Abstract

We extend the notion of a spectral scale to n-tuples of unbounded operators affiliated with a finite von Neumann Algebra. We focus primarily on the single-variable case and show that many of the results from the bounded theory go through in the unbounded situation. We present the currently available material on the unbounded multivariable situation. Sufficient conditions for a set to be a spectral scale are established. The relationship between convergence of operators and the convergence of the corresponding spectral scales is investigated. We establish a connection between the Akemann et al. spectral scale (1999) and that of Petz (1985).

1. Introduction and Preliminaries

The notion of the spectrum of a self-adjoint operator has proved to be of great interest and use in various branches of mathematics. It is natural to try and extend the notion to -tuples of operators. In 1999, Akemann et al. came up with the notion of a spectral scale [1, page 277]. The setting is as follows. Let be a finite von Neumann algebra equipped with a normal, faithful tracial state, . Elements of can be thought of as bounded operators on some Hilbert Space, , [2, page 308]. For a given self-adjoint the corresponding spectral scale, , which we will define below, yields information about the spectrum of in a nice geometric way. Many of the results can be extended to -tuples of self-adjoint operators in . The primary aim of this paper is to explain several of the results on spectral scales, and show how they can be extended when, instead of considering , we consider .

In Section 2, we consider the single-variable case which is fairly well developed. A sequence of technical lemmas culminating in Lemma 2.10 are required before we can make significant progress in the single-variable case. We illustrate with examples. Finally, we establish sufficient conditions to guarantee that a subset of is a spectral scale.

In Section 3, we consider the geometric structure of the -dimensional spectral scale. It turns out that there is little difficulty in generalizing from the bounded situation.

In Section 4, we discuss certain invariance properties of the spectral scale. Significant difficulties arise in the unbounded situation although we believe that, if Conjecture 4.5 is correct, many of the difficulties would be removed.

Section 5 addresses some miscellaneous results. First, we address the natural question of whether the convergence of a sequence of operators implies the convergence of the corresponding spectral scales. Second, we establish a relationship between two logically distinct objects [1, 3] which were both defined by their authors as “spectral scales”.

Finally, in Section 6 we outline some possible future directions of research.

Let us start with some preliminary definitions.

Definition 1.1. Let be a Hilbert space. Let be a subalgebra of . If is closed in the weak operator topology, self-adjoint, and contains 1, then is a von Neumann algebra [2, page 308].

Let denote the set of positive elements of .

Definition 1.2. Let be a function such that for and we have: (The last equation implies that ).
Then is a faithful, finite, normal trace on [4, pages 504-5].

Theorem 1.3. Let be a faithful, finite, normal trace on . Since any element of can be written as a finite linear combination of positive elements of , can be extended to a linear functional on all of [5, page 309].

Two projections in are equivalent if there exists such that and . A projection is finite if . is finite if the projection is finite [5, page 296]. Throughout, we will assume that is finite.

Further, we will assume that there exists a faithful, finite, normal trace of , with ; that is, is a faithful, normal, tracial state on .

A crucial property of is that “things” commute in trace—that is, although, in general , for , we do have the equality [4, page 517].

Let . Let be an -tuple of self-adjoint operators in . Let

Definition 1.4 (see [1, page 260]). is called the spectral scale ofwith respect to .

Now is normal. Further is linear and continuous with respect to the weak operator topology. Moreover, is convex and compact in the weak operator topology: and for . Therefore is a compact, convex subset of .

There have been a large number of results concerning spectral scale. Some papers on the subject include those in [1, 6, 7].

In 2004, Akemann and David Sherman conjectured that, if we replace with the set where each is self-adjoint, we will yield similar results. This paper verifies this, and generalizes much of the first paper on spectral scales [1].

Some results on “noncommutative integration” will prove useful in our exposition. We will use Nelson's 1972 [8] paper on the subject with specific theorem and page references as appropriate.

