Note that p A p need not always be an algebra. Proposition 3.2.
Let an algebra A have logmodularity (resp. factorization) in a von Neumannalgebra M , and let p, q ∈ Lat M A be such that p ≥ q . Then the following statements are true:(1) p A p (= A p ) has logmodularity (resp. factorization) in p M p .(2) p ⊥ A p ⊥ has logmodularity (resp. factorization) in p ⊥ M p ⊥ .(3) ( p − q ) A ( p − q ) has logmodularity (resp. factorization) in ( p − q ) M ( p − q ) .Proof. We shall prove only part (3). Part (1) follows from (3) by taking q = 0, and (2) followsfrom (3) by taking p = 1 and q = p . Also we shall prove only the case of logmodularity. That offactorization follows similarly. So assume that A is logmodular in M .First we show that ( p − q ) A ( p − q ) is an algebra. For all a ∈ A , since ap = pap and aq = qaq ,we note that ( p − q ) aq = ( p − q ) qaq = 0 , and pa ( p − q ) = pap − paq = ap − pqaq = ap − qaq = ap − aq = a ( p − q ) . (3.1)Combining the two expressions above, it follows for all a, b ∈ A that( p − q ) a ( p − q ) b ( p − q ) = ( p − q ) apb ( p − q ) − ( p − q ) aqb ( p − q ) = ( p − q ) ab ( p − q ) , (3.2)which shows that ( p − q ) A ( p − q ) is an algebra.To show that ( p − q ) A ( p − q ) is logmodular in ( p − q ) M ( p − q ), fix a positive and invertibleelement x in ( p − q ) M ( p − q ) and set ˜ x = x + q + p ⊥ . Note that x = ( p − q )˜ x ( p − q ). It is clear that˜ x is positive in M . Since x is positive and invertible in ( p − q ) M ( p − q ), there is some α ∈ (0 , x ≥ α ( p − q ), from which we get˜ x = x + q + p ⊥ ≥ α ( p − q ) + αq + αp ⊥ = α. This shows that ˜ x is invertible in M . We then use logmodularity of A in M to get a sequence { ˜ a n } in A − such that ˜ x = lim n ˜ a ∗ n ˜ a n . So for each n , we have ˜ a n q = q ˜ a n q and ˜ a − n q = q ˜ a − n q . Itthen follows that( q ˜ a n q )( q ˜ a − n q ) = q ˜ a n ˜ a − n q = q and ( q ˜ a − n q )( q ˜ a n q ) = q ˜ a − n ˜ a n q = q, which is to say that q ˜ a n q is invertible in q M q with ( q ˜ a n q ) − = q ˜ a − n q ∈ q A q . In particular,since the sequence { ˜ a − n } is bounded (as { (˜ a ∗ n ˜ a n ) − } is a convergent sequence), it follows that thesequence { ( q ˜ a n q ) − } is bounded. Note that q ˜ x ( p − q ) = 0, and since q ˜ a ∗ n = q ˜ a ∗ n q for all n , we have0 = q ˜ x ( p − q ) = lim n q ˜ a ∗ n ˜ a n ( p − q ) = lim n ( q ˜ a ∗ n q )( q ˜ a n ( p − q )) . Multiplying to the left of the sequence by ( q ˜ a ∗ n q ) − yieldslim n q ˜ a n ( p − q ) = 0 , ATTICES OF LOGMODULAR ALGEBRAS 7 using which and the expression ˜ a n ( p − q ) = p ˜ a n ( p − q ) from (3.1), we get the following: x = ( p − q )˜ x ( p − q ) = lim n ( p − q )˜ a ∗ n ˜ a n ( p − q ) = lim n ( p − q )˜ a ∗ n p ˜ a n ( p − q )= lim n ( p − q )˜ a ∗ n [ q ˜ a n ( p − q )] + lim n ( p − q )˜ a ∗ n [( p − q )˜ a n ( p − q )]= lim n ( p − q )˜ a ∗ n ( p − q )˜ a n ( p − q ) = lim n a ∗ n a n , where a n = ( p − q )˜ a n ( p − q ) ∈ ( p − q ) A ( p − q ). Also for each n , we have from (3.2) that p − q = ( p − q )˜ a − n ( p − q )˜ a n ( p − q ) = ( p − q )˜ a n ( p − q )˜ a − n ( p − q ) , which shows that a n = ( p − q )˜ a n ( p − q ) is invertible with inverse ( p − q )˜ a − n ( p − q ) in ( p − q ) A ( p − q ).Thus we get a sequence { a n } of invertible elements with a n , a − n ∈ ( p − q ) A ( p − q ) for all n suchthat x = lim n a ∗ n a n . Since x is an arbitrary positive and invertible element, we conclude that( p − q ) A ( p − q ) is logmodular in ( p − q ) M ( p − q ). (cid:3) We now recall some basic facts about subspaces in a separable Hilbert space. Following Halmos[10], we say two non-zero subspaces E and F of a Hilbert space are in generic position if all thefollowing subspaces E ∩ F, E ∩ F ⊥ , E ⊥ ∩ F and E ⊥ ∩ F ⊥ are zero. We are going to use the following characterization of subspaces in generic position. Recallthat P E denotes the projection onto a subspace E . Also recall that ker x denotes the kernel of anyoperator x . Lemma 3.3. (Theorem 2, [10] ) Let E and F be two subspaces in generic position in a Hilbertspace H . Then there exist a Hilbert space K , a unitary U : H → K ⊕ K , and commuting positivecontractions x, y ∈ B ( K ) such that x + y = 1 and ker x = ker y = 0 and U P E U ∗ = (cid:20) (cid:21) and U P F U ∗ = (cid:20) x xyxy y (cid:21) . The following lemma is very easy to verify, whose proof is left to the readers.
Lemma 3.4.
Let E and F be two subspaces in a Hilbert space H , and let H denote the subspaceof H given by H = H ⊖ (cid:0) E ∩ F + E ∩ F ⊥ + E ⊥ ∩ F + E ⊥ ∩ F ⊥ (cid:1) . If E = E ∩ H and F = F ∩ H , then exactly one of the following holds true:(1) H = { } , and hence E , F = 0 .(2) E and F are non-zero, and they are in generic position as subspaces of H .Moreover, the projections P E and P F commute if and only if the first condition is satisfied (i.e. H = 0 ). The subspaces E and F as in Lemma 3.4 are called generic part of the subspaces E and F .The structure of general subspaces can now be described using the two lemmas above. Proposition 3.5.
