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Borel structures coming from various topologies on B(H).

Introduction. Let H be a Hilbert space and B(H) be the set of all bounded linear operators on H. In the literature, there are some well-known locally convex topologies on B(H) which are given in the following diagram. For definitions and details see [3].

(0.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [contains] means that the right-hand side is coarser than the left-hand side. In the sequel, the notation [tau] runs over these seven topologies. We just recall the Arens-Mackey topology which is less classical than the six other ones. This locally convex topology is given as the uniform convergence topology on [sigma](B[(H).sub.*], B(H))-compact convex subsets of B[(H).sub.*]. The following two questions will be our concerns:

1) Determine the diagram of the sigma algebras generated by these seven topologies.

It is important to answer this question because, to any sigma algebra on B(H), a particular type of operator valued measurable functions is corresponded. Let [OMEGA] be a measurable space. Naturally, we say that an operator valued function f : [OMEGA] [right arrow] B(H) is [tau]-measurable provided that [f.sup.-1](O) is measurable for any open set O in the topology [tau]. In the classical case (when dim H = 1), the set of complex valued measurable functions forms a complex *-algebra which is also closed under the point-wise limit. In the general case, compatibility of these two algebraic and topological structures on the set of operator valued [tau]-measurable functions will be our second challenge. Indeed the major point is the [tau]-measurability of the product of two operator valued [tau]-measurable functions.

2) Is the product of two [tau]-measurable functions [tau]-measurable as well?

Separable case. When H is a separable Hilbert space, these two questions have been fully studied in [1]. There, it was proved that the [sigma]-algebra generated by all these seven topologies coincide. Also, the set of all measurable functions f : [OMEGA] [right arrow] B(H) forms an *-algebra and is closed under the point-wise product.

Here by an expository discussion, we follow these two problems when H is a non-separable Hilbert space. We will find that:

* In spite of the separable case, two different types of sigma algebras are implemented by the diagram of topologies (0.1).

* Neither addition nor product operations is compatible with the set of operator valued [tau]-measurable functions f : [OMEGA] [right arrow] B(H).

1. Diagram of sigma algebras generated by topologies. We first emphasize that throughout this discussion, H is a non-separable Hilbert space and E = [{[e.sub.i]}.sub.i[member] of]I is fixed as an orthonormal basis for H. We also denote Fin([epsilon]), by the family of all finite subsets of [epsilon]. Authors believe that, a complete and exact analysis concerning relations between sigma algebras generated by topologies [tau]'s, because no attempt on this subject has been done up to now, is much more difficult than the separable case.

Let us denote [M.sub.[tau]], by the sigma algebra generated by [tau]. The elements of [M.sub.[tau]] are also called [tau]-measurable sets. By a simple argument, we first observe that the diagram of the sigma algebras [M.sub.[tau]]'s is reduced to

(1.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To see this, it is enough to combine the following two facts:

* On bounded parts of B (H), the topologies given in each column of the diagram (0.1) are the same. The same result holds for the Arens-Mackey topology and the [sigma]-[strong.sup.*] topology ([3], Theorem III.5.7).

* The norm closed ball B(H)[[parallel]*[parallel].sub.[less than or equal to]n], centered at 0 with radius n, is closed under topologies [tau]'s. Any arbitrary subset E in B(H) can be represented by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

One may now conclude that [M.sub.[sigma]-w] = [M.sub.w], [M.sub.[sigma]-s] = [M.sub.s] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (where [M.sub.a.m] is the sigma algebra generated by the Arens-Mackey topology).

Let us denote [B.sub.[tau]], by the family of all sub-basic neighborhoods that determine the topology [tau]. We also denote [B.sub.[tau]], by the sigma algebra generated by [B.sub.[tau]]. To have a complete evaluation, we have to determine whether the relations between sigma algebras given in the diagram (1.1) are proper or not. We recall in the separable case, the key point which makes the subject move forward well is [M.sub.[tau]] = [B.sub.[tau]] (see [1]). We will show that it is no longer valid in the non-separable case and based on this point, some new (complicated) challenges are raised. In the rest of this discussion, we will observe that:

(1) The sigma algebra [B.sub.[tau]] is properly contained in [M.sub.[tau]], which makes us face two diagrams [M.sub.[tau]]'s and [B.sub.[tau]]'s.

(2) We first simplify the diagram of relations between [B.sub.[tau]]'s and then show that all [B.sub.[tau]]'s takes place in [M.sub.w], the smallest sigma algebra between [M.sub.[tau]]'s.