In his paper, Nelson defines , the predual of [8, Section , pages 112 ff.]. The duality is given by the bilinear form [8, Section , page 112] for and . Now [8, page 112 ff.], and Nelson shows that elements of are closed, densely defined operators affiliated with [8, Theorem , page 107, and Theorem , page 114]. It follows that a bounded linear functional, can be represented by a (possibly unbounded) linear operator affiliated with and we get the equality for every .

2. Spectral Scale Theory for Unbounded Operators—the Single-Variable Case

We are now prepared to discuss how the spectral scale theory generalizes. We start with the single-variable situation.

Definition 2.1. Let be a self-adjoint linear functional. Let Then is the spectral scale of with respect to .

From the theory of noncommutative integration, we see that for some operator affiliated with . Since is self-adjoint, too will be self-adjoint, and hence, as with the original spectral scale, our generalized spectral scale is a compact, convex subset of .

Notation 1. We will often write for .

The following definition was suggested to the author in conversation by Akemann.

Definition 2.2 (Akemann). If is bounded, we will call an operator functional.

Our main goal in this section is to show that is an operator functional if and only if the slopes of the lower boundary function of are all finite. We remark that Akemann et al. have already shown the “only if” part of this statement [1, Section , pages 261–274]. For this reason, we may assume throughout that is unbounded, and show that the lower boundary curve of has, as a consequence, an infinite slope. To get there, we will need a number of preliminary results.

Proposition 2.3. is mapped onto itself by a reflection through the point .

Proof. Let . Then . Therefore Thus the map takes to and hence takes onto itself. The fixed point of is , and the points and lie on a straight line that passes through . The straight line is given by the equation Note also that is the identity map on . Hence, and reflects through the point .

For the next several results, we will need the unbounded spectral theorem for self-adjoint operators. We state it here in the functional calculus form.

Theorem 2.4 (see von Neumann in [9, page 562]). Let be a (densely defined) self-adjoint operator in with domain . Then algebraic -homomorphism takes bounded Borel functions on into such that the following hold.(a) is norm continuous.(b)Let be a sequence of bounded Borel functions with as for each and for every and . Then for , as . The convergence is in norm.(c)If pointwisely, and the sequence is bounded, then strongly.(d)If , then .(e)If , then .

For a given , a bounded Borel function on , it is customary to write as . In other words, the “” is understood. For now it is more convenient to write explicitly.

Definition 2.5. For let and . More generally, if is a characteristic function on a Borel subset of , then is a projection; such projections are referred to as spectral projections [9, pages 234, 267].

For the most part, we will only need spectral projections obtained from intervals. Note that, for , is nonzero on the domain of (and hence all of ) if and only if is an eigenvalue of . Also, since Borel functions commute with respect to multiplication and is a homomorphism, Im is an Abelian subalgebra of . Assume now that is affiliated with our finite von Neumann algebra, . In this case it turns out that Im is an Abelian subalgebra of . This follows from the way that is constructed.

Lemma 2.6. Let . Then

Proof. Using the decomposition , we can write
Hence, (Of course, these equalities only make sense on the domain of .)
For every we get:
Therefore and hence . The other statement in this lemma follows via a similar argument.

Lemma 2.7. Let be a characteristic function of a bounded Borel subset of  . Then .

Proof. Let be a sequence of bounded Borel functions such that for all and . By the Spectral theorem, for every . Note that converges to for every . Since is a bounded Borel function, . For , we have Hence and . Since is self-adjoint, we have the desired result.

Corollary 2.8. The following set relations hold:

Lemma 2.9. The range projection of is . The range projection of is .

Proof. Let be the range projection of . Let be a sequence of bounded Borel functions on such that . Let . Then . From the Spectral theorem, we have
Taking the limit as on the left side, we get and therefore for every .
We have shown that . For let be the range projection of . By the same reasoning as above, . Also, Hence, . Similarly, for , we have .
Now by Lemma 2.7. Therefore, is a bounded operator. In fact, on . We show that is an invariant subspace of under . Suppose that and Im. Then for every , since in , and .
Hence, Im. Thus, as Since for every , But we already know that and so equality holds.
The second statement in the lemma follows from an analogous proof.