Let E and F be two subspaces in a Hilbert space H . Then there is a Hilbertspace K (could be zero), and commuting positive contractions x, y ∈ B ( K ) with x + y = 1 , ker x =ker y = 0 such that, upto unitary equivalence H = E ∩ F ⊕ E ∩ F ⊥ ⊕ E ⊥ ∩ F ⊕ E ⊥ ∩ F ⊥ ⊕ K ⊕ K , and P E = 1 ⊕ ⊕ ⊕ ⊕ ⊕ and P F = 1 ⊕ ⊕ ⊕ ⊕ (cid:20) x xyxy y (cid:21) . Here any of the components in the decomposition could be . Moreover, P E P F = P F P E = P E ∩ F ifand only if K = { } . B.V. RAJARAMA BHAT AND MANISH KUMAR
We are now ready to give proof of our main result through a series of lemmas. The next twolemmas deal with factor von Neumann algebras only, where we use the following comparison theo-rem of projections (see Corollary 47.9, [6]): If M is a factor and p, q are two non-zero projectionsin M , then either p (cid:22) q or q (cid:22) p i.e. there is a non-zero partial isometry v ∈ M such that v ∗ v ≤ p and vv ∗ ≤ q . The same is clearly not true for arbitrary von Neumann algebras. Lemma 3.6.
Let M be a factor, and let p, q be mutually orthogonal projections in M . Then Alg M { p, q } is not logmodular in M .Proof. Since M is a factor and p, q ∈ M are non-zero, it follows that there is a non-zero partialisometry v ∈ M such that v ∗ v ≤ p and vv ∗ ≤ q . In particular, we have v = qv = vp , and since pq = 0 it follows that pv = pqv = 0.Assume to the contrary that Alg M { p, q } is logmodular in M . Let x = 1 + ǫ ( v + v ∗ ) for some ǫ >
0, where ǫ is chosen small enough so that x is positive and invertible in M . Then there existsa sequence { a n } of invertible elements in Alg M { p, q } such that x = lim n a ∗ n a n . Now since v ∗ p = 0and pq = 0, we note that qxp = ǫqvp = ǫv . We also have a n p = pa n p and qa ∗ n = qa ∗ n q for all n ;thus we get ǫv = qxp = lim n qa ∗ n a n p = lim n qa ∗ n qpa n p = 0 , which is a contradiction, as v = 0. (cid:3) We recall here a simple fact that if p and q are commuting projections, then pq is a projectionsuch that p ∧ q = pq and p ∨ q = p + q − pq . Lemma 3.7.
Let M be a factor, and let p, q ∈ M be two commuting projections. If Alg M { p.q } is logmodular in M , then either p ≤ q or q ≤ p holds true.Proof. Since p and q commuting projections, it follows that the operators pq, pq ⊥ and p ⊥ q areprojections. From Lemma 3.6, we know that pq = 0. Note that the required assertion will followby the following argument once we show that either pq ⊥ = 0 or p ⊥ q = 0: say pq ⊥ = 0, then p = p ( q + q ⊥ ) = pq which implies that p ≤ q . Similarly, p ⊥ q = 0 will imply q ≤ p .Assume to the contrary that both the projections pq ⊥ and p ⊥ q are non-zero. Since M is a factor,there is a non-zero partial isometry v ∈ M such that v ∗ v ≤ pq ⊥ and vv ∗ ≤ p ⊥ q ; in particular wehave, v = vpq ⊥ = p ⊥ qv. (3.3)Now let x = 1+ ǫ ( v + v ∗ ) for ǫ >
0, where we choose ǫ small enough so that x is positive and invertiblein M . Since Alg M { p, q } is logmodular in M , there exists a sequence { a n } of invertible elementsin M such that a n , a − n ∈ Alg M { p, q } for all n and x = lim n a ∗ n a n . Note that pqa n pq = a n pq and pqa − n pq = a − n pq for all n ; hence pqa n pq is invertible in pq M pq with inverse pqa − n pq . Inparticular, the sequence { ( pqa n pq ) − } is bounded, as the sequence { a − n } is bounded. Also notefrom (3.3) that vpq = 0 and p ⊥ qv ∗ = 0; hence we get p ⊥ qxpq = p ⊥ qpq + ǫp ⊥ q ( vpq ) + ǫ ( p ⊥ qv ∗ ) pq = 0 . Thus we have 0 = p ⊥ qxpq = lim n p ⊥ qa ∗ n a n pq = lim n p ⊥ qa ∗ n pqa n pq, and so by multiplying by { ( pqa n pq ) − } to right side of the sequence, we get lim n p ⊥ qa ∗ n pq = 0;using which and the expressions qa ∗ n = qa ∗ n q and a n p = pa n p for all n , it follows that p ⊥ qxpq ⊥ = lim n p ⊥ qa ∗ n a n pq ⊥ = lim n ( p ⊥ qa ∗ n qp ) a n pq ⊥ = 0 . On the other hand, we again use the equation p ⊥ qv ∗ = 0 to get p ⊥ qxpq ⊥ = p ⊥ qpq ⊥ + ǫp ⊥ qvpq ⊥ + ǫp ⊥ qv ∗ pq ⊥ = ǫp ⊥ qvpq ⊥ = ǫv = 0 . So we get a contradiction, which arose because we assumed that both pq ⊥ and p ⊥ q are non-zero.Thus one of them is zero and we have the required result. (cid:3) We are going to use the following simple lemma very frequently.
ATTICES OF LOGMODULAR ALGEBRAS 9
Lemma 3.8.
Let { a n } be a sequence of invertible elements in a C ∗ -algebra M such that lim n a ∗ n a n =1 . Then { a − n } is bounded and lim n a n a ∗ n = 1 .Proof. Since lim n a ∗ n a n = 1, it follows that lim n ( a ∗ n a n ) − = 1 and so { ( a ∗ n a n ) − } is bounded. Thisimplies the first assertion that { a − n } is bounded. Further we have lim n a ∗ n a n a ∗ n a n = 1, and hence0 = lim n ( a ∗ n a n a ∗ n a n − a ∗ n a n ) = lim n a ∗ n ( a n a ∗ n − a n . Since the sequence { a − n } is bounded, it follows by multiplying a ∗ n − to the left and a − n to theright of the sequence that lim n ( a n a ∗ n −
1) = 0 , as to be proved. (cid:3) Next we consider lattices of logmodular algebras in arbitrary von Neumann algebras, where ouraim is to prove that the generic part of any two invariant subspaces is zero. Recall that R ( x )denotes the range of an operator x . Lemma 3.9.
