aa r X i v : . [ m a t h . OA ] M a r A NON-DIAGONALIZABLE PURE STATE
PIOTR KOSZMIDER
Abstract.
We construct a pure state on the C*-algebra B ( ℓ ) of all boundedlinear operators on ℓ which is not diagonalizable, i.e., it is not of the formlim u h T ( e k ) , e k i for any orthonormal basis ( e k ) k ∈ N of ℓ and an ultrafilter u on N . This constitutes a counterexample to Anderson’s conjecture withoutadditional hypothesis and improves results of C. Akemann, N. Weaver, I. Farahand I. Smythe who constructed such states making additional set-theoreticassumptions.It follows from results of J. Anderson and the positive solution to theKadison-Singer problem due to A. Marcus, D. Spielman, N. Srivastava thatthe restriction of our pure state to any atomic masa D (( e k ) k ∈ N ) of diagonaloperators with respect to an orthonormal basis ( e k ) k ∈ N is not multiplicativeon D (( e k ) k ∈ N ). Introduction
Recall that a pure state on a C*-algebra is a positive linear functional of normone, i.e., a state, which is not a convex combination of other states. Pure states onthe algebras of all operators on finite dimensional Hilbert spaces ℓ ( n ) for n ∈ N are known all to be vector states, i.e., of the form φ ( T ) = h T ( v ) , v i , where v ∈ ℓ ( n )is a unit vector. Vector states are also pure states in the case of the algebra B ( ℓ )of all linear bounded operators on an infinite dimensional Hilbert space ℓ .There are many other pure states on B ( ℓ ) whose existence is usually provedby means of the Hahn-Banach theorem starting from a pure state on a maximalabelian self-adjoint subalgebra (masa) of B ( ℓ ). If the masa is atomic, that is ofthe form D (( e k ) k ∈ N ) of all diagonal operators with respect to an orthonormal basis( e k ) k ∈ N , then the general form of the initial pure state φ for T ∈ D (( e k ) k ∈ N ) is(D) lim u h T ( e k ) , e k i where u is an ultrafilter on N . J. Anderson showed in [3] that (D) defines a purestate on the entire B ( ℓ ) and conjectured in what became known as Anderson’sconjecture ([5]) that every pure state on B ( ℓ ) is of the above form for some or-thonormal basis ( e k ) k ∈ N of ℓ and an ultrafilter u on N . Our main result (Theorem12) is the construction of a pure state that is non-diagonalizable, that is a coun-terexample to Anderson’s conjecture.Much of the research concerning the relations between pure states on B ( ℓ )and pure states on masas of B ( ℓ ) has been motivated by a seminal paper [9]of Kadison and Singer. The positive solution of one of the problems stated in thepaper and known as the Kadison-Singer problem due to A. Marcus, D. Spielman, N. Mathematics Subject Classification. lim u z k = z for z k , z ∈ C means that for every ε > { k ∈ N : | z k − z | < ε } is in theultrafilter u . Srivastava implies that a non-diagonalizable pure state on B ( ℓ ) necessarily cannothave multiplicative restriction to any atomic masa (because such restrictions extendto pure states on B ( ℓ ) of the form (D) but the extensions are unique by the positivesolution to the Kadison-Singer problem).Another problem from the paper [9] is whether any pure state on B ( ℓ ) has