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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 789182, 24 pages
http://dx.doi.org/10.1155/2011/789182
Research Article

The Spectral Scale of a Self-Adjoint Operator in a Semifinite von Neumann Algebra

Department of Mathematics and Statistics, California State University, Chico, 400 West First Street, Chico, CA 95928-0525, USA

Received 1 November 2010; Revised 10 March 2011; Accepted 11 March 2011

Academic Editor: Ilya M. Spitkovsky

Copyright © 2011 Christopher M. Pavone. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We extend Akemann, Anderson, and Weaver's Spectral Scale definition to include selfadjoint operators from semifinite von Neumann algebras. New illustrations of spectral scales in both the finite and semifinite von Neumann settings are presented. A counterexample to a conjecture made by Akemann concerning normal operators and the geometry of the their perspective spectral scales (in the finite setting) is offered.

1. Introduction

The notion of a “joint spectrum” for an -tuple of operators was created and developed through the 1970s and work continues today. For an account of the various approaches taken and the achievements made see [1, page 3]. One attempt at a geometric interpretation of the “joint spectrum” of an -tuple of self-adjoint operators is developed in spectral scale [26].

Definition 1.1. Let be a finite von Neumann algebra equipped with a finite, faithful, normal, trace such that , and let with and self-adjoint. Define by for all . The spectral scale of with respect to is the set

The spectral scale of an -tuple of self-adjoint operators is defined analogously (in the canonical fashion), becoming a subset of .

The set has proven useful in gaining insight on problems in operator theory and operator algebras. In 2006, for a self-adjoint operator in a finite factor (i.e., and has trivial center), Akemann and Sherman [7] were able to use to characterize the -closure of the unitary orbit of . More precisely, they showed that . Unfortunately, the spectral scale has only been developed for finite von Neumann algebras. In 2010, Wills [6] was able to extend the definition of to include the class of unbounded operators.

The purpose of the work that follows to develop for , where is a semifinite von Neumann algebra, equipped with a semifinite, faithful, normal trace such that . We will show that is an unbounded subset of which may or may not be closed, and that gives a complete description of elements in the spectrum of that lie outside the open interval determined by the minimum and maximum elements of the essential spectrum of .

We begin with a summary of “how to read” in the finite von Neumann algebra setting.

2. Geometric Interpretations of When Is a Finite von Neumann Algebra

The following is a summary of the main results concerning the geometry of [25].

Theorem 2.1. Let , , , , and be as given in Definition 1.1. Then the following hold.(i) is a compact, convex subset of which is symmetric about the point . If , then can be identified with a subset of .(ii) gives a complete description of for all .(iii) can be read from the point .(iv)If , then the set of all slopes of lines tangent to the boundary of is equal to .(v)If , then corners in the boundary of correspond to gaps in .(vi)If , then the extreme points of are images under of spectral projections of .(vii)If , then the extreme points of are images under of spectral projections of the operators for various .(viii)If is normal, then the set of all complex slopes of 1-dimensional faces of (i.e., all the 1-dimensional “edges” in the boundary) is equal to .(ix)The Numerical Range is equal to the set of all slopes (complex when ) of line segments emanating from the origin to points in .(x)If , then for each the cross-section of taken at is isomorphic to the -numerical range

We finish this section with a few examples of spectral scales in the finite setting.

Example 2.2. Let with for all , where is the canonical trace on . Let is given in Figure 1. The MATLAB program SpecGUI [8] was used to generate Figure 1, and by Theorem 2.1 one can see that , , 1 has multiplicity 2, , and .

789182.fig.001
Figure 1: —Example 2.2.

Example 2.3. Let , where is Lebesgue measure on the interval , let for all , and let be defined by for all . Then is positive, with no eigenvalues, and . is given in Figure 2.

789182.fig.002
Figure 2: —Example 2.3.

Example 2.4. Let with for all , and let is given in Figure 3 [8].

789182.fig.003
Figure 3: —Example 2.4.