Let M be a von Neumann subalgebra of B ( H ) for some Hilbert H , and let p, q be twonon-zero projections in M such that R ( p ) and R ( q ) are in generic position in H . Then Alg M { p, q } is not logmodular in M .Proof. Assume to the contrary that the algebra Alg M { p, q } is logmodular in M . Since R ( p ) and R ( q ) are in generic position in H , it follows from Lemma 3.3 that there exist a Hilbert space K and commuting positive contractions x, y ∈ B ( K ) satisfying ker x = 0 , ker y = 0 and x + y = 1such that upto unitary equivalence, we have H = K ⊕ K and p = (cid:20) (cid:21) and q = (cid:20) x xyxy y (cid:21) . (3.4)Since logmodularity is preserved under unitary equivalence by Proposition 2.2, we can assumewithout loss of generality that M is a von Neumann subalgebra of B ( K ⊕ K ), and p, q are of theform as in (3.4).Now let S be an invertible operator such that S, S − ∈ Alg M { p, q } . Then Sp = pSp and S − p = pS − p , which imply that S and S − have the following form: S = (cid:20) a b c (cid:21) and S − = (cid:20) a ′ b ′ c ′ (cid:21) , for some operators a, b, c, a ′ , b ′ , c ′ ∈ B ( K ). It is then clear from the expression SS − = 1 = S − S that a and c are invertible in B ( K ) with respective inverses a ′ and c ′ . Note that Sq = (cid:20) ax + bxy axy + by cxy cy (cid:21) and qSq = (cid:20) x ax + x bxy + xycxy x axy + x by + xycy xyax + xybxy + y cxy xyaxy + xyby + y cy (cid:21) . Since Sq = qSq , we equate (2 ,
1) entries of the matrices Sq and qSq , and use the equation 1 − y = x to get the expression x cxy = xyax + xybxy. Since x is injective (and hence x has dense range,as x is positive), x can be cancelled from both the sides to get the following: xcy = yax + yby. (3.5)Now fix α ≥
1, and let Z = (cid:20) αα α + 1 (cid:21) ∈ B ( K ⊕ K ). It is clear that Z is a positive and invertibleoperator. We claim that Z ∈ M . Since p and q are in M , it follows that (cid:20) x
00 0 (cid:21) = pqp ∈M . Similarly (cid:20) y (cid:21) = p ⊥ qp ⊥ ∈ M . Thus (cid:20) x y (cid:21) ∈ M and hence (cid:20) xyxy (cid:21) ∈ M . Set T = (cid:20) xyxy (cid:21) and let T = U | T | be its polar decomposition, where | T | denotes the square rootof the operator T ∗ T . It is clear that T is one-one (as xy is one-one), and so U is unitary. Itis straightforward to check (using uniqueness of polar decomposition) that | T | = (cid:20) xy xy (cid:21) and U = (cid:20) (cid:21) . Since M is a von Neumann algebra, it follows that U ∈ M and so (cid:20) αα (cid:21) = αU ∈ M .Also since (cid:20) α + 1 (cid:21) = p + ( α + 1) p ⊥ ∈ M , we conclude that Z ∈ M , as claimed.Thus by logmodularity of Alg M { p, q } in M , we get a sequence { S n } of invertible operators with S n , S − n ∈ Alg M { p, q } for all n such that Z = lim n S ∗ n S n . It then follows from above discussionthat for each n , S n is of the form S n = (cid:20) a n b n c n (cid:21) , for a n , b n , c n ∈ B ( K ) such that a n and c n are invertible operators, and from (3.5) we have xc n y = ya n x + yb n y. (3.6)Now we have (cid:20) αα α + 1 (cid:21) = Z = lim n S ∗ n S n = lim n (cid:20) a ∗ n a n a ∗ n b n b ∗ n a n b ∗ n b n + c ∗ n c n (cid:21) . (3.7)So we get lim n a ∗ n a n = 1, and since each a n is invertible, it follows from Lemma 3.8 thatlim n a n a ∗ n = 1 . (3.8)We also get from (3.7) that lim n a ∗ n b n = α , which further yields by multiplying a n to the left sideof the sequence and using (3.8) that lim n ( b n − αa n ) = 0 . (3.9)Set d n = b n − αa n for all n . Then lim n d n = 0, and since lim n a ∗ n a n = 1 we havelim n b ∗ n b n = lim n ( d n + αa n ) ∗ ( d n + αa n ) = lim n α a ∗ n a n = α . We further get from (3.7) that α + 1 = lim n ( b ∗ n b n + c ∗ n c n ), which yields lim n c ∗ n c n = 1. Again aseach c n is invertible, it follows from Lemma 3.8 thatlim n c n c ∗ n = 1 . (3.10)Now from (3.6) and using b n = αa n + d n , we have xc n y = ya n x + yb n y = ya n x + αya n y + yd n y = ya n ( x + αy ) + yd n y = ya n u + yd n y, where u = x + αy . Since α ≥
1, we note that u is positive and invertible (in fact u = 1 + ( α − y + 2 αxy ≥ ya n = xc n yu − − yd n yu − . (3.11)Note that u = ( x + αy ) = x + α y + 2 αxy ≥ α y and since y and u commutes, it follows that y u − ≤ /α . (3.12)Thus we use the expression lim n d n = 0 from (3.9), and equations in (3.8), (3.10), (3.11) and (3.12)to get the following: y = lim n ya n a ∗ n y = lim n ( ya n )( ya n ) ∗ = lim n ( xc n yu − − yd n yu − )( xc n yu − − yd n yu − ) ∗ = lim n ( xc n yu − )( xc n yu − ) ∗ = lim n xc n y u − c ∗ n x ≤ lim n α xc n c ∗ n x = 1 α x . Since α ≥ α tend to ∞ that y = 0, which is clearly not true.Thus our assumption that Alg M { p, q } is logmodular is false completing the proof. (cid:3) Finally we prove our main theorem in full generality, for which we consider the following lemma.
Lemma 3.10.
Let an algebra A have logmodularity (resp. factorization) in a von Neumannalgebra M , and let p, q ∈ Lat M A . If r = ( p ∧ q ) ∨ ( p ⊥ ∧ q ⊥ ) , then r ⊥ A r ⊥ has logmodularity (resp.factorization) in r ⊥ M r ⊥ . ATTICES OF LOGMODULAR ALGEBRAS 11
Proof.
Set r = p ∧ q and r = p ⊥ ∧ q ⊥ . It is clear that r r = 0 and r = r + r . Since p, q ∈ Lat M A , it follows that r ∈ Lat M A . Also we note that p ⊥ , q ⊥ ∈ Lat M A ∗ and hence r ∈ Lat M A ∗ , which is to say that r ⊥ ∈ Lat M A . Note that r ⊥ = 1 − r − r = r ⊥ − r , and so r ≤ r ⊥ . Both the assertions about logmodularity and factorization now follow from part (3) ofProposition 3.2. (cid:3) Proof of Theorem 2.10.