amultiplicative restriction to some masa of B ( ℓ ). Having multiplicative restriction inthis case is equivalent to having the restriction equal to a pure state on the masa.In [1] C. Akemann and N. Weaver provided a negative solution to this problemassuming the continuum hypothesis CH . This, in particular, already showed thatAnderson’s conjecture is consistently false, but as suggested in [1] it could still beconsistent that any pure state on B ( ℓ ) has a multiplicative restriction to a masa.This additional hypothesis in the case of Anderson’s conjecture was weakened to MA ([14]) or to cov ( M ) = c or to d ≤ p ∗ (12.5 of [7]), or to another one in [13].Our counterexample to Anderson’s conjecture shows that the additional hypoth-esis in the result of Akemann and Weaver is not needed when we limit ourself toatomic masas. However, we do not know if our non-diagonalizable pure state canhave a multiplicative restriction to a non-atomic masa. An improvement of a resultfrom [9] due to J. Anderson from [2] says that any pure state on a non-atomic masahas many extensions to pure states on B ( ℓ ). So we can say that our pure state isnot “determined” by any pure restriction to any masa, i.e., either the restriction isnot pure or if it is pure it does not uniquely extends to our pure state.Our construction is entirely different than that of Akemann and Weaver whichused properties of separable C*-subalgebras of B ( ℓ ) and a well-ordering of all masasin the first uncountable type ω based on the continuum hypothesis to approximatethe desired pure state with separable fragments. Let us describe the main idea of ourconstruction here. In a sense, instead of using separable approximations we obtainthe desired pure state by approximating it with finite dimensional fragments. Let { , } m denote the set of all sequences of zeros and ones of length m ∈ N and let { , } N denote the set of all infinite sequences of zeros and ones. We fix a function d : N → N which will be specified later and identify ℓ with(I ) M m ∈ N O σ ∈{ , } m ℓ ( d ( m )) . This can be done by considering a partition of N into finite sets of sizes d ( m ) (2 m ) for m ∈ N . Recall that there is a canonical isomorphism(I ) B (cid:0) O σ ∈{ , } m ℓ ( d ( m )) (cid:1) ≡ O σ ∈{ , } m B ( ℓ ( d ( m ))) . If for every σ ∈ { , } m we choose any non-zero projection P σ ∈ B ( ℓ ( d ( m ))) and I denotes the identity, then(P) Y σ ∈{ , } m ( I ⊗ ... ⊗ σ P σ ⊗ ... ⊗ I ) = O σ ∈{ , } m P σ . is a projection that is dominated by any of the projections ( I ⊗ ... ⊗ σ P σ ⊗ ... ⊗ I ).So for any α ∈ { , } N and any choice v α = ( v αm ) m ∈ N ∈ Q m ∈ N ( ℓ ( d ( m )) \ { } ) wecan define rank one projections R v αm : ℓ ( d ( m )) → ℓ ( d ( m )) onto the direction of v αm and distribute them along the restrictions α | m = α |{ , ..., d ( m ) − } ∈ { , } m NON-DIAGONALIZABLE PURE STATE 3 for m ∈ N defining P α,v α = M m ∈ N (cid:0) I ⊗ ... ⊗ α | m R v αm ⊗ ... ⊗ I (cid:1) . Under the identifications (I ) and (I ) the operator P α,v α is a projection in B ( ℓ ).It follows from (P) that for any choices v α for α ∈ { , } N any finite productformed by the projections ( P