(3) By some calculations, we will find the diagram of the sigma algebra [B.sub.[tau]]'s, in comparison to [M.sub.[tau]]'s, is much more secretive.

Let us recall a fact which will be used several times in the sequel.

Remark 1.1. Let [OMEGA] be a non-empty set and [GAMMA] be a subset of [2.sup.[OMEGA]]. Let us denote [sigma]-([GAMMA]), by the sigma algebra generated by T. For a given A in [sigma]([GAMMA]), one may find a sequence [A.sub.1],[A.sub.2],... in [GAMMA] such that A is contained in the sigma algebra generated by the sequence [{[A.sub.n]}.sup.[infinity].sub.n=1] (just check that [union]{[sigma]([DELTA]) : [DELTA] [??] [GAMMA], [DELTA] is countable} forms a sigma algebra).

Proposition 1.2. Let A be a proper subset of B(H).

(1) Assume A is a w-measurable set in [B.sub.w]. There exists a countable set E in [epsilon] such that A = [E.sup.[perpendicular to].sub.w] + A where,

[E.sup.[perpendicular to].sub.w] = {x [member of] B(H) : <xe, f> = 0 for all e, f [member of] E}.

(2) Assume A is a s-measurable set in [B.sub.s]. There exists a countable set E in E such that A = [E.sup.[perpendicular to].sub.w] + A where,

[[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(3) Assume A is a [s.sup.*]-measurable set in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. There exists a countable set E in [epsilon] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(4) Assume A is an a.m-measurable (Arens-Mackey measurable) set in [B.sub.a.m]. There exists a countable set E in [epsilon] such that A = [E.sup.[perpendicular to].sub.a.m] + A where,

[E.sup.[perpendicular to].sub.a.m] = {x [member of] B(H) : [x.sup.*]e = xe = 0 for all e [member of] E}.

Proof. For a given [tau]-measurable set A in [B.sub.[tau]], there exists a sequence of sub-basic neighborhoods {[A.sub.n]} (for the topology [tau]) such that A is generated by [A.sub.n]'s (see Remark 1.1). Indeed by a countably many operations uunion and intersection" on the union {[A.sub.n]}[union]{[A.sub.n]}, one may construct A. Therefore it is enough to prove these assertions when A or [A.sup.c] is itself a sub-basic neighborhood. At first, assume that A is a sub-basic neighborhood in the weak operator topology. There exist f and p in H and [x.sub.0] [member of] B(H) with

A = {x [member of] B(H) : [absolute value of (<x[zeta],[eta]>-<[x.sub.0][zeta],[eta]>)] < [epsilon]}

Let E be a countable subset in [epsilon] such that both [zeta] and [eta] are generated by E. One may directly check that [E.sup.[perpendicular to].sub.w] works for both A and [A.sup.c], that is, [E.sup.[perpendicular to].sub.w] + A = A and [E.sup.[perpendicular to].sub.w] + [A.sup.c] = [A.sup.c]. Except the Arens-Mackey topology, others are obtained by a similar argument as well as the weak operator topology. As for the last one, let us consider the sub-basic neighborhood

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [x.sub.0] [member of] B(H) and K is a weakly compact convex subset of the predual B[(H).sub.*]. To any element F of Fin([epsilon]), a finite subset of [epsilon], we correspond to the finite rank projection whose range is linearly spanned by F, say [p.sub.F]. Clearly [{[p.sub.F]}.sub.Fin([epsilon])] forms an increasing net of projections which is strongly convergent to the identity operator. A deep result due to Akemann ([3], Theorem III.5.4) says that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It implies that there exists (at most) a countable subset E in [epsilon] such that

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [p.sub.E] is the orthogonal projection whose range is generated by E. We check that E works for this assertion. To do this, assume ye = [y.sup.*]e = 0 for all e [member of] E. Equivalently, y[p.sub.E] = [y.sup.*][p.sub.E] = 0 which implies that y = (1 - [p.sub.E])y(1 - [p.sub.E]). We now apply (1.2) to conclude [phi](y) = 0 for all [phi] in K which finishes the proof.

Let A be a [tau]-measurable set in [B.sub.[tau]]. By the previous result, there exists a countable subset E of [epsilon], called a refiner set of A, such that A + [E.sup.[perpendicular to].sub.[tau]] = A.

Corollary 1.3. Every [tau]-measurable set A in [B.sub.[tau]] is unbounded.

Proof. Let E be a refiner set of A. Since H is non-separable Hilbert space, then [E.sup.[perpendicular to].sub.[tau]] is unbounded. It implies A is unbounded too.