Let A be a logmodular subalgebra of a von Neumann algebra M , and let p, q ∈ Lat M A . We have to show that pq = qp . The second assertion that p ≤ q or q ≤ p whenever M is a factor, will then follow from Lemma 3.7.Set r = ( p ∧ q ) ∨ ( p ⊥ ∧ q ⊥ ). Then r ⊥ A r ⊥ is a logmodular algebra in r ⊥ M r ⊥ by Lemma 3.10.Note that the projections p and q commute with r , and hence with r ⊥ . So if we set p ′ = r ⊥ pr ⊥ and q ′ = r ⊥ qr ⊥ , then it is immediate that p ′ , q ′ are projections in r ⊥ M r ⊥ , and we have p ′ = p ∧ r ⊥ and q ′ = q ∧ r ⊥ . Note that pq ( p ∧ q ) = p ∧ q = qp ( p ∧ q ) and pq ( p ⊥ ∧ q ⊥ ) = 0 = qp ( p ⊥ ∧ q ⊥ ); hence pqr = p ∧ q = qpr , which further yields pq = pq ( r + r ⊥ ) = pqr + pqr ⊥ = p ∧ q + ( r ⊥ pr ⊥ )( r ⊥ qr ⊥ ) = p ∧ q + p ′ q ′ ,qp = qpr + qpr ⊥ = p ∧ q + ( r ⊥ qr ⊥ )( r ⊥ pr ⊥ ) = p ∧ q + q ′ p ′ . Therefore, in order to show the required assertion it is enough to prove that p ′ q ′ = q ′ p ′ . Also wenote that p ′ ∧ q ′ = p ∧ q ∧ r ⊥ ≤ r ∧ r ⊥ = 0 , and( r ⊥ − p ′ ) ∧ ( r ⊥ − q ′ ) = ( r ⊥ − pr ⊥ ) ∧ ( r ⊥ − qr ⊥ ) = p ⊥ r ⊥ ∧ q ⊥ r ⊥ = ( p ⊥ ∧ q ⊥ ) ∧ r ⊥ ≤ r ∧ r ⊥ = 0 . Here r ⊥ − p ′ and r ⊥ − q ′ are the orthogonal complement of the projections p ′ and q ′ in r ⊥ M r ⊥ respectively. Thus if necessary, by replacing the algebras M and A by r ⊥ M r ⊥ and r ⊥ A r ⊥ respectively, and the projections p, q by p ′ , q ′ respectively we assume without loss of generalitythat p ∧ q = 0 = p ⊥ ∧ q ⊥ , (3.13)so that r = 0 and M = r ⊥ M r ⊥ . The purpose of reducing M to r ⊥ M r ⊥ is just to avoid multiplecases, and work with 4 × × pq = qp . Then the generic part of R ( p ) and R ( q ) in H arenon-zero by Proposition 3.5, where H is the Hilbert space on which the von Neumann algebra M acts. Further if both p ∧ q ⊥ and p ⊥ ∧ q are also zero, then (as p ∧ q = 0 = p ⊥ ∧ q ⊥ ) R ( p ) and R ( q )will be in generic position, which is not possible by Lemma 3.9 since Alg M { p, q } is logmodular in M as well. Therefore at least one of the projections p ∧ q ⊥ and p ⊥ ∧ q is non-zero. Thus for theremainder of the proof, we assume that both p ⊥ ∧ q and p ∧ q ⊥ are non-zero (the case of exactlyone of them being non-zero follows similarly). It then follows from Proposition 3.5 that there exista non-zero Hilbert space K and commuting positive contractions x, y ∈ B ( K ) satisfying x + y = 1and ker x = 0 = ker y such that upto unitary unitary equivalence, we have H = R ( p ∧ q ⊥ ) ⊕ K ⊕ K ⊕ R ( p ⊥ ∧ q ) (3.14)and p = and q = x xy xy y
00 0 0 1 . (3.15)Since logmodularity is preserved under unitary equivalence by Proposition 2.2, we assume withoutloss of generality that M is a von Neumann subalgebra of B ( R ( p ∧ q ⊥ ) ⊕ K ⊕ K ⊕ R ( p ⊥ ∧ q )), and p, q have the form as in (3.15). Now set e K = R ( p ∧ q ⊥ ) ⊕ K and e K = K ⊕ R ( p ⊥ ∧ q )so that H = e K ⊕ e K . (3.16) Throughout the proof, we make use of both the decomposition of H in (3.14) and (3.16), whichshould be understood according to the context. Now fix α ≥ Z ∈ B ( H )by Z = α α α + 1 00 0 0 1 = (cid:20) Z Z ∗ Z (cid:21) , where Z = (cid:20) α (cid:21) and Z = (cid:20) α + 1 00 1 (cid:21) . (3.17)It is clear that Z is a positive and invertible operator. In the similar fashion as in Lemma 3.9, it iseasy to show, by using p, q ∈ M , that Z ∈ M . Since A is logmodular in M , we get a sequence { S n } of invertible operators in A − such that Z = lim n S ∗ n S n . Then for each n , we have S n p = pS n p and S − n p = pS − n p ; hence the operators S n and S − n have the form S n = a n b n p n q n c n d n r n s n e n f n g n h n =: (cid:20) A n B n C n (cid:21) and S − n = a ′ n b ′ n p ′ n q ′ n c ′ n d ′ n r ′ n s ′ n e ′ n f ′ n g ′ n h ′ n =: (cid:20) A ′ n B ′ n C ′ n (cid:21) , (3.18)for appropriate operators a n , b n , ., a ′ n , b ′ n , .. etc. In particular, we have A n A ′ n = 1 = A ′ n A n i.e. A n is invertible in B ( e K ). Similarly C n is invertible in B ( e K ). Now (cid:20) Z Z ∗ Z (cid:21) = Z = lim n S ∗ n S n = lim n (cid:20) A ∗ n A n A ∗ n B n B ∗ n A n B ∗ n B n + C ∗ n C n (cid:21) . (3.19)Then we have lim n A ∗ n A n = 1 and since A n is invertible, it follows from Lemma 3.8 thatlim n A n A ∗ n = 1 . (3.20)We also have lim n A ∗ n B n = Z , which after multiplied by A n to left side of the sequence and using(3.20) yields lim n ( B n − A n Z ) = 0; but we have B n − A n Z = (cid:20) p n q n r n s n (cid:21) − (cid:20) a n b n c n d n (cid:21) (cid:20) α (cid:21) = (cid:20) p n − αb n q n r n − αd n s n (cid:21) , and thus we get the following equations:lim n ( p n − αb n ) = 0 , (3.21)lim n ( r n − αd n ) = 0 . (3.22)Also if D n = B n − A n Z for all n , then lim n D n = 0 and since lim n A ∗ n A n = 1, we havelim n B ∗ n B n = lim n ( D n + A n Z ) ∗ ( D n + A n Z ) = lim n Z ∗ A ∗ n A n Z = Z ∗ Z . Further, we have from (3.19) that lim n ( B ∗ n B n + C ∗ n C n ) = Z which yieldslim n C ∗ n C n = Z − lim n B ∗ n B n = Z − Z ∗ Z = (cid:20) α + 1 00 1 (cid:21) − (cid:20) α
00 0 (cid:21) = (cid:20) (cid:21) . (3.23)In particular, by computing the matrices C ∗ n C n , we get lim n ( e ∗ n e n + g ∗ n g n ) = 1; hence there exists m ∈ N such that k e ∗ n e n k ≤
2, which in turn yields e n e ∗ n ≤ n ≥ m. (3.24)Now S n q = b n x + p n xy b n xy + p n y q n d n x + r n xy d n xy + r n y s n e n xy e n y f n g n xy g n y h n ATTICES OF LOGMODULAR ALGEBRAS 13 and qS n q = x d n x + x r n xy + xye n xy x d n xy + x r n y + xye n y x s n + xyf n xyd n x + xyr n xy + y e n xy xyd n xy + xyr n y + y e n y xys n + y f n g n xy g n y h n . Since S n q = qS n q for each n , by equating (3 ,