Lemma 2.10. Let . If , then .
If , then .

Proof. Write , using the decomposition
Note that for . Assume that The diagonal entries of are for . Hence, . Thus, for all , such that , . Since on which contains , we get Hence and so . Since is not an eigenvalue of , . Therefore and so for . From the Spectral theorem we can then conclude that Thus, for every . But and for every and every . Hence, is dense in Im. Since Im, we have for . Thus, so The other statement in the lemma follows from an analogous argument.

We remark that in the original paper on spectral scales [1, Lemma , pages 262, 263], the above conclusion was obtained with a little less work, since, in that situation, was bounded and so we did not have to worry about the domain of . The proofs of the next several results, however, are virtually identical to the original proofs. In other words, much of the hard work has now been done.

Lemma 2.11. Fix , , and . Suppose that . Then the following hold:().()If , then .()If , then .

Proof. Note that and from Lemma 2.6. Hence since is faithful. Similarly, .
() We compute and so .
() Suppose that . Then Similarly, Therefore and hence equality holds throughout. Thus From (2.33), while from (2.34),
Since is faithful and the arguments are positive, the arguments are in fact equal to zero. By Lemma 2.10, .
()Suppose that and . Then . Since is comparable to , and is faithful, .

We next state a theorem proved by Akemann and Pedersen [10, Theorem ].

Theorem 2.12. If and are von Neumann algebras, is a normal linear map from to , and a face of , then there are unique projections and in with such that and .

The following results are generalizations of the main theorems for the case from the first paper on spectral scales [1, Theorems 1.5–1.7, pages 266–274]. We will introduce some new notation at this time.

Notation 2. Recall that we are assuming that is a finite von Neumann algebra equipped with a faithful, normal, tracial state . The operator is unbounded and self-adjoint on affiliated with obtained from a linear functional (i.e., for each ). is the spectral scale of . The lower boundary of is given by The upper boundary of is given by The endpoints of the lower boundary are and .
Let denote the spectrum of , and let be the point spectrum of . Let be the function on whose graph is the lower boundary. For , let Let be the positive half-plane determined by .

Our next result describes the faces of the lower boundary of . We do not include the endpoints at this time.

Theorem 2.13. The zero-dimensional faces in the lower boundary of are precisely the points of the form for . Also,
The one-dimensional faces in the lower boundary of are the sets of the form for . For each face , The slope of is .

Proof. We have the following steps.Step 1. We show that are zero-dimensional faces.
Fix . If and , then by Lemma 2.11. Hence, is on the lower boundary of .
But again by Lemma 2.11. Hence,
We now show that is an extreme point of . Suppose that for . Since Since projections are extreme points in , .
Step 2. For , are faces of .
Fix so that Then If , then for each such that . Hence is on the graph of which is the lower boundary curve.
Write . Then , and is a typical point on the line segment connecting and . Hence graph contains this line segment. The slope is
Let denote the line segment in the graph of that contains and consider the endpoints of . By Theorem 2.12, there are projections such that , and hence If , then, since , is in the interior of , contradicting Step 1. Hence and similarly . Thus, is a line segment in graph with slope and
Step 3. We show that we have accounted for all of the graph of , except possibly the endpoints.
Fix a point with , and assume that for every . Write We would like to show that . Since is closed, for .
By definition, Since for , If , , then by definition Similarly, and
Suppose that . Then . If for some , we would have which is clearly false. Hence, But then and , which again is a contradiction. Hence, Since so is an eigenvalue of . From Step 2, is a line segment in graph. Hence is on the interior of that line segment.

Corollary 2.14. The extreme points on the upper boundary (excluding the endpoints) are precisely the points of the form for and
The line segments on the upper boundary are precisely the sets of the form , for . The slope of is and

Proof. This result is a direct consequence of applying Proposition 2.3 to Theorem 2.13.

Let be the left endpoint of . Note that may be . Let be the right endpoint of . Note that may be .