Corollary 1.4. The sigma algebra [B.sub.[tau]] is properly contained in [M.sub.[tau]].

Proof. Norm closed balls are contained in [M.sub.[tau]]. The previous corollary finishes the proof.

Proposition 1.5. Concerning the sigma algebras [B.sub.[tau]]'s, we have that:

(1) [B.sub.w] = [B.sub.[sigma]-w] [??] [B.sub.a.m],

(2) [B.sub.s] = [B.sub.[sigma]-s] and

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. We first show that [B.sub.[sigma]-w] [??] [B.sub.w] which implies [B.sub.w] = [B.sub.[sigma]-w]. Let us consider the following sub-basic neighborhood in the [sigma]-weak operator topology

{x [member of] B(H) : [absolute value of ([phi](x) - [phi]([x.sub.0]))] < [epsilon]},

where [phi] is a normal functional on B(H) and [x.sub.0] is in B(H). There exist two square summable sequences {[[zeta].sub.n]} and {[[eta].sub.n]} in H with [phi](x) = [SIGMA]<x[[zeta].sub.n],[[eta].sub.n]>.

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us consider [[phi].sub.n]: (B(H), [B.sub.w]) [right arrow] C given by [[phi].sub.n](x) = [[SIGMA].sup.n.sub.k=1]<x[[zeta].sub.k],[[eta].sub.k]>. Obviously, [[phi].sub.n]'s are all measurable functions. Since f is the point-wise limit of the sequence {[[phi].sub.n]}, then [phi] is measurable with respect to the sigma algebra [B.sub.w] too. Therefore, the sub-basic open set {x [member of] B(H) : [absolute value of ([phi](x) - [phi]([x.sub.0]))] < [epsilon]} is contained in [B.sub.w]. By a similar argument, one may show that [B.sub.s] = [B.sub.[sigma]-s] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It remains to prove [B.sub.w] is properly contained in [B.sub.a.m]. To do this, we make an example of a sub-basic neighborhood in the Arens-Mackey topology which is not in [B.sub.w]. Let us fix e [member of] [epsilon] and consider

[K.sub.e] := {[[omega].sub.[eta],e] [member of] B[(H).sub.*] : [eta] is in the unit ball of H},

where [[omega].sub.[eta],e](x) = <x[eta], e>. Consider the map [F.sub.e] : [H.sub.1] [right arrow] B[(H).sub.*] which maps [eta] [right arrow] [[omega].sub.[eta],e,] where [H.sub.1] denotes the unit ball of H. One may directly check that [F.sub.e] is weak-weak continuous. Then [K.sub.e], the range of [F.sub.e], is weakly compact.

Since [K.sub.e] is also convex, then

[O.sub.e] = {x [member of] B(H) : [absolute value of ([phi](x))] < 1 for all [phi] [member of] [K.sub.e]}

forms a sub-basic neighborhood containing 0, in the Arens-Mackey topology. We apply the first item of Proposition 1.2 and show that [O.sub.e] is no longer in [B.sub.w]. If [O.sub.e] is in [B.sub.w], then there exists a refiner set E [??] [epsilon] with

[E.sup.[perpendicular to].sub.w] = {x [member of] B(H) : <xf, g> = 0 : f, g [member of] E} [??] [O.sub.e].

Let us select e' [member of] [epsilon] - E. The rank one operator x = 2e [cross product] e' is in [E.sup.[perpendicular to].sub.w] and so should be contained in [O.sub.e] (0 [member of] [O.sub.e]) which is not true since [absolute value of ([[omega].sub.e',e](x))] = 2.

Remark 1.6. In the previous proposition, as for weakly compactness of [K.sub.e], we have replaced the proof with the one suggested by the referee since it was really much easier.

Let us consider A = {x [member of] B(H) : [parallel] (x - [x.sub.0])[zeta][[parallel].sup.2] < [epsilon]} which is a sub-basic neighborhood in the strong operator topology.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If one follows a similar argument for [strong.sup.*] operator topology and the Arens-Mackey topology, then the following facts will be obtained:

* Any sub-basic neighborhood in the strong operator topology forms a [F.sub.[sigma]]-set in [M.sub.w].

* Any sub-basic neighborhood in the [strong.sup.*] operator topology forms a [F.sub.[sigma]]-set in [M.sub.w].

* Any sub-basic neighborhood in the Arens-Mackey topology forms a [F.sub.[sigma]]-set in [M.sub.[sigma]-w].

Based on these three points, we infer the following fact.

Proposition 1.7. The sigma algebras [B.sub.[tau]] 's are all contained in [M.sub.w].