2) entries of the respective matrices and then using1 − y = x we get the expression x e n xy = xyd n x + xyr n xy ; but since x is one-one and hence x has dense range, x cancels from both sides of the equation to yield xe n y = yd n x + yr n y. If we set t n = r n − αd n for all n , then above equation further yields xe n y = yd n x + yr n y = yd n x + y ( αd n + t n ) y = yd n ( x + αy ) + yt n y, which in other words says that yd n = xe n yu − − yt n yu − (3.25)where u = x + αy , which is clearly positive and invertible as u ≥
1. In a similar vein as in (3.12)in Lemma 3.9, u and y commute and we have y u − ≤ /α . (3.26)Also by equating (1 ,
2) entries of S n q and qS n q , we get b n x + p n xy = 0; again since x has denserange, it follows that b n x + p n y = 0 for all n , which by using (3.21) further yields0 = lim n ( b n x + p n y ) = lim n b n ( x + αy ) + lim n ( p n − αb n ) y = lim n b n ( x + αy ) . But x + αy is invertible as seen before, so the above equation yieldslim n b n = 0 . (3.27)Similarly since S − n also has all these properties, we havelim n b ′ n = 0 . (3.28)Note that the (2 ,
2) entry of the matrix S n S − n (with respect to the decomposition R ( p ∧ q ⊥ ) ⊕ K ⊕K ⊕ R ( p ⊥ ∧ q ))) is c n b ′ n + d n d ′ n ; hence we have c n b ′ n + d n d ′ n = 1 for all n . Since lim n b ′ n = 0 from(3.28), it follows that lim n d n d ′ n = 1 . Hence there exists n ∈ N such that k d n d ′ n − k < n ≥ n , which in particular says that d n d ′ n is invertible for all n ≥ n . Thus d n d ′ n ( d n d ′ n ) − = 1,which is to say that d n is right invertible for all n ≥ n . Likewise, from (2 ,
2) entry of S − n S n andusing lim n b n = 0 from (3.27), we get lim n d ′ n d n = 1. Again this implies that d ′ n d n is invertible,and hence d n is left invertible for large n . Thus we have shown that d n is both left and rightinvertible, which is to say that d n is invertible for large n .Now for each n , note that the (2 ,
2) entry of the matrix S ∗ n S n (with respect to the decomposition R ( p ∧ q ⊥ ) ⊕ K ⊕ K ⊕ R ( p ⊥ ∧ q ))) is b ∗ n b n + d ∗ n d n . Since lim n S ∗ n S n = Z , it then follows thatlim n ( b ∗ n b n + d ∗ n d n ) = 1 , and since lim n b n = 0 from (3.27), we get lim n d ∗ n d n = 1. But d n isinvertible for large n , so it follows from Lemma 3.8 thatlim n d n d ∗ n = 1 . (3.29)Now using lim n t n = 0 from (3.22), and equations (3.24), (3.25), (3.26) and (3.29), we get thefollowing: y = lim n yd n d ∗ n y = lim n ( yd n )( yd n ) ∗ = lim n ( xe n yu − − yt n yu − )( xe n yu − − yt n yu − ) ∗ = lim n ( xe n yu − )( xe n yu − ) ∗ = lim n xe n y u − e ∗ n x ≤ α lim n xe n e ∗ n x ≤ α x . Since α ≥ α → ∞ that y = 0, which is a contradiction. Theproof is now complete. (cid:3) Reflexivity of algebras with factorization
Our main result of this article says that lattice of any algebra with factorization property ina factor is a nest. A natural question that arises is whether algebras with factorization are alsonest subalgebras i.e. are they reflexive? Certainly, we cannot always expect automatic reflexivityof such algebras (See Example 4.10). But then what extra condition can be imposed in orderto show that they are reflexive? A result due to Radjavi and Rosenthal [22] says that a weaklyclosed algebra in B ( H ) whose lattice is a nest, is a nest algebra if and only if it contains a maximalabelian self-adjoint algebra (masa). In this section, we show that if the lattice of an algebra withfactorization in B ( H ) has finite dimensional atoms, then it contains a masa and hence it is reflexive.We recall some terminologies to this end. Let M be a von Neumann algebra, and let E be acomplete nest in M . For any projection p ∈ E , let p − = ∨{ q ∈ E ; q < p } and p + = ∧{ q ∈ E ; q > p } . An atom of E is a nonzero projection of the form p − p − for some p ∈ E with p = p − . Clearlytwo distinct atoms are mutually orthogonal. The nest E is called atomic if there is a finite orcountably infinite sequence { r n } of atoms of E such that P n r n = 1, where the sum convergesin weak operator topology (WOT). Further an algebra A in M is called M -transitive (simply transitive when M = B ( H )) if Lat M A = { , } . We now consider the following simple lemma. Lemma 4.1.
Let A be an algebra in a von Neumann algebra M such that Lat M A is a nest, andlet p, q ∈ Lat M A with p < q . If r = q − p , then Lat r M r ( r A r ) = { s ∈ r M r ; p + s ∈ Lat M A} . Inparticular, if p = q − then r A r is r M r -transitive.Proof. As seen in Proposition 3.2, r A r is a subalgebra of r M r . Now let s ∈ Lat r M r ( r A r ), and let a ∈ A . Note that ( rar ) s = s ( rar ) s , and since rs = s , it follows that ras = ( rar ) s = s ( rar ) s = sas, using which, and the expression aq = qaq , we have as = aqs = qaqs = qas = pas + ras = pas + sas = ( p + s ) as. (4.1)Also since sp = 0 and ap = pap , we have sap = spap = 0, which along with (4.1) yield( p + s ) a ( p + s ) = pap + sap + ( p + s ) as = ap + as = a ( p + s ) . Since a is arbitrary in A , it follows that p + s ∈ Lat M A . Conversely let s ∈ r M r be a projectionsuch that p + s ∈ Lat M A , and fix a ∈ A . Then a ( p + s ) = ( p + s ) a ( p + s ), and since ps = 0 = pr and rs = s , we have ( rar ) s = ras = ra ( p + s ) s = r ( p + s ) a ( p + s ) s = s ( rar ) s. Again as a ∈ A is arbitrary, we conclude that s ∈ Lat r M r ( r A r ). Thus we have proved the firstassertion. Note that if p = q − then for any s ∈ r M r , p + s ∈ Lat M A if and only if s = 0 or s = r .The second assertion then follows from the first. (cid:3) The following lemma is the crux of this section. Recall our convention that all algebras areunital and norm closed.
