HYPERREFLEXIVITY

1. HYPERREFLEXIVITY Analysis Seminar November 2, 2008 D. P. Dwiggins University of Memphis

2. 1-Hyperreflexivity and Complete Hyperreflexivity Davidson and Levine, JFA 2006 • Tools used in Operator Theory: • Invariant Subspaces • (Algebraic) Reflexivity Hyperreflexivity is a significantly stronger version of algebraic reflexivity. It implies algebraic reflexivity, but it also provides an alternate method of comparing distances in operator space. • Applications: • Nest Algebras • Von Neumann Algebras • Toeplitz Algebras Many Von Neumann algebras are hyperreflexive, “but it is not known if they all are.”

3. 1-Hyperreflexivity and Complete Hyperreflexivity Davidson and Levine, JFA 2006 Let M be a subspace of B(H) where H is a Hilbert space. Then M is 1-hyperreflexive if This gives an alternate way to calculate dist(T,M) without having to calculate operator norms. That is, instead of using , one may use where

4. Hyperreflexivity of Finite-Dimensional Spaces Müller and Ptak, JFA 2005 X is a Banach Space, and K is a closed subspace of B(H). If K is algebraically reflexive and dim(K) <  thenK is hyperreflexive. Moreover, if dim(K) = n andthen K is k-hyperreflexive. Note: the definition of k-hyperreflexivity in this article does not appear to correspond with the definition of 1-hyperreflexivity given in the preceding article. Every hyperreflexive space is also algebraically reflexive, but there are infinite-dimensional spaces which are algebraically reflexive but not hyperreflexive. (Kraus and Larson, J. Operator Theory 1985)

5. Hyperreflexivity of Finite-Dimensional Spaces Müller and Ptak, JFA 2005 Algebraic reflexivity was introduced by D. Sarason in 1966:Invariant subgroups and unstarred operator algebras, Pacific J. Math. v 17 The subspace version of algebraic reflexivity was given in 1975 by Loginov and Shulman: Hereditary and intermediate reflexivity of W* algebras, Math. USSR-Izv. v 9 Hyperreflexivity was introduced by N. T. Arveson in 1975:Interpolation problems in nest algebras, JFA v 20 The subspace version of hyperreflexivity was given in 1986 by Kraus and Larson: Reflexivity and distance formulae, Proc. London Math. Soc. v 53

6. Motivation for Algebraic Reflexivity The Double Commutant Theorem Let F be a subset of B(H) where H is a Hilbert space. F´ is called the commutant of F, and is weak-operator closed. Double Communtant Theorem: If A B(H) is a self-adjoint algebra containing the identity then both the weak and strong operator closures of A coincide with (A´)´. Kadison and Ringrose, Fundamentals of the Theory of Operator Algebras. vol. I Another way to restate this theorem is as follows: If A is a normal operator on a Hilbert space H, and if the operator B commutes with every projection operator that commutes with A, then B belongs to the weakly closed star-algebra generated by A and the identity.

7. Motivation for Algebraic Reflexivity The Double Commutant Theorem Another way to restate this theorem is as follows: If A is a normal operator on a Hilbert space H, and if the operator B commutes with every projection operator that commutes with A, then B belongs to the weakly closed star-algebra generated by A and the identity. However, this condition on B also implies every A-invariant subspace must also be B-invariant. Thus, yet another way to restate the double communtant theorem is as follows: If A is a normal operator on a Hilbert space H, and if the operator B leaves invariant every invariant subspace of A, then B belongs to the weakly closed star-algebra generated by A and the identity. D. Sarason, Invariant subspaces and unstarred operator algebras, Pac. J. Math. 1966

8. Motivation for Algebraic Reflexivity The Double Commutant Theorem If A is a normal operator on a Hilbert space H, and if the operator B leaves invariant every invariant subspace of A, then B belongs to the weakly closed star-algebra generated by A and the identity. Sarason notes it is easy to construct a non-normal operator (using 2x2 matrices) for which the above result does not hold, but also notes there are non-normal operators for which the above result does hold (namely, analytic Toeplitz operators). He does not however refer to spaces of such operators as reflexive, that particular term came later. (First cited instance?)