α,v α : α ∈ { , } N ) dominates a nonzero projectionbecause eventually α | m, ..., α n | m ∈ { , } m are all distinct if α , ..., α n ∈ { , } N are distinct. This guarantees that for any choices v α for α ∈ { , } N there is a purestate φ on B ( ℓ ) such that φ ( P α,v α ) = 1 for all α ∈ { , } N .To make sure that φ is not diagonalized by any orthonormal basis we need toshow that there is a constant 0 < c < e k ) k ∈ N of ℓ there is α ∈ { , } N such that(ND) |h P α,v α ( e k ) , e k i| < c for every k ∈ N . To obtain the above property we manipulate the choice of v α .Here we exploit the fact that if f ( d ) points on d -dimensional real sphere form an ε -net on the sphere for ε <
1, then f ( d ) must grow exponentially in the dimension d ∈ N . Using this with the choice of d satisfying for each m ∈ N (i) d ( m ) ≥ ,(ii) 32 m ( d ( m ) (2 m ) ) d ( m ) (2 m − < (100 / d ( m ) we can obtain v α satisfying (ND)for c = 19 /
20 and a fixed orthonormal basis ( e k ) k ∈ N of ℓ . As there are as manyorthogonal bases in ℓ as elements α ∈ { , } N , we can make sure that φ is notdiagonalized by any basis.The structure of the paper is as follows. In the second section we discuss thepreliminaries including the above mentioned tensor products of finite dimensionalHilbert spaces and the exponential growth of the above mentioned function. Inthe third section we construct the required family of projections (Theorem 10)and include the final argument. The main result is Theorem 12. The last sectioncontains additional remarks.The notation should be standard. When X is a set, then ℓ ( X ) denotes theHilbert space whose orthonormal basis is labeled by elements of X . All norms are ℓ -norms or operator norms on Hilbert spaces. | X | denotes the cardinality of a setand | z | denotes the absolute value of a complex number z , it should be always clearfrom the context which meaning of | | is used. We also often identify n ∈ N withthe set { , ..., n − } . For sets A, B by B A we mean the set of all functions from A into B . The restriction σ = x | m of an infinite sequence x ∈ { , } N for m ∈ N is asequence σ ∈ { , } m of length m such that σ ( k ) = x ( k ) for all k < m .The author would like to thank C. Akemann, I. Farah, P. Wojtaszczyk for valu-able comments which were used to improve the previous version of the paper.2. Preliminaries
Projections and the inner product.Lemma 1.
Suppose that P is an orthogonal projection in B ( ℓ ) and x ∈ ℓ . Then h P ( x ) , x i = k P ( x ) k . Proof.
Using the facts that P = P = P ∗ we obtain h P ( x ) , x i = h P ( x ) , x i = h P ( x ) , P ( x ) i = k P ( x ) k . (cid:3) PIOTR KOSZMIDER
Lemma 2.
Suppose that ( e k ) k ∈ N is an orthonormal basis of ℓ and F ⊆ ℓ is an n -dimensional linear subspace of ℓ . Let ε > . There is X ⊆ N of cardinality notbigger than n /ε such that k P F ( e k ) k < ε for every k ∈ N \ X .Proof. Let { e ′ , ..., e ′ n − } be an orthonormal basis of F . We have 1 = k e ′ j k =Σ k ∈ N |h e ′ j , e k i| for each j < n . So there are A j ⊆ N of cardinality not bigger than n/ε such that |h e ′ j , e k i| ≤ ε/n for every k ∈ N \ A j . Let X = S j The purpose of this