Lemma 4.2.
Let an algebra A have factorization in a von Neumann algebra M , and let p, q ∈ Lat M A such that p < q . If q − p has finite dimensional range, then q − p ∈ A . In particular, ifeither p or p ⊥ has finite dimensional range, then p ∈ A .Proof. The second assertion clearly follows from the first. For the first assertion, set r = q − p .Let M be a von Neumann subalgebra of B ( H ) for some Hilbert space H . Note that H = R ( p ) ⊕ R ( r ) ⊕ R ( q ⊥ ) , and we consider operators of B ( H ) with respect to this decomposition. For each n ∈ N , set X n = r + 1 n r ⊥ = /n /n . It is clear that each X n is a positive and invertible operator, and since r ∈ M it follows that X n ∈ M . So by factorization property of A in M , there exists an invertible operator S n ∈ A − such that X n = S ∗ n S n . Then each S n leaves R ( p ) and R ( q ) invariant, which equivalently says that S n has the form S n = a n b n c n d n e n f n , (4.2)for appropriate operators a n , b n .. etc. We claim that the off diagonal entries b n , c n , e n are 0 for all n . Note that since S − n ∈ A , each S − n leaves R ( p ) and R ( q ) invariant, meaning that S − n is alsoupper triangular. Consequently, the diagonal entries a n , d n , f n of S n are invertible. Now for all n ,we have /n /n = X n = S ∗ n S n = a ∗ n a n a ∗ n b n a ∗ n c n b ∗ n a n b ∗ n b n + d ∗ n d n b ∗ n c n + d ∗ n e n c ∗ n a n c ∗ n b n + e ∗ n d n c ∗ n c n + e ∗ n e n + f ∗ n f n . By equating (1 ,
2) entries of the matrices above, we have a ∗ n b n = 0 and since a n is invertible, itfollows that b n = 0. Similarly from (1 ,
3) entries, we have a ∗ n c n = 0, from which we again useinvertibility of a n to get c n = 0. Further from (2 ,
3) entries, we have b ∗ n c n + d ∗ n e n = 0. But since b n = 0 and d n is invertible, it follows that e n = 0. This proves the claim that for all n , theoperators b n , c n and e n are 0.Next equating (1 ,
1) entries of S ∗ n S n and X n we have a ∗ n a n = 1 /n for all n , which implies thatlim n a n = 0. Also from equating (3 ,
3) entries we have c ∗ n c n + e ∗ n e n + f ∗ n f n = 1 /n , so it follows that f ∗ n f n ≤ /n for all n ; hence lim n f n = 0. Further since R ( r ) is finite dimensional by hypothesis,from d ∗ n d n = 1, d n is a unitary for every n. By compactness of the unitary group in finite dimensionswe get a subsequence { d n k } converging to a unitary d ∈ B ( R ( r )) . Thus we have lim k S n k = S ,where S = d
00 0 0 . Since each S n k ∈ A and A is norm closed, it follows that S ∈ A . Note that lim k d − n k = lim k d ∗ n k = d ∗ = d − , using which we havelim k S − n k S = lim k a − n k d − n k
00 0 f − n k d
00 0 0 = lim k d ∗ n k d
00 0 0 = = r. Since S − n k S ∈ A (as S − n k and S ∈ A ) for all k , we conclude that r ∈ A , as required to prove. (cid:3) We now discuss a sufficient criterion imposed on the dimension of atoms of the lattice to provethe reflexivity of an algebra having factorization in B ( H ). It is clearly not necessary as any nestalgebra arising out of a countable nest has factorization and is reflexive.To this end, let E be a complete nest in B ( H ). Let { r n } be the collection of all atoms of E , andlet r = P n r n in WOT convergence. If r = 1, then it is straightforward to check that the nest { p ∧ r ⊥ ; p ∈ E} in B ( R ( r ⊥ )) is complete and has no atom (such nests without any atom are called continuous ). But then any continuous complete nest has to be uncountable (in fact indexed by[0 , E is countable, then r = 1 and hence E isatomic. Thus we have the following lemma: Lemma 4.3.
Let an algebra A have factorization in B ( H ) . Then Lat A is an atomic nest.Proof. Since A has factorization in B ( H ), Alg Lat A also has factorization in B ( H ) as it contains A . Consequently Lat A is a countable nest by Corollary 2.14, so it is atomic. (cid:3) Theorem 4.4.
Let A be a weakly closed algebra having factorization in B ( H ) . If all atoms of thelattice Lat A has finite dimensional range, then A is reflexive and hence A is a nest algebra.Proof. We shall show that A contains a masa. As noted above, this claim along with the fact thatLat A is a nest (from Corollary 2.11) will imply the required assertion that A is reflexive and a nestalgebra (see Theorem 9.24, [22]). Let { r i } i ∈ Λ be the collection of all the atoms of Lat A for some at most countable indexing set Λ. Since Lat A is atomic from Lemma 4.3, it follows that P i ∈ Λ r i = 1in WOT; hence H = ⊕ i ∈ Λ H i , where H i = R ( r i ) which satisfies H i ⊥ H j for all i = j . For each i ∈ Λ since r i is an atom, we note that r i = p i − q i for some p i , q i ∈ Lat A (where q i = p i − ), andsince r i has finite dimensional range by hypothesis, it follows from Lemma 4.2 that r i ∈ A .Now recognize the von Neumann algebra r i B ( H ) r i with B ( H i ) for each i ∈ Λ. Since r i is anatom, we know from Lemma 4.1 that r i A r i is a transitive subalgebra of B ( H i ). Therefore, as H i is finite-dimensional, it follows from Burnside’s Theorem (Corollary 8.6, [22]) that r i A r i = B ( H i ).In particular, this implies that r i B ( H ) r i = r i A r i , and since r i ∈ A , it follows that r i B ( H ) r i ⊆ A . (4.3)Now for each i , let L i be a masa in B ( H i ), and let L = L i ∈ Λ L i , which is considered a subalgebraof B ( H ). It is clear that L is a masa in B ( H ). Note that L r i = r i L for all i ∈ Λ. Also it followsfrom (4.3) that L i = r i L r i ⊆ A , and since A is WOT closed we have L = X i ∈ Λ L r i = X i ∈ Λ r i L r i ⊆ A , where the sum above is taken in WOT. Thus we have shown our requirement that A contains amasa, completing the proof. (cid:3) A nest of projections on a Hilbert space is called maximal or simple if it is not contained inany larger nest. It is easy to verify that a nest E is maximal if and only if all atoms in E areone-dimensional. Thus we have the following corollary. Corollary 4.5.
Let an algebra A have factorization in B ( H ) , and let Lat A be a maximal nest.Then A is reflexive, and so it is a nest algebra. We emphasize the importance of above corollary in the following example.