We end this section with two unsolved problems:

Problem 1.8. (1) Let b(H) be the set of all closed balls in B(H). Is [M.sub.w] generated by the union [B.sub.w] [union] b(H)?

(2) Let [{[e.sub.j]}.sub.j[member of]J] be a proper subset of E whose cardinal is larger than continuum and then consider [O.sub.s] = [U.sub.j[member of]j]{x [member of] B(H) : [parallel][xe.sub.j][parallel] < 1} which forms an open set in the strong operator topology. Is [O.sub.s] a w-measurable set?

2. Sigma algebras generated by the base of topologies. In this section, we focus on the sigma algebras [B.sub.[tau]]'s. We saw that discussion on this sigma algebras is reduced to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [B.sub.w] [??] [B.sub.a.m]. We first follow this question, what do we know more concerning relations between these sigma algebras? Secondly, we show that the algebraic operations, addition and product, do not fit on the set of operator valued [tau]-measurable functions.

Let {[phi]T} be the family of semi-norms that determines the locally convex topology [tau]. By a [tau]-sub-basic neighborhood centered at 0, we mean a subset of the form of {x [member of] B(H) : [[phi].sub.T](x) < [epsilon]}. Let us denote [B.sup.0.sub.[tau]] by the sigma algebra generated by all [tau]-sub-basic neighborhoods centered at 0.

Remark 2.1. As for the sigma algebras [B.sup.0.sub.[tau]]'s, we have that any [tau]-sub-basic neighborhood centered at 0 is invariant under any rotation (A = [e.sup.i[theta]]A, for any arbitrary angle [theta]). Therefore, any element of the sigma algebra [B.sup.0.sub.[tau]] is also invariant under any rotation. It implies that [B.sup.0.sub.[tau]] is properly contained in [B.sub.[tau]].

Proposition 2.2. We have that:

(1) [B.sub.w] [intersection] [B.sup.0.sub.s] = {[??], B(H)},

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. (1) Let A be in the intersection [B.sub.w] [intersection] [B.sup.0.sub.s] which is neither empty set nor B(H). We may assume A contains 0, otherwise [A.sup.c] is considered.

i) Since A is contained in [B.sup.0.sub.s], then there exists a sequence {[N.sub.n]} of sub-basic neighborhoods centered at 0 in the strong operator topology such that A is generated by [N.sub.n]'s (see Remark 1.1). Based on definition of [N.sub.n]'s, there are a sequence {[[zeta].sub.n]} of vectors in H and a sequence of positive numbers {en} with [N.sub.n] = {x [member of] B(H) : [parallel]x[[zeta].sub.n][parallel] < [[epsilon].sub.n]}. We put

[[GAMMA].sub.A] = {[gamma](x) : x [member of] A},

where [gamma](x) is the sequence {[parallel]x[[zeta].sub.n][parallel]}. Notice that

x [member of] A [??] [gamma](x) [member of] [GAMMA](A).

Since A is properly contained in B(H) then, there exists an operator y such that [gamma](y) is no longer in [[GAMMA].sub.A].

ii) Since A is a w-measurable set in [B.sub.w] then, there is a refiner set E = {[e.sub.n]} in E with A = [E.sup.[perpendicular to].sub.w] + A. Any countable set in [epsilon] containing E also forms a refiner set of A, then we may also assume that y[[zeta].sub.n]'s are all in the closed subspace generated by E.

Let us select a sequence {[f.sub.n]} [??] [epsilon] such that <[f.sub.n],[e.sub.m]> = 0 for all n and m (H is non-separable), and then consider the operator q = [sigma]2 [f.sub.n] [cross product] [e.sub.n]. As for the operator [>>] = qy, we have that:

* A direct calculation shows [gamma]([??]) = [gamma](y). Since A does not contain y, then item i) implies that [??] is not in A too.

* [??] is clearly contained in [E.sup.[perpendicular to].sub.w]. Since A contains 0, then A = [E.sup.[perpendicular to].sub.w] + A forces [??] should be in A.

This is a contradiction and so the intersection [B.sub.w] [intersection] [B.sup.0.sub.] should be trivial.

(2) To prove this case, one may exactly repeat the proof given in (1), when some notations in the proof are changed as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(3) To obtain the last one, again we may exactly repeat the proof given in (1), and the list of notations should be changed as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Finally by some examples we show how the addition and product of two [tau]-measurable functions may be no longer [tau]-measurable. We need to recall a fact from a classical set theory. Let X be a measurable space whose cardinal is larger than continuum. Let us consider the cartesian product X x X equipped with the product sigma algebra. Assume that S [??] X x X is a measurable set. Then the family of sections [S.sub.x] = {y [member of] X : (x, y) [member of] X} contains at most continuum of distinct sets and consequently the diagonal D = {(x, x) : x [member of] X} is no longer a measurable set (see [2], p. 231 Problem 3.10.44).