subsection is to proveLemma 5 which roughly says that there is an absolute constant such that if in d -dimensional Hilbert space we have less then “exponentially in d ”-many directions,then there is another direction whose inclination to all the original ones is at leastthe constant. Lemma 3. Suppose that X is a collection of unit vectors in R d for d ≥ , suchthat for every unit vector y ∈ R d there is x ∈ X with k x − y k ≤ / . Then thecardinality of X is at least (100 / d / .Proof. Let B r ( a ) denote the ball of radius r > a ∈ R d . Let V d ( r )denote the d -dimensional volume of B r ( a ) for any a ∈ R d . Recall that V d ( r ) is equalto r d V d (1) which follows from the formula for integration by substitution with thesubstitution sending a ∈ R d to ra .The hypothesis on X implies that the sphere in R d is covered by S { B / ( x ) : x ∈ X } . So whenever 99 / ≤ k y k ≤ y ∈ R d , then there is x ∈ X such that d ( x, y ) ≤ d ( x, y/ k y k ) + 1 / ≤ / 10 + 1 / 100 = 91 / 100 and so set S { B / ( x ) : x ∈ X } covers B (0 d ) \ B / (0 d ), where 0 d denotes the origin in R d .The latter set has volume V d (1) − V d (99 / − (99 / d ) V d (1) and theunion which covers it has volume not bigger than | X | (91 / d V d (1). It follows that(1 − (99 / d ) V d (1) ≤ | X | (91 / d V d (1) and so (100 / d (1 − (99 / d ) ≤ | X | .As (99 / (2 ) ≈ , / d ≤ / d ≥ and so | X | ≥ (100 / d / d s, as required. (cid:3) The above argument is a version of well known fact concerning ε -nets of the n -dimensional ball, e.g. Proposition 15.1.3 of [10]. Lemma 4. Suppose that d ∈ N \ { } and x, y ∈ C d are unit vectors. Suppose that ε > and k x ± y k , k x ± iy k ≥ ε . Then |h x, y i| ≤ √ − ε / . Proof. Let α ∈ { , − , i, − i } . By the parallelogram law 2( k v k + k w k ) = k v + w k + k v − w k we conclude that k x + αy k = 4 −k x − αy k and k x + iαy k = 4 −k x − iαy k .Using the above and the polarization identity h u, v i = ( k u + v k − k u − v k − i k u − iv k + i k u + iv k ) we obtain |h x, y i| = |h x, αy i| = p (1 − k x − αy k / + (1 − k x − iαy k / ≤ √ − ε / NON-DIAGONALIZABLE PURE STATE 5 for ε / , k x − αy k / , k x − iαy k / ≤ − β ) isdecreasing below β = 1 and we have ε / ≤ k x − αy k / , k x − iαy k / 2. The restof the proof consist of noting that under our hypothesis that k x k = k y k = 1 thereis α ∈ { , − , i, − i } such that k x − αy k , k x − iαy k ≤ √ C d we may assume that y = (1 , , ..., x ∈ C such that | x | ≤ α ∈ { , − , i, − i } such that | x − α | , | x − iα | ≤ 1. Now k x − αy k = | x − α | + X Suppose that d, n ∈ N satisfy d ≥ and n < (100 / d / and that X = { x j : j < n } is a collection of vectors in C d . Then there is a unit vector x ∈ C d suchthat |h x j , x i| ≤ (9 / k x j k for every j < n . In particular k R x ( x j ) k ≤ (9 / k x j k for every j < n , where R x is the orthogonal projection onto the direction of x .Proof. First assume that all x j s are unit vectors. Identifying R with C we canconsider Y ( l ) = { y j ( l ) : j < n } ⊆ R d for l ∈ { , , , } , satisfying x jk = y j k (1) + iy j k +1 (1) , − x jk = y j k (2) + iy j k +1 (2) ,ix jk = y j k (3) + iy j k +1 (3) , − ix