Extension of Spectral Scales to Unbounded Operators

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 .


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 n-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 M be a finite von Neumann algebra equipped with a normal, faithful tracial state, τ.Elements of M can be thought of as bounded operators on some Hilbert Space, H, 2, page 308 .For a given self-adjoint b ∈ M the corresponding spectral scale, B, which we will define below, yields information about the spectrum of b in a nice geometric way.Many of the results can be extended to n-tuples of self-adjoint operators in M. 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 b ∈ M, we consider g ∈ M * .
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 R 2 is a spectral scale.
In Section 3, we consider the geometric structure of the n-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.The last equation implies that ∞ / ∈ Im τ .Then τ is a faithful, finite, normal trace on M 4, pages 504-5 .
Theorem 1.3.Let τ be a faithful, finite, normal trace on M .Since any element of M can be written as a finite linear combination of positive elements of M, τ can be extended to a linear functional on all of M [5, page 309].
Two projections p, q in M are equivalent p ∼ q if there exists u ∈ M such that uu * p and u * u q.A projection p is finite if p ∼ q ≤ p ⇒ p q. M is finite if the projection 1 is finite 5, page 296 .Throughout, we will assume that M is finite.Further, we will assume that there exists a faithful, finite, normal trace τ of M, with τ 1 1; that is, τ is a faithful, normal, tracial state on M. A crucial property of τ is that "things" commute in trace-that is, although, in general ab / ba, for a, b ∈ M, we do have the equality τ ab τ ba 4, page 517 .Let Now τ is normal.Further Ψ is linear and continuous with respect to the weak operator topology.Moreover, M 1 is convex and compact in the weak operator topology: τ M 1 ∈ R and τ b i a τ a 1/2 b i a 1/2 ∈ R for i 1, . . ., n. Therefore B is a compact, convex subset of R n 1 .
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 B with the set τ a , g 1 a , g 2 a , . . ., g n a | a ∈ M 1 , 1.3 where each g i ∈ M * 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 L 1 M , the predual of M 8, Section 3, pages 112 ff. .The duality is given by the bilinear form a, b → τ ab τ ba 8, Section 3, page 112 for a ∈ M and b ∈ L 1 M .Now ba ∈ L 1 M 8, page 112 ff., and Nelson shows that elements of L 1 M are closed, densely defined operators affiliated with M 8, Theorem 1, page 107, and Theorem 5, page 114 .It follows that a bounded linear functional, g ∈ M * can be represented by a possibly unbounded linear operator b affiliated with M and we get the equality g a τ ba for every a ∈ M.

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 g ∈ M * be a self-adjoint linear functional.Let Then B g is the spectral scale of g with respect to τ.
From the theory of noncommutative integration, we see that B g { τ a , τ ba | a ∈ M 1 } for some operator b affiliated with M. Since g is self-adjoint, b too will be selfadjoint, and hence, as with the original spectral scale, our generalized spectral scale is a compact, convex subset of R 2 .Notation 1.We will often write B for B g .
The following definition was suggested to the author in conversation by Akemann.Definition 2.2 Akemann .If b is bounded, we will call g an operator functional.

International Journal of Mathematics and Mathematical Sciences
Our main goal in this section is to show that g is an operator functional if and only if the slopes of the lower boundary function of B are all finite.We remark that Akemann et al. have already shown the "only if" part of this statement 1, Section 1, pages 261-274 .For this reason, we may assume throughout that b is unbounded, and show that the lower boundary curve of B has, as a consequence, an infinite slope.To get there, we will need a number of preliminary results.
Thus the map υ : The fixed point of υ is 1/2, τ b /2 , and the points x 0 , x 1 and 1 − x 0 , τ b − x 1 lie on a straight line that passes through 1/2, τ b /2 .The straight line is given by the equation Note also that υ 2 is the identity map on R 2 .Hence, υ B B and υ reflects B through the point 1/2, τ b /2 .
For the next several results, we will need the unbounded spectral theorem for selfadjoint operators.We state it here in the functional calculus form.c If h n x → h x pointwisely, and the sequence For a given h, a bounded Borel function on R, it is customary to write φ h as h b .In other words, the "φ" is understood.For now it is more convenient to write φ explicitly.
p − s and φ χ −∞,s p s .More generally, if h is a characteristic function on a Borel subset of R, then φ h 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 s ∈ R, p s − p − s is nonzero on the domain of b and hence all of H if and only if s is an eigenvalue of b.Also, since Borel functions commute with respect to multiplication and φ is a homomorphism, Im φ is an Abelian subalgebra of B H . Assume now that b is affiliated with our finite von Neumann algebra, M. In this case it turns out that Im φ is an Abelian subalgebra of M. This follows from the way that φ is constructed.