Since is normal, by Theorem 2.1 we know that the eigenvalues of can be read from as the slopes (complex) of 1-dimensional faces of (i.e., slopes of 1-dimensional “edges” in the boundary of ). Alternatively, if we were given without knowing what actually was, we could conclude by [3, Theorem 6.1] that is normal. The numerical range of can be seen as the slopes (complex) of secant lines between the origin and points in .

Our last example is a counterexample to conjecture made by Akemann in 2005 which stated: is normal if the cross-section of taken at is a polygon. The following is a spectral scale that has no 1-dimensional faces, and the cross-section taken is not a polygon, yet the operator it corresponds to is normal.

Example 2.5. Let be the unit circle, and let denote Lebesgue measure on . Put , and note that is a finite von Neumann algebra equipped with the finite, faithful, normal trace defined by for all .

Consider the function on . The spectral scale of with respect to is the collection of points in . From [2, Theorem 2.3] we know that applied to spectral projections of the operators for various are extreme points of .

Let , and let . If is the characteristic function on the arc from to , then, for every (to see this think of the case where ). But for . Therefore, every is a spectral projection of some . Conversely, if and , then for if is chosen so that and and is chosen so that , then we have

Since the extreme points of come from the spectral projections described above, we can calculate the extreme points of explicitly: Since is the convex hull of its extreme points, we see that can be recovered by rotating the function about the -axis (see Figure 4).

789182.fig.004
Figure 4: —Example 2.5.

3. The Spectral Scale of a Self-Adjoint Operator in a Semifinite von Neumann Algebra

For the remainder of what follows, let be a semifinite von Neumann algebra, equipped with a faithful, semifinite, normal trace , and let be self-adjoint. In order to avoid the finite setting (for which the spectral scale has already been developed), we also assume that .

Define and . The by [9, Lemma 2.16, page 318], we can define the function by

Definition 3.1. The spectral scale of (with respect to ) is the set

Since is convex, is linear, and is positive, we have that is a convex subset of contained in the right half-plane. Since is semifinite and , is an unbounded subset of . We will see that may or may not be closed.

Note 1. If and are projections in and then we write for the order interval determined by and . Given a real number , we let denote the spectral projection of corresponding to the interval . Similarly, we let denote the spectral projection of corresponding to the interval . We use to indicate either of these projections and we write when . We let and denote the spectral projections corresponding to the intervals and , respectively. We also use to indicate either of these projections and we write when . Note that for every . The spectral projections and will be used throughout the remainder of this text, and will be instrumental in deciphering .

The following lemma was proven in [2, Lemma 1.2] for any type of von Neumann algebra.

Lemma 3.2. Let be a self-adjoint element in , , and . Then the following statements hold:(1) and the range projection of ;(2)if , then ;(3) and the range projection of is ;(4)if then .

The following two lemmas tell us that points of the form lie on the boundary when (or ) for some . A slight variation of these lemmas was proved in [2, Lemma 1.3].

Lemma 3.3. If is self-adjoint and , then the following statements hold for each and .(1)If , then .(2)If and , then ; moreover, if , then equality in the second equation obtains only for .

Proof. (1) From the previous lemma we have that Therefore Note that since and is an ideal. Since and is order preserving on [6], we have that Since , an identical argument yields It follows from (*), (**), and our hypothesis that and so .
To prove (2), suppose that and . Then Combining these two results with (*) and (**) we get that Therefore, all the above terms must be equal, and so
From this it follows that
Since is faithful we get that By parts (2) and (4) of the previous lemma, .
If it is the case that , , and then by what we just showed, , so , which implies that since is faithful.

Remark 3.4. Notice the importance of our choice of and as elements of ideal in the above proof. Without this assumption, many of the equalities and inequalities above would have been either trivial or undefined because is not necessarily finite on all of .

Lemma 3.5. If is self-adjoint and , then the following statements hold for each and .(1)If , then .(2)If and , then ; moreover, if , then equality in the second equation obtains only for .

Proof. (1) Similarly, by Lemma 3.2 (since ), we have that Therefore Using the fact that the trace is faithful and finite on , we get that It follows that
To prove (2), suppose that and . Then
From these two observations, (*), and (**) we get that But then all these terms must be equal. Therefore and . It follows that But since and we get By Lemma 3.2 we get . If , then since is faithful we must have .