jk = y j k (4) + iy j k +1 (4)for all k < d and j < n and 1 ≤ l ≤ 4. It is clear that y j ( l ) are unit vectors for all j < n and 1 ≤ l ≤ | Y (1) ∪ Y (2) ∪ Y (3) ∪ Y (4) | = 4 n < (100 / d / < (100 / d / z ∈ R d such that k z − y j ( l ) k ≥ / 10 for all j < n and1 ≤ l ≤ x ∈ C d whose coordinates are complex numbers whose real and imagi-nary parts are formed from the 2 d real coordinates of z , i.e., x k = z k + iz k +1 for any k < d . It is clear that x is a unit vector. We have k x − x j k = sX k 10 for all j < n .It follows from Lemma 4 that for any j < n we have |h x, x j i| ≤ √ − (9 / / √ − (9 / / ≤ q ≤ , ≤ = 9 / 10, so |h x, x j i| ≤ / j < n .If x j have arbitrary norms, we have |h x, x j i| = k x j kh x, x j / k x j ki ≤ (9 / k x j k . Also by Lemma 1 k R x ( x j ) k = h R x ( x j ) , x j i = hh x j , x i x, x j i = h x j , x ih x j , x i = |h x j , x i| ≤≤ (9 / k x j k ≤ (9 / k x j k . (cid:3) Obtaining inclined intersecting subspaces in tensor products. For sets A, B as usual B A denotes the set of all functions from A to B . { ( a, b ) } will standfor a function whose domain is { a } and which assumes value b at a . So any t ∈ B A can be written uniquely as t = s ∪ { ( a, b ) } , where s ∈ B A \{ a } . We will view theHilbert space ℓ ( B A ) as the tensor product of Hilbert spaces N a ∈ A ℓ ( B { a } ), where h⊗ a ∈ A x a , ⊗ a ∈ A y a i = Q a ∈ A h x a , y a i ([12, 6.3.1]). This notation will allow us tohandle many-fold tensor products with precision and a relatively modest amount ofindices. For example e { a,b } ⊗ e s = e s ∪{ ( a,b ) } = e s ⊗ e { a,b } and we do not need to worryabout the order of factors in tensor products of Hilbert spaces. However in the caseof tensors of operators we will be using a more standard notation S ⊗ ... ⊗ a T ⊗ ... ⊗ S to indicate with a letter above T at which coordinate we put the operator T . Recallthat R v denotes the rank one orthogonal projection onto the direction of a nonzerovector v . Definition 6. Suppose that A, B are nonempty sets and v ∈ ℓ ( B { a } ) , then wedefine the orthogonal projection R a,v ∈ B ( ℓ ( B A )) onto the subspace ℓ ( B A \{ a } ) ⊗ C v of dimension | B | | A |− by P A,Bα,v = I ⊗ ... ⊗ a R v ⊗ ... ⊗ I. More explicitly for each x = P t ∈ B A x t e t ∈ ℓ ( B A ), a ∈ A and s ∈ B A \{ a } wedefine x ( s ) = P x ( s ) ( a,b ) e { ( a,b ) } ∈ ℓ ( B { a } ) by(1) x ( s ) ( a,b ) = x s ∪{ ( a,b ) } . That is we arrange the coordinates of x into | A | | B |− blocks x ( s ) for s ∈ B A \{ a } .Then given v = P b ∈ B v b e { ( a,b ) } ∈ ℓ ( B { a } ) we define P A,Ba,v : ℓ ( B A ) → ℓ ( B A ) by(2) P A,Ba,v ( X t ∈ B A x t e t ) = X s ∈ B A \{ a } X b ∈ B h x ( s ) , v i v b e s ∪{ ( a,b ) } . That is to each block x ( s ) of the coordinates of x we apply the projection R v ontothe direction of v . To check that this corresponds to Definition 6 one can checkthis for basic vectors e t = e s ∪{ ( a,b ) } , namely X b ∈ B h e { ( a,b ) } , v i v b e s ∪{ ( a,b ) } = (cid:0) R v ( e { ( a,b ) } ) (cid:1) ⊗ e s . NON-DIAGONALIZABLE PURE STATE 7 Lemma 7. Let A, B be finite sets and let v a ∈ ℓ ( B { a } ) be nonzero for each a ∈ A .Then