Example 4.6.
Consider the Hilbert space H = ℓ (Γ), for Γ = N or Z , and let A be the reflexivealgebra of upper-triangular matrices in B ( H ) with respect to the canonical basis { e n } n ∈ Γ . Notethat Lat A = { p n ; n ∈ Γ } , where p n is the projection onto the subspace span { e m ; m ≤ n } . ClearlyLat A is a maximal nest. So if B is any subalgebra of A with Lat B a nest, then Lat A ⊆
Lat B ,which implies by maximality that Lat A = Lat B . Thus it follows from Corollary 4.5 that the onlysubalgebra of A that has factorization in B ( H ) is A .Next we consider the consequence of above results for subalgebras of finite dimensional vonNeumann algebras. Let M n denote the algebra of all n × n complex matrices for some naturalnumber n . Let A be a logmodular subalgebra of M n . It can easily be verified by the compactnessargument of unit ball of M n that the algebra A also has factorization in M n . Since all atomsof Lat A are clearly finite dimensional, it follows from Theorem 4.4 that A is a nest algebra in M n . Thus we have shown that upto unitary equivalence, A is an algebra of block upper-triangularmatrices in M n . This assertion was put as a conjecture in [18], and an affirmative answer wasgiven in [14]. We have provided a different solution, and we state it below. Corollary 4.7.
Let A be a logmodular algebra in M n . Then A is an algebra of block uppertriangular matrices upto unitary equivalence. Moreover, the corollary above generalizes to any logmodular subalgebras of finite dimensionalvon Neumann algebras.
Corollary 4.8.
Let M be a finite dimensional von Neumann algebra, and let A be a logmodularalgebra in M . Then A is a nest subalgebra of M , and A is M -reflexive.Proof. Since M is a finite dimensional von Neumann algebra, there exist natural numbers n , . . . , n k such that M is ∗ -isomorphic to M n ⊕ . . . ⊕ M n k . In view of Proposition 2.2, we assume without lossof generality that M = M n ⊕ . . . ⊕ M n k , which acts on the Hilbert space H = C n ⊕ . . . ⊕ C n k . Usingcompactness of the unit ball in finite dimensional algebras, we note that A also has factorizationin M . ATTICES OF LOGMODULAR ALGEBRAS 17
Now for i = 1 , . . . , k , let p i denote the orthogonal projection of H onto the subspace C n i (considered as a subspace of H ), and let A i = p i A p i . We claim that A = A ⊕ . . . ⊕ A k . Firstlynote that p i ∈ M ∩ M ′ ; hence p i ∈ Lat M A . This in particular says that A i is an algebra. Since p i has finite dimensional range, it follows from Lemma 4.2 that p i ∈ A . This implies that A i ⊆ A for each i ; hence we have A ⊕ . . . ⊕ A k ⊆ A . On the other hand, since P ki =1 p i = 1 we get A = A k X i =1 p i ⊆ k X i =1 A p i = k X i =1 p i A p i = ⊕ ki =1 A i , proving our claim that A = ⊕ ki =1 A i . Now if we recognize M n i as a subalgebra of M ( ∗ -isomorphicto p i M p i ) for each i , then the algebra A i has factorization in M n i by Proposition 3.2; hence thelattice E i = Lat M ni A i is a nest in C n i by Corollary 2.11, and that A i = Alg M ni E i by Theorem4.4. Now consider the lattice E = ⊕ ki =1 E i = {⊕ ki =1 q i ; q i ∈ E i , ≤ i ≤ k } in M . It is clear that E = Lat M A , which implies A ⊆
Alg M E . Note that E is not a nest if k ≥ F , of E such that F is a nest and each element q i in E i appears atleast once as the i th coordinate of an element of F . Such F can always be chosen: for exampleconsider the nest F i for each i given by F i = { e ⊕ . . . ⊕ e i − ⊕ q i ⊕ ⊕ . . . ⊕ q i ∈ E i } ⊆ E , where e i denotes the identity of M n i , and let F = ∪ ki F i . Since each E i is a nest and F ⊆ F ⊆ . . . ⊆ F k , it follows that the sublattice F is a nest in M , and note that F fulfils the requirement.We claim that A = Alg M F , which will prove that A is a nest subalgebra of M . Clearly as F ⊆ E , we have
A ⊆
Alg M E ⊆
Alg M F . Conversely let x ∈ Alg M F , and let x = ⊕ ki =1 x i for some x i ∈ M n i , 1 ≤ i ≤ k. The way F has been chosen, each element of E i appears as the i th coordinateof some element of F , so it follows that x i q = qx i q for all q ∈ E i and for each 1 ≤ i ≤ k . Thisshows that x i ∈ Alg M ni E i = A i ; hence x ∈ A . Thus we conclude that Alg M F ⊆ A proving theclaim.Finally since
F ⊆ E = Lat M A , it follows that Alg M Lat M A ⊆
Alg M F = A . Since the otherinclusion is obvious, we have A = Alg M Lat M A which is to say that A is M -reflexive. (cid:3) More generally, Corollary 4.8 can easily be extended to any direct sum of finite-dimensional vonNeumann algebras, whose proof goes on the same lines. We record the statement here.
Corollary 4.9.
Let M be a (possibly countably infinite) direct sum of finite dimensional vonNeumann algebras, and let A be a logmodular algebra in M . Then A is a nest subalgebra of M and A is M -reflexive. In general, Corollary 4.8 fails to be true for algebras having factorization (or logmodularity) ininfinite dimensional von Neumann algebra, as the following example suggests.
Example 4.10.