Proposition 2.15. If , then the left derivative of the lower boundary function at exists and is given by the formula If , then the right derivative of at exists and is given by the formula

Proof. Since is convex, is a convex function, and so the left and right derivatives exist. Fix with . Then . Define . Since is closed, .Case 1 ( for some ). If , which contradicts the choice of . Thus , and hence is an isolated point in the spectrum, that is, is an eigenvalue of . Moreover, and so is the right-hand endpoint of a line segment in graph with slope by Theorem 2.13. Hence .Case 2 ( for every ). Choose such that and for . Then is on the graph of . Furthermore, in the weak- topology. Since is normal, . Since is faithful, . We have for every . Hence, for every . Thus, Letting gives the desired result.The statement regarding right derivatives is proved in a similar way.

Proposition 2.16. The corners of are in one-to-one correspondence with the gaps of , that is, the maximal bounded intervals in the complement of the spectrum. (One is not currently concerned with unbounded maximal intervals in the complement of the spectrum, that is, those which take the form or .)

Proof. Let be an interior gap of the spectrum. Then for every we have . Fix . Then Hence, is not differentiable at , and so a gap in the spectrum corresponds to a corner. Conversely, we have already seen that is differentiable at for .

Proposition 2.17. For each , The line is a line of support for such that In this case, passes through . Moreover, one has

Proof. Fix . If is an eigenvalue, then Otherwise, . Either way, and so Let . Then so lies in . We now wish to show that is a line of support for . There are several cases to consider.Case 1 (). In this situation, and are endpoints of a line segment in graph whose slope is . passes through both points and has slope . Thus, contains this line segment and is tangent to . Hence .Case 2 (). Note that is not an isolated point in . Moreover, . At least one of the one-sided derivatives of takes the value at . Hence, admits a line of support at with slope . As with Case 1, the line is and .Case 3 ( is an interior gap in the spectrum). In this case . Let and . Then and are lines of support passing through whose slope lies between and . If is a line of support for whose slope lies between and , then . But is such a line for . Hence, the statement is true for any .Case 4 ( or ). Since is unbounded, at least one of and has infinite magnitude. Suppose that is finite (and so must be infinite). Then . Moreover, . is a line of support for at and by Case 1. Suppose . Then is also a line of support for at and .
The case for is dealt with similarly. Hence, for every , is a line of support for and .
Conversely, for fixed , the lines are all parallel as varies over . Hence, there exists a unique for which is a line of support and . But has these properties and hence .
For the last statement, consider . Then is a face of . Hence, is an extreme point or a line segment on graph.
If is an extreme point, then , then . Since is an extreme point, then , and so
Similarly, if is a line segment, then for some , and so

From the above results, if , then the right derivative of approaches as . If in addition , then the left derivative of approaches as . By Proposition 2.3, the graph of the upper boundary curve of is vertical at . Hence, the only line of support at is vertical. Similarly, the only line of support at is vertical. Therefore, if both and are nonreal, then and are not corners of .

Conversely, if one of and is finite, then and are corners of .

Here the bounded and unbounded spectral scale theories do not coincide, since, in the bounded situation, and are always corners.

In both situations, we can read spectral data of the lower boundary curve as follows()1-dimensional faces correspond to eigenvalues of . The slope of each face is the corresponding eigenvalue.()Other places where the lower boundary curve is differentiable correspond to elements of the continuous spectrum. The slope at such a given point is the corresponding element of the spectrum.()Corners on the lower boundary curve correspond to gaps in the spectrum.

We now exhibit two examples. In both examples, we will take and . The trace is integration with respect to Lebesgue measure and for , , and . Then and makes sense on .

Example 2.18. Define almost everywhere. Then is densely defined and self-adjoint on . It turns out that the equation of the lower boundary function is . This was obtained by integrating multiplied by appropriate characteristic functions. Observe that and . Hence the center of is and we get Figures 1 and 2.

Example 2.19. Define for and for . Then the lower boundary curve for is given by . And was chosen so that we would get a spectral scale that is invariant under the reflection . Note that , , , , and . The resulting pictures are shown in Figures 3 and 4.