for any nonzero choice of v a ∈ ℓ ( B { a } ) for a ∈ A the product Q a ∈ A P A,Ba,v a ≤ P A,Ba,v a is a nonzero projection.Proof. Y a ∈ A P A,Ba,v a = Y a ∈ A ( I ⊗ ... ⊗ a R v a ⊗ ... ⊗ I ) = O a ∈ A R v a . (cid:3) More explicitly if v a = P v ( a,b ) e { ( a,b ) } for a ∈ A then we consider v = X t ∈ B A Y a ∈ A v a,t ( a ) e t ∈ ℓ ( B A ) . It is enough to show that each of the projections P A,Ba,v a for a ∈ A leaves v intact.Indeed by (1) and (2) we have v ( s ) = ( Q a ′ ∈ A \{ a } v a ′ ,s ( a ′ ) ) v a , so P A,Ba,v a ( v ) = P A,Ba,v a ( X t ∈ B A Y a ∈ A v a,t ( a ) e t ) == X s ∈ B A \{ a } X b ∈ B ( Y a ′ ∈ A \{ a } v a ′ ,s ( a ′ ) ) h v a , v a i v ( a,b ) e s ∪{ ( a,b ) } == X t ∈ B A Y a ∈ A v a,t ( a ) e t = v. Lemma 8. Suppose that A, B are finite sets such that | A | = m > , | B | = d ≥ and { x j : j < n } are vectors of ℓ ( B A ) for some n ∈ N . Moreover let us assumethat nd m − < (100 / d / . Then for every a ∈ A there is a nonzero v a ∈ ℓ ( B { a } ) such that k P A,Bv a ,a ( x j ) k ≤ (9 / k x j k for each j < n .Proof. Fix A, B, a and { x j : j < n } as in the lemma. Let x j = P t ∈ B A x jt e t . As in(1) we can write it as x j = X s ∈ B A \{ a } x j ( s ) ⊗ e s . Apply Lemma 5 to the collection { x j ( s ) : j < n, s ∈ B A \{ a } } of cardinality nd m − and obtain a unit vector v = X b ∈ B v b e ( a,b ) ∈ ℓ ( B { a } )such that k R v ( x j ( s )) k ≤ (9 / k x j ( s ) k for all s ∈ B A \{ a } and j < n . For each s ∈ B A \{ a } we have k P A,Bv,a ( x j ) k = k (cid:0) I ⊗ ... ⊗ a R v ⊗ ... ⊗ I (cid:1) ( X s ∈ B A \{ a } x j ( s ) ⊗ e s ) k == k X s ∈ B A \{ a } R v ( x j ( s )) ⊗ e s k = X s ∈ B A \{ a } k R v ( x j ( s )) k ≤≤ (9 / X s ∈ B A \{ a } k x j ( s ) k = (9 / k x j k . (cid:3) PIOTR KOSZMIDER More explicitly using (1) and (2) k P A,Bv,a ( x j ) k = X s ∈ B A \{ a } k X b ∈ B h x j ( s ) , v i v b e s ∪{ ( a,b ) } k == X s ∈ B A \{ a } k R v ( x j ( s )) k ≤ (9 / X s ∈ B A \{ a } k x j ( s ) k = (9 / k x j k . A family of projections and the pure state For this section we fix d : N \ { } → N such that for any m ∈ N , m > • d ( m ) ≥ , • m ( d ( m ) (2 m ) ) d ( m ) (2 m − < (100 / d ( m ) .Such d can be easily constructed as for each m ∈ N the polynomial p m ( x ) = 32 m ( x (2 m ) ) x (2 m − is smaller than the exponential function (100 / x for sufficiently big x ∈ R .In the rest of this section we will identify d ( m ) with the set { , ..., d ( m ) − } .Define O = [ m> d ( m ) ( { , } m ) . Note that the summands of this union are pairwise disjoint as they consist offunctions with different domains { , } m for m ∈ N . In this section instead of theusual ℓ = ℓ ( N ) we will work with ℓ ( O ). For m > Q m : ℓ ( O ) → ℓ ( O ) bethe orthogonal projection onto ℓ ( d ( m ) ( { , } m ) ) considered as a subspace of ℓ ( O )consisting of vectors whose coordinates in O \ d ( m ) ( { , } m ) are zero. We will bedealing with algebras B m = Q m B ( ℓ ( O )) Q m for m > 0, they will be identified with B ( ℓ ( d ( m ) ( { , } m ) )). The projections we will be constructing will be elements of M m> B m So for operators T m ∈ B ( ℓ ( d ( m ) ( { , } m ) )) for m > M m ∈ N T