2.4
Proof.Using the decomposition Of course, these equalities only make sense on the domain of b.
For every N > s we get:

2.10
Lemma 2.9.The range projection of b Proof.Let q be the range projection of b − s1 1 − p s .Let {h n } n∈N be a sequence of bounded Borel functions on R such that lim n → ∞ h n x xfor all x ∈ R, n ∈ N. Let h x χ s,∞ x .Then φ h 1 − p s .From the Spectral theorem, we have

2.11
Taking the limit as n → ∞ on the left side, we get We have shown that q ≤ 1−p s .For t > s let q t be the range projection of b−s1 p − t −p s .By the same reasoning as above, q t ≤ p − t − p s .Also, Hence, q t ≤ q.Similarly, for t 1 ≥ t 2 > s, we have q t as t → ∞.Since q t ≤ q for every t, 2.17 But we already know that 1 − p s ≥ q and so equality holds.The second statement in the lemma follows from an analogous proof.
Proof.Write a

2.20
Hence b 2 − s1 1/2 a * i3 ψ 0 and so b 2 − s1 a * i3 ψ 0. Since s is not an eigenvalue of b 2 , a * 13 ψ 0. Therefore and so From the Spectral theorem we can then conclude that Thus,

2.27
The other statement in the lemma follows from an analogous argument.
We remark that in the original paper on spectral scales 1, Lemma 1.2, pages 262, 263 , the above conclusion was obtained with a little less work, since, in that situation, b was bounded and so we did not have to worry about the domain of b.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 s ∈ R, c ∈ p − s , p s , and a ∈ M 1 .Suppose that τ a τ c .Then the following hold: and so τ ba ≥ τ bc .

2.32
and hence equality holds throughout.Thus

2.34
International Journal of Mathematics and Mathematical Sciences From 2.33 , while from 2.34 , Since τ is faithful and the arguments are positive, the arguments are in fact equal to zero.By Lemma 2.10, p − s ≤ a ≤ p s .
3 Suppose that c p ± s and τ ba τ bc .Then τ a τ p ± s .Since a is comparable to p ± s , and τ is faithful, a p ± s .
We next state a theorem proved by Akemann and Pedersen 10, Theorem 2.2 .
Theorem 2.12.If M and N are von Neumann algebras, Ψ is a normal linear map from M to N, and F a face of Ψ M 1 , then there are unique projections p and q in M with p ≤ q such that Ψ −1 F ∩ M 1 p, q and F Ψ p, q .
The following results are generalizations of the main theorems for the n 1 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.

2.37
The upper boundary of B is given by

2.38
The endpoints of the lower boundary are f 0 and f 1 .
Let σ b denote the spectrum of b, and let σ p be the point spectrum of b.Let f be the function on 0, 1 whose graph is the lower boundary.For s, α ∈ R, let

2.39
Let L ↑ s, α be the positive half-plane determined by L s, α .
Our next result describes the faces of the lower boundary of B. We do not include the endpoints at this time.

Theorem 2.13. 1 The zero-dimensional faces in the lower boundary of B are precisely the points of the form Ψ p ±
s for s ∈ σ b .Also, The one-dimensional faces in the lower boundary of B are the sets of the form F Ψ p − s , p s for s ∈ σ b .For each face F,

2.41
The slope of F is s.
Proof.We have the following steps.

2.44
We now show that Ψ p ± s is an extreme point of B. Suppose that

2.46
Since projections are extreme points in M 1 , a 1 a 2 p ± s . Step

2.49
Let F denote the line segment in the graph of f that contains Ψ p − s , p s and consider the endpoints of F. By Theorem 2.12, there are projections p < q such that p, q Ψ −1 F ∩ M 1 , and hence s and similarly q p s .Thus, Ψ p − s , p s is a line segment in graph f with slope s and

2.51
Step 3. We show that we have accounted for all of the graph of f, except possibly the endpoints.Fix a point with 0 < x 0 < 1, and assume that Ψ a / Ψ p ± s for every s ∈ σ b .Write

2.53
We would like to show that r

2.55
International Journal of Mathematics and Mathematical Sciences 13 If s < r 2 , s ∈ σ b , then by definition τ p s < x 0 < τ p r 2 .