We now examine a question posed to the author by Crandall. We start by stating the necessary properties that must have in order for it to be a spectral scale for an operator functional.

Definition 2.20. A prespectral scale is a set contained in which satisfies the following properties.(i) is compact and convex.(ii) and there are no other points of the form in .(iii).
Further, if and , then the following are given.
(iv) is invariant under the reflection .(v)The set is the graph of a function on [], which we will call the lower boundary curve of .

Lemma 2.21. Let be a prespectral scale with lower boundary curve . Then is a continuous convex function.

Proof. Since is closed, . Let . Since is convex, the line segment is a subset of . Now From the definition of , Hence, is convex on , and therefore continuous on [11, pages 61, 62]. Then and exist as extended real numbers [12, page 116]. Since is compact, then is finite and . Therefore, . By (ii) in Definition 2.20, . Applying (iv) from Definition 2.20, . Thus, is continuous and convex on .

Since is convex on , the left and right derivatives of exist for all as extended real numbers, and is differentiable almost everywhere [12, pages 113, 114].

It is easy to see that a spectral scale must be a prespectral scale: condition (i) is noted on page 3 of this paper, condition (ii) follows from Definition 2.1, condition (iii) follows from the fact that is a state, condition (iv) follows from Proposition 2.3, and condition (v) follows from the definition of the lower boundary (Notation 2.14).

Crandall asked whether a prespectral scale is automatically a spectral scale. In the next theorem, we show that the answer is yes.

Theorem 2.22. Let and let be the Lebesgue integral on . Given a prespectral scale , there exists self-adjoint such that .

Proof. From the symmetry required for (condition (v)) it is sufficient to examine the lower boundary curve, , of . The function has the following properties (as noted in Lemma 2.21):(i) is continuous and convex,(ii).
Let us denote as the right derivative of at . Similarly, denote as the left derivative of at . Let on [0,1) and . Since is convex, then is nondecreasing. Hence, has at most a countable number of discontinuities. Since is convex, then is of bounded variation. By Exercise .H in [13, page 244], is absolutely continuous. By Theorem in [11, page 148], , and the fundamental theorem of calculus holds. Let , with . Since is increasing, Hence, is the lower boundary curve of .

3. The Geometry of Spectral Scales in Higher Dimensions

This section is devoted to further generalizations of results from the original paper on spectral scales by Akemann et al. [1, Section , pages 276–280]. Often, with some modifications, the proofs are the same as in the original paper. Recall that is a Hilbert space, is a finite von Neumann algebra equipped with faithful, normal, tracial state, .

Notation 3. In Section 2 of this paper, we considered . We now consider an -tuple of self-adjoint linear functionals, . Let . Let . Then is also self-adjoint since each is self-adjoint and each is real. For each ( there is an associated self-adjoint, densely defined operator in , , and for every we have .

Definition 3.1. Let for every . Let for each . Then is the spectral scale of with respect to and is the spectral scale of with respect to .

Essentially the motivation for the introduction of is it allows us to reduce the -dimensional case to the 1-dimensional case by studying “2-dimensional cross-sections” of the spectral scale. We note that . Indeed, the right hand side may have trivial domain. However, as we will see, equality “almost” holds; that is, equality holds in trace.

Define , where .

Proposition 3.2. The equality holds.

Proof. For we have

Corollary 3.3. As a consequence of this calculation, .

We next introduce some additional notation.

Notation 4. Let be the spectral projection of determined by .
Let be the spectral projection of determined by .
Let .
Let .
Let .

The following results discuss the geometrical properties of .

Proposition 3.4. If is an extreme point of , the n there exists a projection , such that and . Further, is an extreme point of .

Proof. Fix an extreme point . Since is a face of , by Theorem 2.12 there are unique projections, in , such that . Thus, and so . Since is faithful, we have that , and so .
Next, suppose that for some , . Then Thus, By Lemma 2.11, . Since projections are extreme points in , then , and hence is an extreme point of .