Let A be an algebra having factorization in a von Neumann algebra M such that A 6 = M and D = A ∩ A ∗ is a factor. We claim that A is not M -reflexive. Assume otherwise that A = Alg M Lat M A . Note that Lat M A ⊆ D . Also it is easy to verify that Lat M A ⊆ D ′ and thuswe have Lat M A ⊆ D ∩ D ′ = C . It then follows that Lat M A = { , } , so A = Alg M { , } = M which is not true.There are plenty of such algebras. To see one, let G be a countable discrete ordered group. Let ℓ ( G ) = { f : G → C ; P g ∈ G | f ( g ) | < ∞} , and for each g ∈ G , let U g : ℓ ( G ) → ℓ ( G ) be theunitary operator defined by U g f ( g ′ ) = f ( g − g ′ ) for f ∈ ℓ ( G ) and g ′ ∈ G . Let M be the finite vonNeumann algebra in B ( ℓ ( G )) generated by the family { U g } g ∈ G , called the group von Neumannalgebra of G . Note that each element X of B ( ℓ ( G )) has a matrix representation ( x gh ) with respectto the canonical basis of ℓ ( G ). Let A = { X = ( x gh ) ∈ M ; x gh = 0 for g > h } . Then A is an example of a finite maximal subdiagonal algebra in M with respect to the expectation φ : M → M given by φ (( x gh )) = x ee x gh ) ∈ M , where e denotes the identity of G (see Example 3, [2]). In particular, A has factorization in M (Theorem 4.2.1, [2]). But note that A ∩ A ∗ = C (in fact if X = ( x gh ) ∈ A ∩ A ∗ , then x gh = 0 for all g = h and x gg = x g ′ g ′ for all g, g ′ ∈ G ), so A cannot be M -reflexive as discussed above. Moreover, we can choose the orderedgroup G to be countable with ICC property (e.g. G = F , the free group on two generators), sothat M is a factor. In this case although Lat M A is a nest, A cannot be a nest subalgebra of M ,otherwise A ∩ A ∗ will contain the nest and so cannot be equal to C .5. Concluding remarks
In this paper we have discussed ‘universal’ or ‘strong’ factorization property for subalgebras ofvon Neumann algebras. But there are weaker notions of factorization which can also be explored.Say a subalgebra A has weak factorization property (WFP) in a von Neumann algebra M if forany positive element x ∈ M , there is an element a ∈ A such that x = a ∗ a . Here the invertibilityassumption is relaxed.Power [20] has studied WFP of nest algebras where he proved that if a nest E of projections ona Hilbert space H is well-ordered (i.e. p = p + = ∩ q>p q for all p ∈ E and p = 1), then Alg E hasWFP in B ( H ). Inspired from our result on lattices of algebras with factorization, it appears thatlattices of algebras with WFP in a factor should also be a nest. But it is not clear to us at thispoint. However, for a subalgebra in a finite von Neumann algebra we can certainly say so. We canfollow the similar lines of proof along with the fact that any left (or right) invertible element in afinite von Neumann algebra is invertible. We record it here. Theorem 5.1.
Let A be a subalgebra of a finite von Neumann algebra (resp. factor) M satisfyingWFP. Then Lat M A is a commutative subspace lattice (resp. nest). So a natural question is the following:
Question 5.2.
Is the lattice of a subalgebra satisfying WFP in a von Neumann algebra (resp.factor) is a commutative subspace lattice (resp. nest)?
We conclude with the question of reflexivity of algebras with factorization. We showed that analgebra with factorization in B ( H ) has masa and hence is reflexive if we impose some dimension-ality condition on the atoms of its lattice. But we still do not know whether every algebra withfactorization in B ( H ) has masa. Thus the following question is open. Question 5.3.
Is a weakly closed algebra having factorization in B ( H ) automatically reflexive? Inparticular, is a weakly closed transitive algebra with factorization equal to B ( H ) ? Acknowledgments.
The first author thanks SERB(India) for financial support through J C BoseFellowship.
References [1] M. Anoussis and E. Katsoulis,
Factorisation in nest algebras , Proc. Amer. Math. Soc. 125 (1997), no. 1, 87–92.[2] W. B. Arveson,
Analyticity in operator algebras , Amer. J. Math. 89 (1967), 578–642.[3] W. Arveson,
Interpolation problems in nest algebras , J. Funct. Anal. 20 (1975), no. 3, 208–233.[4] D.P. Blecher and L.E. Labuschagne,
Logmodularity and isometries of operator algebras , Trans. Amer. Math.Soc. 355 (2003), no. 4, 1621–1646.[5] D.P. Blecher and L.E. Labuschagne,
Von Neumann algebraic H p theory , Function spaces, 89–114, Contemp.Math., 435, Amer. Math. Soc., Providence, RI, 2007.[6] J.B. Conway, A Course in Functional Analysis , second edition, Graduate Texts in Mathematics, 96. Springer-Verlag, New York, 1990.[7] K.R. Davidson,
Nest algebras , Pitman Research Notes in Mathematics Series, 191. Longman Scientific andTechnical, Harlow, New York, 1988.[8] C. Foia¸s and I. Suciu,
On operator representation of logmodular algebras , Bull. Acad. Polon. Sci. S´er. Sci. Math.Astronom. Phys. 16 (1968), 505–509.[9] I.C. Gohberg and M.G. Krein,
Theory and applications of Volterra operators in Hilbert space , Trans. Math.Monographs, Vol. 24, Amer. Math. Soc. Providence, 1970.[10] P.R. Halmos,
Two subspaces , Trans. Amer. Math. Soc. 144 (1969), 381–389.[11] K. Hoffman,
Analytic functions and logmodular Banach algebras , Acta Math. 108 (1962), 271–317.[12] K. Hoffman and H. Rossi,
Function theory and multiplicative linear functionals , Trans. Amer. Math. Soc. 116(1965), 536–543.
ATTICES OF LOGMODULAR ALGEBRAS 19 [13] G. Ji and K.-S. Saito,
Factorization in subdiagonal algebras , J. Funct. Anal. 159 (1998), no. 1, 191–202.[14] K. Juschenko,
Description of logmodular subalgebras in finite-dimensional C ∗ -algebras , Indiana Univ. Math.J. 60 (2011), no. 4, 1171–1176.[15] D.R. Larson, Nest algebras and similarity transformations , Ann. of Math. 121 (1985), no. 2, 409–427.[16] D.R. Larson,
Triangularity in operator algebras , Surveys of some recent results in operator theory, Vol. II,121–188, Pitman Res. Notes Math. Ser., 192, Longman Sci. Tech., Harlow, 1988.[17] M. McAsey, P.S. Muhly and K.-S. Saito,
Nonselfadjoint crossed products (invariant subspaces and maximality) ,Trans. Amer. Math. Soc. 248 (1979), no. 2, 381–409.[18] V.I. Paulsen and M. Raghupathi,
Representations of logmodular algebras , Trans. Amer. Math. Soc. 363 (2011),no. 5, 2627–2640.[19] D.R. Pitts,
Factorization problems for nests: factorization methods and characterizations of the universalfactorization , J. Funct. Anal. 79 (1988), no. 1, 57–90.[20] S.C. Power,
Factorization in analytic operator algebras , J. Funct. Anal. 67 (1986), no. 3, 413–432.[21] S. Power,
Analysis in nest algebras , Surveys of some recent results in operator theory, Vol. II, 189–234, PitmanRes. Notes Math. Ser., 192, Longman Sci. Tech., Harlow, 1988.[22] H. Radjavi and P. Rosenthal,
Invariant subspaces , Second edition. Dover Publications, Inc., Mineola, NY, 2003.[23] B. Solel,
Analytic operator algebras (factorization and an expectation) , Trans. Amer. Math. Soc. 287 (1985),no. 2, 799–817.[24] B. Solel,
Cocycles and factorization in analytic operator algebras , J. Operator Theory 20 (1988), no. 2, 295–309.[25] T.P. Srinivasan and J-K. Wang,
Weak ∗ -Dirichlet algebras , Function Algebras (Proc. Internat. Sympos. onFunction Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, 1966, pp. 216–249. Email address : [email protected] Email address : manish [email protected] [email protected]