2.56
Similarly, r 1 < s and for some s ∈ σ b , we would have which is clearly false.Hence,

2.60
But then p r 1 p − r 2 and τ p r 1 τ p − r 2 , which again is a contradiction.Hence, r 1 r 2 : r.

2.61
Since τ p − r < x 0 < τ p r , 2.62 p − r < p r so r is an eigenvalue of b.From Step 2, Ψ p − r , p r is a line segment in graph f .Hence x 0 , x 1 is on the interior of that line segment.

2 The line segments on the upper boundary are precisely the sets of the form
The slope of F is s and

2.64
Proof.This result is a direct consequence of applying Proposition 2.3 to Theorem 2.13.
Let s min be the left endpoint of σ b .Note that s min may be −∞.Let s max be the right endpoint of σ b .Note that s max may be ∞.

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Proposition 2.15.If s > s min , then the left derivative of the lower boundary function f at τ p − s exists and is given by the formula

2.65
If s < s max , then the right derivative of f at τ p s exists and is given by the formula Proof.Since B is convex, f is a convex function, and so the left and right derivatives exist.Fix ∅ which contradicts the choice of r.Thus r < s, and hence r is an isolated point in the spectrum, that is, r is an eigenvalue of b.Moreover, p r p − s and so Ψ p − s Ψ p r is the right-hand endpoint of a line segment in graph f with slope r by Theorem 2.13.Hence 2.67 for every n.Hence, for every n.Thus, Letting n → ∞ gives the desired result.
The statement regarding right derivatives is proved in a similar way.
Proposition 2.16.The corners of f are in one-to-one correspondence with the gaps of σ b , 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 −∞, s or s, ∞ .)

International Journal of Mathematics and Mathematical Sciences 15
Proof.Let r, t be an interior gap of the spectrum.Then for every s 1 , s 2 ∈ r, t we have p s 1 p − s 2 .Fix s ∈ r, t .Then

2.70
Hence, f is not differentiable at τ p − s , and so a gap in the spectrum corresponds to a corner.Conversely, we have already seen that f is differentiable at p ± s for s ∈ σ b .

2.71
The line L s, α is a line of support for B such that

2.72
In this case, L s, α passes through Ψ p ± s .Moreover, one has so Ψ p ± s lies in L s, α .We now wish to show that L s, α is a line of support for B. There are several cases to consider.
Case 1 s ∈ σ p b .In this situation, Ψ p − s and Ψ p s are endpoints of a line segment in graph f whose slope is s.L s, α passes through both points and has slope s.Thus, L s, α contains this line segment and is tangent to f. Hence B ⊂ L ↑ s, α .

International Journal of Mathematics and Mathematical Sciences
Case 2 s ∈ σ b \ σ p b .Note that s is not an isolated point in σ b .Moreover, p − s p s p s .At least one of the one-sided derivatives of f takes the value s at τ p s .Hence, B admits a line of support at Ψ p s with slope s.As with Case 1, the line is L s, α and B ⊂ L ↑ s, α .
Case 3 s 1 , s 2 is an interior gap in the spectrum .In this case p s Then L s 1 , α 1 and L s 2 , α 2 are lines of support passing through Ψ p s 1 whose slope lies between s 1 and s 2 .If L is a line of support for B whose slope lies between s 1 and s 2 , then L ↑ ⊃ B. But L s, α is such a line for s ∈ s 1 , s 2 .Hence, the statement is true for any s ∈ s 1 , s 2 .
Case 4 s < s min or s > s max .Since b is unbounded, at least one of s min and s max has infinite magnitude.Suppose that s min is finite and so s max must be infinite .Then s min ∈ σ b .Moreover, p − s min 0. L s min , 0 is a line of support for B at Ψ 0 and b ⊂ L ↑ s min , 0 by Case 1. Suppose s < s min .Then L s, 0 is also a line of support for B at Ψ 0 and B ⊂ L ↑ s, 0 .
The case for s > s max is dealt with similarly.Hence, for every s ∈ R, L s, α is a line of support for B and B ⊂ L ↑ s, α .
Conversely, for fixed s, the lines L s, β are all parallel as β varies over R. Hence, there exists a unique β 0 for which L s, β 0 is a line of support and B ⊂ L ↑ s, β 0 .But L s, α has these properties and hence α β 0 .
For the last statement, consider F L s, α ∩ B. Then F is a face of B. Hence, F is an extreme point or a line segment on graph f .
If F is an extreme point, then F {Ψ p ± s }, then s ∈ σ b .Since F is an extreme point, then F {ψ p − s } {Ψ p s }, and so