Proposition 3.5. Suppose that . Then is a hyperplane of support for with In this case, .

Proof.
Let . Then
Therefore . Let . By Proposition 2.17, is a line of support for and . Fix . Then . Hence, and .

Fix and , and let vary over . The hyperplanes are all parallel and hence there exists a unique such that supports and . But we have seen that satisfies these conditions and so .

Proposition 3.6. If , then is a face of . Further, and .

Proof. Let . By our previous result, is a supporting hyperplane for . Hence, is a face of . By Theorem 2.12, there are unique projections in such that and . If and , then , and therefore
We would like to show that and .
Since , we have But and . Hence,
Therefore, , and so and .

Proposition 3.7. Let . Then supports , and passes through and .

Proof. If , then is a hyperplane of support for that contains and . But , so and . Therefore, Therefore,

4. Invariance Properties of the Spectral Scale

The main goal in this section is to establish the circumstances required for the spectral scale to determine (up to equivalence of tracial representations) the algebra and the -tuple, . Let be the algebra generated by 1 and where ranges over the bounded Borel subsets of and ranges from 1 to .

Observe that To see this, note that strongly. Hence, for a fixed , strongly and so weakly. Since is a bounded linear functional on , we have the desired convergence.

We now show that we only need to generate the spectral scale for the -tuple with respect to . By Proposition from [5, page 232], there exists a faithful normal projection , with such that . Hence for , Since is faithful and normal, as desired.

We now introduce additional notation and change some of the old notation.

Notation 5. Let and be finite von Neumann algebras equipped with faithful, normal, tracial states and , respectively. Let and be the associated Hilbert spaces obtained by the tracial Gelfand-Naimark-Segal (GNS) construction [2, pages 278, 279]. Let and be self-adjoint. Then there exist closed, densely defined, self-adjoint operators affiliated with such that for all . Similarly, there exist closed, densely-defined, self-adjoint operators affiliated with such that for all . Let be the von Neumann algebra generated by 1 and , where and ranges over the bounded Borel subsets of . Similarly, let be the von Neumann algebra generated by 1 and . Note that and . When we are concerned only with objects restricted to and , we will write and , respectively.

Let be the spectral scale for relative to determined by and the spectral scale for relative to determined by . Let and be the GNS representations of and . Let and be the canonical cyclic vectors that arise from this tracial GNS construction.

Definition 4.1. Suppose that there exists a surjective unitary transformation such that and for and all bounded Borel subsets of . Then the tracial representations of and are said to be equivalent.

This definition is unsatisfying since it requires uncountably many conditions. We believe that there exists a more satisfactory definition of equivalence using the 's and the 's. We have not to date been able to formulate such a definition.

Proposition 4.2. Suppose that . Then there exists an isometry, , from to such that for and .

Proof. Let us temporarily denote and . For , define .
We would first like to show that is well defined and can be extended linearly to the span of the 's. Suppose one of the 's is a linear combination of the others. Without loss of generality, . Let . Since , there exists such that for every . Thus, If is not zero, then . There exists such that for every . Therefore, Since any element in is a finite linear combination of elements in , it follows that if , then . Hence, is well-defined and we can therefore extend it linearly to linear combinations of the 's and hence to all of .
We now show that is an isometry. For let us denote . Let be the set of points in with norm 1, and let be the set of points in with norm 1. We need to show that
Consider . Then . There exists such that for , and . Let , and let . Hence, A similar calculation shows the reverse inequality and therefore

Notation 6. Recall that is the spectral projection of corresponding to and is the spectral projection of corresponding to . Let denote the spectral projections of on the same intervals.

Proposition 4.3. The following are equivalent: , for , for and , for and a bounded Borel function on , for every is a bounded Borel subset of .