9. Motivation for Algebraic Reflexivity Dual Spaces Given topological vector spaces X and Y, let L(X,Y) denote the set of linear operators F : XY. If Y is one-dimensional then L(X,Y) is called a dual space of X. In particular, L(X,R) (where R is the set of real numbers) is called the algebraic dual of X. (Since L(X,R) is also called the set of linear functionals on X, this might also be called a functional dual.) Now suppose Y = X. A linear operator F : XX is bounded on X if and only if it is (uniformly) continuous on X, and these properties of F depend upon the topology on X. Thus, the set of linear bounded operators B(X) = {F  L(X,X) : F is bounded/continuous} might be called the topological dual of X. Thus there are two types of functional duals, algebraic and topological, where the latter depends on the topology on X and the former does not.

10. Motivation for Algebraic Reflexivity Dual Spaces Now let X be a Banach space and let X* denote its topological dual. (T X* means T : XX is linear and bounded.) Let X** (the topological dual of X*) denote the second dual of X. This then defines a mapping j : X X** by j(x0) = j0 as defined above. This mapping is also an isomorphism, and is called the canonical mapping on X. (This mapping is also isometric on X.) A Banach space X is said to be reflexive if X is (isometrically) isomorphic to its second dual, where the isomorphism is given by the canonical mapping. (James space: There is an isometric mapping between X and its second dual which need not be isomorphic.)

11. Motivation for Algebraic Reflexivity Dual Spaces A Banach space X is said to be reflexive if X is isomorphic to its second dual, where the isomorphism is given by the canonical mapping (and hence is also isometric). However, this definition of reflexivity has nothing to do with the double communtant version mentioned earlier, which is now referred to as algebraic reflexivity. To distinguish between the two types, Hadwin suggests using the term classical reflexivity when referring to the above definition. Since this definition comes from first considering functional dual spaces, we might also refer to the classical version as functional reflexivity.

12. Motivation for Algebraic Reflexivity Dual Spaces Earlier it was mentioned the double communtant theorem could be restated in terms of operators which commute with every projection that commutes with a given operator algebra. Thus, given an operator algebra A (a subalgebra of B(H)), define Lat(A) = {P : P is a projection and PA = PAP A  A}, and define A to be algebraically reflexive if BP = BPB  P  Lat(A) implies B A . Another way to state this is to define, for any collection P of projections, Alg(P) = {T : TP = TPT  P  P}. Thus A is algebraically reflexive if Alg(Lat(A)) = A. This property of recovering A by performing (in a sense) a double commutant operation on A is similar (again, in a sense) to the property of recovering a Banach space X by performing a second dual operation on X. Thus, one might think of Lat(A) and Alg(Lat(A)) as the first and second operator duals of A.

13. Motivation for Algebraic Reflexivity Dual Spaces Because there is an automatic relation between projections in B(H) and closed subspaces of H, one could also define algebraic reflexivity in terms of subspace duals as follows: Given an operator algebra A (a subalgebra of B(H)), define A* to be the collection of closed subspaces of H which are A-invariant  A  A (so that A* corresponds to Lat(A)), and define A** = (T B(H) : K is T-invariant  K  A*}. Then A** is equal to Alg(Lat(A)), and A is algebraically reflexive if and only if A** = A. One reason for using this approach to checking for algebraic reflexivity is it may be easier to calculate the invariant subspaces (and which operators leave them invariant) than it is to directly calculate first Lat(A) and then Alg(Lat(A)).