m ∈ B ( ℓ ) . Lemma 9. Let ( e k : k ∈ N ) be an orthonormal basis of ℓ ( O ) and let σ m ∈ { , } m for each m > . For each m > there is v m ∈ ℓ ( d ( m ) { σ m } ) such that for each k ∈ N we have |h (cid:0) M m> P { , } m ,d ( m ) σ m ,v m (cid:1) ( e k ) , e k i| ≤ / . Proof. For m > X m = { k ∈ N : k Q m ( e k ) k > /π m } . As P m> m = π ,note that for every k ∈ N we have( ∗ ) X { m> k X m } k Q m ( e k ) k ≤ π X m> m ≤ / . For m ∈ N by Lemma 2 applied for ε = 3 /π m knowing that the dimension of B m is d ( m ) m we have that | X m | ≤ π m ( d ( m ) m ) / ≤ m ( d ( m ) m ) .Now for m > { Q m ( e k ) : k ∈ X m } . Since 8 | X m | d ( m ) (2 m − < (100 / d ( m ) using Lemma 8 we can find v m ∈ ℓ ( d ( m ) { σ m } ) such that k P { , } m ,d ( m ) σ m ,v m ( e k ) k = k P { , } m ,d ( m ) σ m ,v m ( Q m ( e k )) k ≤ (9 / k Q m ( e k ) k for each k ∈ X m since the range of P { , } m ,d ( m ) σ m ,v m is included in the range of Q m .For each k ∈ N we have k M m ∈ N P { , } m ,d ( m ) σ m ,v m ( e k ) k ≤ X { m> k ∈ X m } k P { , } m ,d ( m ) σ m ,v m ( e k ) k + X { m> k X m } k Q m ( e k ) k ≤≤ (9 / X { m> k ∈ X m } k Q m ( e k ) k + X { m> k X m } k Q m ( e k ) k . Putting α = P { m> k X m } k Q m ( e k ) k we have that P { m> k ∈ X m } k Q m ( e k ) k =1 − α as e k = P m> Q m ( e k ). So k M m> P { , } m ,d ( m ) σ m ,v m ( e k ) k ≤ (9 / − α ) + α However α ∈ [0 , / 2] by (*) and (9 / − α ) + α = (1 / α + (9 / 10) assumes itsmaximum on [0 , / 2] at α = 1 / 2. The maximum is 19 / 20 and so for each k ∈ N byLemma 1 we have |h M m> P { , } m ,d ( m ) σ m ,v m ( e k ) , e k i| = k M m> P { , } m ,d ( m ) σ m ,v m ( e k ) k ≤ / , as required. (cid:3) Theorem 10. There is a collection ( P α : α ∈ { , } N ) of infinite dimensionalorthogonal projections in B ( ℓ ) such that for any α , ..., α n ∈ { , } N and n ∈ N there is a nonzero projection P α ,...,α n ≤ P α , ..., P α n and for every orthonormalbasis ( e k ) k ∈ N of ℓ there is α ∈ { , } N such that for each k ∈ N we have |h P α ( e k ) , e k i| ≤ / . Proof. We will prove the theorem for ℓ ( O ) instead of ℓ = ℓ ( N ). Since O iscountably infinite, this makes no difference. Enumerate all orthonormal bases of ℓ ( O ) as { ( e αk ) k ∈ N : α ∈ { , } N } . This is possible since by the cardinal equality(2 ω ) ω = 2 ω both { , } N and the collection of all orthonormal bases of ℓ have thesame cardinality equal to the continuum. For α ∈ { , } N consider P α = M m ∈ N P { , } m ,d ( m ) α | m,v αm , where v αm s are chosen according to Lemma 9 for the basis ( e αk ) k ∈ N and σ m = α | m which is an element of { , } m formed by the first m terms of α . Hence we have |h P α ( e αk ) , e αk i| ≤ / 20 for each k ∈ N and each α ∈ { , } N .Now let α , ..., α n ∈ { , } N . Let m ∈ N be such that α j | m = α j ′ | m for any two1 ≤ j < j ′ ≤ m . By Lemma 7 Y ≤ j ≤ n P { , } m ,d ( m ) α j | m,v αjm is a projection dominated by each P { , } m ,d ( m ) α j | m,v αjm s for 1 ≤ j ≤ m hence the same istrue for the projections P α , ..., P α n . (cid:3) To construct our pure state we a result relating certain collections of projectionsin B ( ℓ ) and pure states on B ( ℓ ). Based on Chapter 6 of [6] it seems that thefollowing result is due to N. Weaver. We provide the proof for the convenience ofthe reader. Lemma 11. Suppose that ( P j ) j ∈ J is a collection of projections in B ( ℓ ) such thatfor any j , ..., j n ∈ J there is a nonzero projection P such that P ≤ P j i for each ≤ i ≤ n . Then there is a pure state φ on B ( ℓ ) such that φ ( P j ) = 1 for all j ∈ J .Proof. Let S denote the set of states on B ( ℓ ). Let P denote the family of allfinite subsets of J and let P a ≤ P j , ..., P j n be the projection as in the lemma for a = { j , ...j n } . As k P a k = 1 for every a ∈ P , there are states φ a ∈ S such that φ a ( P a ) = 1 ([12, 5.1.11]) which satisfy φ ( P j ) = 1 for each j ∈ a as φ ( P a ) ≤ φ ( P j ) ≤ 1. For a ∈ P consider F a = { φ ∈ S : φ ( P j ) = 1 for all j ∈ a } F a s are convex, nonempty, weak ∗ closed and form a centered family as F a ∪ a ′ ⊆ F a ∩ F a ′ for all a, a ′ ∈ P , so by the compactness of the dual ball of B ( ℓ ) in theweak ∗ topology we have T a ∈P F a = ∅ . Moreover T a ∈P F a = ∅ is convex as theintersection of convex sets. By the Krein-Milman theorem T a ∈P F a has an extremepoint φ . We claim that φ is the desired pure state. If φ = αψ + (1 − α ) ψ ′ forsome ψ, ψ ′ ∈ S and α ∈ (0 , αψ ( P a ) + (1 − α ) ψ ′ ( P a ) = φ ( P a ) = 1for any a ∈ P . But this implies that ψ ( P a ) = ψ ′ ( P a ) = 1 for all a ∈ P , and so ψ, ψ ∈ T a ∈P F a . However, in such a case, ψ = ψ ′ = φ as φ was an extreme pointof T a ∈P F a . (cid:3) Theorem 12. There is a non-diagonalizable pure state in B ( ℓ ) .Proof. Let ( P α : α ∈ { , } N ) be the collection of orthogonal projections fromTheorem 10. By Lemma 11 there is a pure state φ on B ( ℓ ) such that φ ( P α ) = 1for each α ∈ { , } N .However, by Theorem 10 for every orthonormal basis ( e k ) k ∈ N of ℓ there is α ∈ { , } N such that | lim u h P α ( e k ) , e k i| ≤ / = 1 = φ ( P α )which shows that φ is not diagonalizable. (cid:3) Remarks. u on N one can construct a pure state φ as inTheorem 12 which additionally satisfies φ ( L m ∈ X Q m ) = 1 for all X ∈ u . This isbecause the projections L m ∈ X Q m can be added to the family of projections fromTheorem 10 maintaining the hypothesis of Lemma 11. It follows that such statescan be multiplicative on a big abelian subalgebras of B ( ℓ ) of the form A [ K ] ofSection 12.5 of [7]. Here A [ K ] is the von Neumann subalgebra of B ( ℓ ) generatedby a pairwise orthogonal collection of finite dimensional orthogonal projections in ℓ whose supremum is the identity.I. Farah and N. Weaver showed that under an additional set-theoretic hypothesis d ≤ p ∗ (12.5.10 of [7]) there is a pure state whose restriction to any algebra A [ K ]is not multiplicative. I. Farah conjectures (p. 336 of [7]) that it is consistent thatany pure state has a multiplicative restriction to a subalgebra of the form A [ K ].So our pure state is compatible with this conjecture. NON-DIAGONALIZABLE PURE STATE 11 P α , P α ] of projections from Theorem10 for distinct α , α ∈ { , } N are finite dimensional, namely they belong to L 1. 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