Proof. ()(). Consider
()().
Suppose that . Then (relabeling if necessary) there exists a vector . Since is compact and convex and , there exists a hyperplane that strictly separates from . Thus, there exists , and such that for , we have Hence, . Therefore, for every , and so .
()().
Fix . There exists unique such that is a hyperplane of support for and . Since , then has the same properties with respect to . Hence, . Therefore, and so .
()().
Given (), () holds when is a characteristic function of an interval or . These intervals generate the Borel structure of . Now and are normal and linear. Since any bounded Borel function, , is uniformly approximated by linear combinations of characteristic functions, () holds for all such .
()().
Since characteristic functions on intervals are bounded and Borel, this is immediate.
()().
Take and . Then . Therefore, . Define on by for . is a bounded Borel function, and . Hence, for every nonzero . We have These are the extreme points of the lower boundaries of and , and so the lower boundaries coincide; hence the upper boundaries coincide by Proposition 2.3, and so .
()().
Define for . Then is a bounded Borel function, and hence, by assumption, .
()().
Define for . By assumption, , for every , a bounded Borel subset of , and every . But the 's are weakly dense in , and the 's are weakly dense in . Also, and are normal, and so , for being a bounded Borel function on .

Lemma 4.4. The tracial representations of and are equivalent if and only if for every , a monomial in variables, and , a bounded Borel subset of .

Proof. We begin by making the notation a little less cumbersome. Let = ,, and let .

Suppose that for every and .
Define . Extend this definition by linearity to polynomials. Let , be two such polynomials. Then for we have Such polynomials are dense in and , so extends to a unitary transformation from to with the desired properties.

Suppose that the tracial representations of and are equivalent. Then

Suppose that is Abelian and , are closed, densely defined operators affiliated with . Then and are closable, densely defined operators whose closures are affiliated with and , the set of operators affiliated with is an Abelian -algebra [2, pages 351, 352]. If in addition and are self-adjoint, then and are also self-adjoint and hence closed [14, page 536]. Further, we have for every . Thus, for every . Let be the positive part of and the negative part. Choose , where is any bounded Borel set. Then is positive since is commutative and But , since is faithful . Thus on its domain. Similarly , and so . Thus .

To proceed with the theory as given in [1, Section , page 281 ff.], it would be convenient if the following conjecture were true.

Conjecture 4.5. If and are Abelian and , then the tracial representations of and are equivalent.

By Lemma 4.4, it is enough to show that, if denotes a monomial in the commuting variables , then for every , a bounded Borel subset of . By part () of Proposition 4.3, we know that for every being a bounded Borel subset of . Let us fix and , and let be the set of all monomials in commuting variables such that . Routine computations show that Similarly, Since then we have In the bounded case, no characteristic functions are present and so we can equate coefficients of the polynomials. Even if we could do that here, we still do not have the desired result since we want something independent of .

5. Miscellaneous Results

A natural question is to ask whether convergence of -tuples of self-adjoint operators implies convergence of the corresponding spectral scales. Since spectral scales are compact and convex, the Hausdorff Metric is a natural metric to work with. The following definition is taken from [15, page 274].

Definition 5.1. Let be a metric space, with and being nonempty subsets of . Define . For , let us define Define Then is the Hausdorff distance between and .

We first establish a result for the original definition of a spectral scale, that is the spectral scale from Definition 1.4.

Theorem 5.2. Let . Suppose that strongly in each coordinate for where and are self-adjoint (in each coordinate). If and are the corresponding spectral scales, then in the Hausdorff metric induced by the usual topology on .

Proof. Write for the th coordinate of and for the th coordinate of . Let . Now and . Fix a . Define and note that is self-adjoint. Further, strongly, and so strongly as well. Since is normal, . Therefore, there exists such that . Since is a weight defined on all of [4, page 486], the map () is a positive-definite inner product on [4, page 489]. (The map is definite because is faithful.) Hence, the Cauchy-Schwarz inequality applies. For and , we have Let , and fix . For , let Note that all points in are of the form and all points in are of the form . Then by the inequality in (5.3) we have for every Hence for . Since was arbitrary, the result follows.