14. Other Versions of Algebraic Reflexivity Subspace Version A is a subalgebra of B(H) if A is closed under operator composition. (If A is also closed under the adjoint operation and is topologically closed under the topology on H then A is called a C*-algebra, and a C*-algebra with identity which is also closed under the weak operator topology is called a Von Neumann algebra.) These additional properties are not generally needed in studying algebraic reflexivity, and Loginov and Shulman (1975) gave a definition that does not require the closure under composition property, although they still retained the subspace property (closure under the vector operations on H). Let M be a subspace of B(H), and for x H define Mx = {Sx : x  M}.Let K(x) denote the closure of Mx. Then K(x) is a closed subspace of H which is T-invariant  T  M, and in fact M* = {K(x) : x H} consists precisely of those closed subspaces of H which are T-invariant  TM.

15. Other Versions of Algebraic Reflexivity Subspace Version Now let M** = Ref(M) = {T  B(H) : Tx K(x)  x H}, and define M to be algebraically reflexive if Ref(M) = M. This is not quite the same as saying Ref(M) consists of those operators which leave invariant the subspaces left invariant by the elements of M, since T  Ref(M) might not imply K(x) is T-invariant. However, this will be true if M is also closed under operator composition, and in fact if M is a subalgebra of B(H) then the subspace definition of algebraic reflexivity is equivalent to the subalgebra definition. Note also if T  M then Tx  Mx is automatically true  x H, and so M  Ref(M) is always true. Thus to show a subspace is algebraically reflexive it suffices to show T  Ref(M) implies T  M.

16. Other Versions of Algebraic Reflexivity Subset Version The following definitions are given in a 1996 paper by Batty and Molnár (Archiv der Mathematik v 67): Given a Banach space X, a subset M B(X) is algebraically reflexive if Given a Banach space X, a subset M B(X) is topologically reflexive if Note again, as with the difference between the algebraic and topological duals, here the difference between algebraic and topological reflexivity is that the latter uses the topology on X (to define closure) while the former does not.

17. A General View of Reflexivity D. Hadwin, Trans. AMS v 344 (1994) K is a topological field with a Hausdorff topology, e.g. K = C or R or a subfield of C with the usual topology, or K could be an arbitrary field with the discrete topology. X is a vector space over K, and Y is a vector space of linear maps F : XK which separates points in X. That is, given x1  x2  f  Y with f(x1)  with f(x2). Another way to state this is Given these conditions, (X,Y) is called a dual pair over K. Now define s(X,Y) as the smallest topology on X for which f is continuous  f  Y. Given x0  X define j0 : YK by j0(f) = f(x0), which defines the canonical map j : XY* . Then s(Y,X) is the smallest topology on Y for which j0 = j(x0) is continuous on Y for every x0  X .

18. A General View of Reflexivity D. Hadwin, Trans. AMS v 344 (1994) Proposition: In the proof the author notes that half the directions in 1) and 2) are trivial, and that what needs to be shown is f Y whenever ker(f) is s(X,Y)-closed. 3) follows easily from 2), and after giving the proof of 4) the author states, “This is the same as 4), especially if you are dyslexic.”

19. A General View of Reflexivity D. Hadwin, Trans. AMS v 344 (1994) Given a dual pair (X,Y), let E be a nonempty subset of Y which is closed under scalar multiplication and has E = {0}. The collection (X,Y,E) is called a reflexivity triple. Given M a subspace of X define RefE(M) = Then M is E-reflexive if M = RefE(M). Algebraic Reflexivity:H = Hilbert space, X = B(H), Y is the set of linear functionals F : XK which are continuous under the weak operator topology, and E is the set of rank-one tensors, i.e. F  E  Y iff  u,v  H  F(T) = Tu,v  T  X. Then a subalgebra A of X is algebraically reflexive iff A is E-reflexive, since RefE(A) = Alg(Lat(A)). Functional Reflexivity:Y is any Banach space, X = Y* is the topological dual of Y, and E = Y defines the reflexivity triple (Y*,Y,Y). If M is a subspace of Y* then RefE(M) = (M)  is the weak* closed span of M, and Y is functionally reflexive iff every closed subspace of Y* is Y-reflexive.