Notice that this is one major difference between in the case where is finite, and when is semifinite. In the finite setting was convex and compact. Next, we define the boundary of , and then study some of its properties.

Definition 3.6. Let denote the closure of in . The lower boundary of is defined as The upper boundary of is defined analogously as Since is a closed convex set, and takes on every value in , we let defined by for each denote the lower boundary function determined by b. Similarly, let defined by for all denote the upper boundary function determined by b.
It is clear that is a convex function and is a concave function since is convex.

Theorems 3.7 and 3.8 describe the possibilities for the boundary of . These theorems describe the extreme points , the line segments in and , respectively, and the rays in and (resp.). Rays contained in and will be described in Theorems 3.9 and 3.10 (resp.). Differentiability of the functions and will be discussed in Theorems 3.12 and 3.13 (resp.).

Note 2. For a convex set we let denote the set of extreme points of .

Theorem 3.7. Let . Then one has the following. (1)If with , then .(2)If with , then is a line segment in having slope .(3)If and for all , then for some with .(4)If , then for some with .(5) is a ray in having slope if and only if and with .

Proof. (1) Fix such that . Then by part (1) of Lemma 3.3 we have that By part (2) of Lemma 3.3  we have that To see that are extreme in , suppose that for some and . But then and by part (2) of Lemma 3.3 we have that . Since projections are extreme points of , we must have . Therefore, .
(2) Fix so that . Then we have that , and . Note that if , then by part (1) of Lemma 3.3, so For each let . Then, each is in , and so . Furthermore, is a typical point on the line segment between and and the slope of this line segment is Therefore is a line segment in with slope . Note that since are extreme points of contained in , we know that the endpoints of any line segment containing must be and .
(3) Suppose and for all . We will show that for some with . Let We want to show that . Note that and are in since is closed, and by lower semicontinuity of .
We first show that . If with , and , then by the positivity of we must also have , so . This fact, along with Lemma 3.3, contradicts our assumption about . Thus, everything in is less than or equal to everything in , hence .
Now, either or not. Suppose . Since , and since is an upper bound for , we must have . But . Therefore when .
Assume that . Then, . But we are assuming that for all , and that . Therefore, by Lemma 3.3 we get Observe that, since is a lower bound for the set , if and then Similarly, if with , then Suppose that . Then since we have that On the other hand if then by (3.32) and (3.33) we would have . This is not possible, so it must be that . Therefore , and we have , contradicting (3.33). Hence, it must be that and so . Therefore and by Lemma 3.3.
(4) Since is convex, every point in the boundary of must either be an extreme point, or lie on a flat spot (face). Therefore, (1), and (2), (3), imply (4).
(5) () Suppose for some . We claim that is a ray in with slope , emanating from the point . We will first show that every point in lies on the ray through with slope .
Let , and write for some . Then , so . This implies that Therefore lies on the ray with slope passing through the point . Furthermore, by the convexity of , we know there can only be one ray having slope in the lower boundary of , and so if is any other point in that lies on a ray with slope in the lower boundary, then we must have .
(5) () If is a ray in with slope , emanating from , then is an extreme point of (if not, then does not emanate from since would then lie on a flat portion of ), so we must have for some by part (4). But is a ray, so cannot contain any extreme points of . So by part (3) we know that if with , then for some with . Furthermore, since any such is a line segment (or ray) with slope , and since , we must have . Since there can only be one ray with slope in the lower boundary of , we may conclude that and .

Theorem 3.8. Let . Then one has the following.(1)If with , then .(2)If with , then is a line segment in having slope .(3)If and for all , then for some with .(4)If then for some with .(5) is a ray in having slope if and only if and with .

Proof. Note that for each we have that Since is self-adjoint, the previous theorem applies. Alternatively, substitute Lemma 3.5 for Lemma 3.3 in the proof of Theorem 3.7, the claim follows immediately.

Until this point we have focused on points that lie in . As mentioned earlier, it may be the case that is not closed. The next four results describe this phenomenon, and tell us exactly when is closed, when it is not, and, when we get a “dotted” (open) portion in the boundary of .