Theorem 5.2 is false if we replace strong convergence with weak convergence. Sherman came up with the following example in conversation with the author of this paper, and kindly gave permission for it to be included here.

Example 5.3. Let and . Then there exist , self-adjoint, such that weakly, but in the Hausdorff metric.

Proof. Let the th binary digit of is . Let where is the characteristic function on the set . Each is clearly self-adjoint, and standard analysis arguments show that the sequence converges weakly to . Now for every . Since for every , is the ball in centered at with radius 0.5 (see Figure 5). On the other hand, the spectral scale for is simply the line segment . Hence, for every . (The base metric for is the Euclidean norm.)

The next result concerns the unbounded situation that we have been dealing with for most of this paper.

Theorem 5.4. Let , be self-adjoint -tuples of bounded linear functionals on our finite von Neumann algebra . Suppose that in the dual norm. Then in the Hausdorff metric.

Proof. Let . Then there exists such that Let . We want to find that (for ) such that .
Since , then there exists such that . Define . Then Note that does not depend on . Hence taking supremums over all the 's, we have that . Hence, . Similarly, . Since was arbitrary, we have convergence in the Hausdorff metric as desired.

The spectral scale as given in Definition 1.4 is not the only object that has been called a spectral scale. The following definition of a spectral scale was formulated by Petz.

Definition 5.5 (see [3, page 74]). Let be a finite von Neumann algebra equipped with a faithful, normal, tracial state, . Let be a self-adjoint operator affiliated with . The Spectral Scale is defined for as follows: where as before.

Notation 7. We shall call this spectral scale the Petz spectral scale. We will call the spectral scale from Definition 2.1 the AAW spectral scale.

We now show how the two notions are related. To this end, we first find what values can take for a given . To this end, fix and note that we can write There are 3 cases to consider.

Case 1. There exists such that .
In this case, and hence, .

Case 2. For every , but there exists such that .
Since is closed, and is weak-* continuous, we may choose so that . Hence, is the smallest real value such that , and so .

Case 3. For every , .

Note that, in this case, and must be an unbounded operator, with unbounded spectrum on the right. In this case, .

The following result was proposed by Pavone in conversation with the author of this paper.

Proposition 5.6. Let be the function whose graph is the upper boundary curve of the AAW spectral scale. Then, for , .

Proof. By the rotational symmetry of the AAW spectral scale, where is the function whose graph is the lower boundary curve of the AAW spectral scale. If and , then . If , a gap in the spectrum, with then the right-hand derivative of at is .

At , the slope of is . If this is a finite number , then . If , then, as we saw in Case 3 above, . This completes our discussion of the relationship between the AAW spectral scale and the Petz spectral scale.

6. Future Research

A great deal of further work has been done with the spectral scale in the bounded situation. For us, the first question to ask is whether Conjecture 4.5 is in fact true. If so, we believe that many of the remaining results in [1] can be extended to the unbounded case fairly readily.

Additionally, we believe that the idea of a spectral scale of an unbounded operator can be used in the discussion of numerical range.

Definition 6.1. Let be a (bounded) linear operator on . Define
Then is the -numerical range of . When , we simply write , and refer to it as the numerical range [6, page 226].

We can write , where and are self-adjoint and so we can define the spectral scale of to be . It turns out that the boundary of is exactly the set of radial complex slopes on at the origin.

In the unbounded situation, we start with , where is a finite von Neumann algebra equipped with , a finite, faithful, normal, tracial state. We can certainly find such that and for every , but (except in the Abelian case) it is not obvious that there is any relationship between and the 's. We define the numerical range for by making the additional assumption in Definition 6.1 that the 's are in the domain of . At the moment, it is not clear that there is any relationship between and . However, if we can establish some kind of relationship, it is natural to ask how much of the theory developed in [6, 7] can be extended in this context.

Finally, we ask whether Theorem 2.22 can be extended beyond the single-variable situation?

Acknowledgments

The author wishes to thank Christopher Pavone, Roger Roybal, David Sherman, and especially Charles Akemann for several valuable conversations in connection with the material presented here.