Example 2.3. Let [[OMEGA].sub.j] = (B(H), [M.sub.[tau]]) for j = 1,2. We consider the cartesian product space [OMEGA] = [[OMEGA].sub.1] x [[OMEGA].sub.2] equipped with the product sigma algebra.

(1) Let us consider functions f and g on [OMEGA] given by f(x,y) = x and g(x,y) = -y which are clearly [tau]-measurable. We have then

[(f + g).sup.-1] (0) = {(x,x) : x [member of] B(H)}.

The cardinal number of B(H) is larger than continuum (H is non-separable). Therefore, the diagonal [(f + g).sup.-1] (0) is not measurable in [OMEGA] which implies that f + g is not [tau]-measurable.

(2) In this example, we assume that the cardinal number [{[e.sub.i]}.sub.i[member of]I] is 2c. The projections f (x, y) = x and g(x, y) = y on [sigma] are clearly [tau]-measurable functions. We now verify that the product f x g is not [tau]-measurable. To do this, we put

S = [(f x g).sup.-1]([id.sub.H]) = {(x, y) [member of] [OMEGA] : xy = [id.sub.H]},

and check that the family of sections [S.sub.x] = {y [member of] B(H) : (x,y) [member of] S} contains [2.sup.c] of distinct sets. Let [I.sup.0] consist of those subsets in I which are infinite and countable. Then I and [I.sub.0] have the same cardinal. Let J be the subset in Io and [H.sub.J] be the subspace generated by {[e.sub.j] : j [member of] J}. Let [v.sub.J] be a bilateral shift on the Hilbert space [H.sub.J] and consider the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in B(H). We have then ([x.sub.J], [y.sub.J]) is in S where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Obviously, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not the same when [J.sub.1] and [J.sub.2] are different sets in [I.sub.0].

(3) Let us consider the measurable space Ow = (B(H), [B.sub.w]) and the inclusion mapping [iota] : [[OMEGA].sub.w] [right arrow] (B(H), [B.sub.w]). We check that [[iota].sup.2] is not measurable. Let us fix e [member of] [epsilon]. If [[iota].sup.2] is measurable function, then A = [([[iota].sup.2]).sup.-1] ([N.sub.e]) is a measurable set, where [N.sub.e] = {x [member of] B(H) : [absolute value of (<xe, e>)] < 1}. Let E be a [B.sub.w]-refiner set for A containing e. We select f [member of] [epsilon] - E and then consider the rank one operators a = 2e [cross product] f and b = f [cross product] e. One may directly check that a [member of] [E.sup.[perpendicular to].sub.w] and b [member of] A. Therefore, a + b [member of] [E.sup.[perpendicular to].sub.w] + A = A. But we have that

[absolute value of (<[[iota].sup.2](a + b)e, e>)] = 2,

which is a contradiction. This means that the set of measurable functions {f : [[OMEGA].sub.w] [right arrow] (B(H), [B.sub.w])} is not closed under product.

(4) If the sigma algebra [B.sub.w] is replaced by any of the sigma algebras [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or [B.sub.a.m] in the Example (2) above, then the same result is obtained, i.e., the product dose not work well again. To prove them, one may exactly repeat the proof when some notions are changed. The list of changes are given as follow:

changes when [B.sub.w] is replaced by

[B.sub.s] : [N.sub.e] = {x [member of] B(H) : [parallel]xe[parallel] < 1}

changes when [B.sub.w] is replaced by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

changes when [B.sub.w] is replaced by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

doi: 10.3792/pjaa.93.7

Acknowledgment: We thank the referee for he review and highly appreciate the comments and uggestions, which significantly contributed to mproving the quality of the publication.

References

[1] G. A. Bagheri-Bardi, Operator-valued measurable functions, Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 1, 159 163.

[2] V. I. Bogachev, Measure theory. Vol. I, Springer, Berlin, 2007.

[3] M. Takesaki, Theory of operator algebras. I, Springer, New York, 1979.

Ghorban Ali BAGHERI-BARDI and Minoo KHOSHEGHBAL-GhoRABAYI

Department of Mathematics, Faculty of Sciences, Persian Gulf University, Bushehr (7516913817), Iran

(Communicated by Masaki KASHIWARA, M.J.A., Jan. 12, 2017)
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