20. A General View of Reflexivity D. Hadwin, Trans. AMS v 344 (1994) Hyperreflexivity. Assume K = C (or R) and assume X and Y are normed linear spaces over K. Also assume Y is a subspace of the topological dual of X. Then any reflexivity triple (X,Y,E) is called a normed reflexivity triple. Given a subspace M of X define the seminorm dY = dY ( , M) on X by It is clear that dE (x,M)<dY (x,M) x X, and also almost as clear that this inequality can not in general be reversed. Also note that M is E-hyperreflexive if  k> 1 such that dY (x,M)<kdE (x,M) x X. It is clear that E-hyperreflexivity implies E-reflexivity, since

21. Motivation for Hyperreflexivity An Interpolation Theorem Given operators A1, . . . , AN on a Hilbert space H, there will exist operators B1, . . . , BN such that B1A1 + . . . + BNAN = 1 iff The question now is, if the Ai’s come from a subalgebra A of B(H), will the corresponding Bi’s also come from A? Positive results were obtained if A is both self-adjoint (e.g. normal operators) and commutative, but otherwise the “lower bound” property given above will not be sufficient. However, the more general result can be obtained if instead  e > 0  holds for every P Lat(A). W. Arveson, Interpolation Problems in Nest Algebras, JFA 1975

22. Motivation for Hyperreflexivity An Interpolation Theorem Now, the following inequality always holds for T B(H): If this inequality could be reversed then the approach outlined on the preceding slide will work. Actually, reversing this inequality (and thus obtaining equality) is a bit too much, since all that is actually needed is the existence of k> 1 such that holds for T B(H). Arveson determined this property holds for nest algebras, wondered if there might be other spaces where this property holds (answering yes, Toeplitz algebras), suggesting this might lead to new applications. However, he did not use the term “hyperreflexive” in this paper. W. Arveson, Interpolation Problems in Nest Algebras, JFA 1975

23. Motivation for Hyperreflexivity Projective Properties Given a closed subspace Y of a Hilbert space H, for every x H there exists a unique c = c(x)  Y such that In fact, c(x) is the projection of x under the projection defined by the subspace Y. If H is replaced by a Banach space X, then this projective property will also hold provided X is (functionally) reflexive. However, for algebraic (or topological) reflexivity, this distance property is not guaranteed. It is when a variant of this distance property is added to the subalgebra (or subspace) that the term hyperreflexivity is used. (As noted earlier, hyperreflexivity will always imply reflexivity.) However, as seen above, there is not corresponding version of “functional hyperreflexivity.”

24. Hyperreflexivity and the Distance Constant The definition of hyperreflexivty for subalgebras was given two slides earlier, so now consider the definition for subspaces. Given a Banach space X and a closed subspace M of B(X), define the function bM : B(X)R by Thus bM(T) <dist(T,M) by the definition of bM, and if bM(T) = dist(T,M)  T  B(X) then M is said to be 1-hyperrreflexive. More generally, if  k > 1  dist(T,M) < kbM(T)  T  B(X) then M is said to be hyperreflexive, and the smallest such value of k is called the distance constant, kM.

25. Problems for Future Research Every finite-dimensional space is hyperreflexive, but there is no way to relate the value of the distance constant to the dimension n.However, it is possible to classify which types of subspaces will be 1-hyperreflexive (have a distance constant = 1), but the work has only just begun (for matrix spaces so far up to about 3x3). Also, much work could be done in calculating the distance constant for specific types of subspaces. For infinite-dimensional spaces, the models for hyperreflexive subalgebras and subspaces still use upper triangular matrices, but there are some questions in this area remaining open. For example it would be nice to nail down the classification Von Neumann algebras in terms of hyperreflexivity. Finally, I have not seen any papers yet discussing hyperreflexivity for topologically reflexive subsets. (Almost every article on this topic is from the algebraic approach.) It might be good to see how much has been done on the classification of sequence spaces and function spaces in terms of topological hyperreflexivity.