Theorem 3.9. Let and let . Then if and only if . Furthermore, if then is a ray in emanating from .

Proof. Suppose that . Theorem 3.7 tells us that every has the property that either for some with , or for some with . Note that by the definition of , for all . Furthermore, for every we can choose , so . Therefore, for every we have . Suppose, for the sake of contradiction, that . Then, if with and , then for some with . Therefore it must be that . Clearly this is a contradiction since . Hence, it must be that .
Suppose , and let . We want to show that . Now, either or not. If then since for all , we get that for every . For each let . Then for every , and as . Therefore, we can find such that . Let To show that , we will show that , which will force for some by Theorem 3.7. Note that by lower semicontinuity of . We will first show that .
If with , and , then by the positivity of we must also have , which implies . Thus, everything in is less than or equal to everything in , hence .
Suppose . Either or not. Suppose . Since , and since is an upper bound for , we get . But this can not be, so it must be that . This implies . If equality holds on either side, then we would get (again, by Theorem 3.7).
Suppose . Observe that, since is a lower bound for the set , if and then Similarly, if with then Since , we have Thus, if then by (3.32) and (3.33) we would have . This is not possible, so it must be that . Therefore , and we have , contradicting (3.33). Hence, it must be that and so , and our desired result holds. Therefore, in this case we have .
If then by Theorem 3.7   will be a ray in . In both cases we have .
For the second conclusion of the theorem, suppose . Plugging into we get , so the point lies on the line . We will now show that, for , the ray forms the lower boundary of , but no point on the ray lies in .
Suppose with (see Figure 5).
We will show that lies above the line , and, for any there exists such that and is within of .
We will first show that lies above when , then prove the general case (). Suppose . We want to show that Suppose for the sake of contradiction that . Since is convex there must be such that and (see Figure 6).
Write with and positive and . Note that , so we have . On the other hand we have that Therefore, adding to both sides we get . This implies . Therefore, since is faithful, By part (2) of Lemma 3.2 we have But our assumption is that . Therefore, we have reached a contradiction to our assumption that , so it must be that .
Now suppose that . As before, we want to show that Suppose that is not. Then as before we can find such that
Note that for every we have that Let and consider . From (3.46) and (3.47) we have that Using (3.47) and the argument concluded at (3.44) we get . But this is a contradiction since . Therefore .
To prove comes arbitrarily close to the line , let . Using the definition of and our assumption that we have . Since is semifinite we can find a nonzero, positive such that . Furthermore, we may scale so that . Let . Then since , we have that

789182.fig.005
Figure 5: Ray in —Theorem 3.9.
789182.fig.006
Figure 6: Ray with slope 0 in —Theorem 3.9.

Theorem 3.10. Let and let . Then if and only if . Furthermore, if then is a ray in emanating from .

Proof. Substitute Theorem 3.8 for Theorem 3.7 in the proof of Theorem 3.9.

Corollary 3.11. Let and be as in Theorems 3.9 and 3.10 (resp.). Then if and only if .

Proof. Note that if and only if if and only if (since for each ). Thus, the claim follows immediately from Theorems 3.9 and 3.10.

Note that Theorems 3.7, 3.8, 3.9, and 3.10 tell us that the boundary of (which contains information about ) is completely determined by spectral projections of the form and where . To summarize, the following information is contained in the boundary of . (i)Extreme points contained in the lower boundary are of the form with .(ii)Line segments having slope in the lower boundary are of the form with and .(iii)Rays having slope in the lower boundary are of the form with and and .(iv)A “dotted” ray in the lower boundary occurs if . Moreover, this ray emanates from and has as its equation .(v)Extreme points contained in the upper boundary are of the form with .(vi)Line segments having slope in the upper boundary are sets of the form with and .(vii)Rays having slope in the upper boundary are sets of the form with and and .(viii)A “dotted” ray in the upper boundary occurs if . Moreover, this ray emanates from and has as its equation .

Recall that for each we let defined by denote the lower boundary function generated by . Similarly, denotes the upper boundary function generated by . For each , let () and () denote the left-sided and right-sided, (respectively) derivatives of and at . Note that by Theorem 3.7, every in has the property that for some with