2.78
Similarly, if F is a line segment, then F Ψ p − s , p s for some s ∈ σ p b , and so

2.79
From the above results, if s min −∞, then the right derivative of f x approaches −∞ as x ↓ 0. If in addition s max ∞, then the left derivative of f x approaches ∞ as x ↑ 1.By Proposition 2.3, the graph of the upper boundary curve of B is vertical at x 0. Hence, the only line of support at Ψ 0 is vertical.Similarly, the only line of support at Ψ 1 is vertical.Therefore, if both s min and s max are nonreal, then Ψ 0 and Ψ 1 are not corners of B.
Conversely, if one of s min and s max is finite, then Ψ 0 and Ψ 1 are corners of B.
Here the bounded and unbounded spectral scale theories do not coincide, since, in the bounded situation, Ψ 0 and Ψ 1 are always corners.
In both situations, we can read spectral data of the lower boundary curve as follows We now exhibit two examples.In both examples, we will take H L 2 0, 1 and M L ∞ 0, 1 .The trace τ is integration with respect to Lebesgue measure and a ψ x : a x ψ x for a ∈ M, ψ ∈ H, and x ∈ 0, 1 .Then M * L 1 0, 1 and τ makes sense on M * .2. Hence the center of B is P 0.5, 1 and we get Figures 1 and 2.
We now examine a question posed to the author by Crandall.We start by stating the necessary properties that U ⊂ R 2 must have in order for it to be a spectral scale for an operator functional.
Definition 2.20.A prespectral scale is a set U contained in R 2 which satisfies the following properties.
i U is compact and convex.
ii 0, 0 ∈ U and there are no other points of the form 0, y in U.
iii   Proof.

2.81
From the definition of G, tf a Hence, f is convex on 0, 1 , and therefore continuous on 0, 1 11, pages 61, 62 .Then lim x↓0 f x and lim x↑1 f x exist as extended real numbers 12, page 116 .Since U is compact, then lim x↓0 f x is finite and lim x↓0 x, f x ∈ U. Therefore, 0, lim x↓0 f x ∈ U.By ii in Definition 2.20, lim x↓0 f x 0 f 0 .Applying iv from Definition 2.20, lim x↑1 f x f 1 .Thus, f is continuous and convex on 0, 1 .
Since f is convex on 0, 1 , the left and right derivatives of f exist for all x ∈ 0, 1 as extended real numbers, and f 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 τ is a state, condition iv follows from Proposition 2.3, and condition v follows from the definition of the lower boundary Notation 2.14 .

International Journal of Mathematics and Mathematical Sciences
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 M L ∞ 0, 1 and let τ be the Lebesgue integral on 0, 1 .Given a prespectral scale U, there exists g ∈ L 1 0, 1 M * self-adjoint such that U B g .
Proof.From the symmetry required for U condition v it is sufficient to examine the lower boundary curve, f, of U. The function f has the following properties as noted in Lemma 2.21 : Let us denote f R t as the right derivative of f at t ∈ 0, 1 .Similarly, denote f L t as the left derivative of f at t ∈ 0, 1 .Let g t f R t on 0,1 and g 1 f L 1 .Since f is convex, then g is nondecreasing.Hence, g has at most a countable number of discontinuities.Since f is convex, then f is of bounded variation.By Exercise 14.H in 13, page 244 , f is absolutely continuous.By Theorem 7.20 in 11, page 148 , g ∈ L 1 0, 1 M * , and the fundamental theorem of calculus holds.Let a ∈ M 1 , with 1 0 a s ds t ∈ 0, 1 .Since g is increasing, Hence, f is the lower boundary curve of B g .

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.Essentially the motivation for the introduction of g t is it allows us to reduce the ndimensional case to the 1-dimensional case by studying "2-dimensional cross-sections" of the spectral scale.We note that b t / n i 1 t i b i .Indeed, the right hand side may have trivial domain.However, as we will see, equality "almost" holds; that is, equality holds in trace.
Define π t x 0 , x 1 , . . ., x n x 0 , n i 1 t i x i , where x 0 , x i ∈ R.
The following results discuss the geometrical properties of B.
Proposition 3.4.If x is an extreme point of B, the n there exists a projection p ∈ M, such that Ψ p x and Proof.Fix an extreme point x ∈ B. Since {x} is a face of B, by Theorem 2.12 there are unique projections, p ≤ q in M, such that Ψ −1 x ∩ M 1 p, q .Thus, Ψ p Ψ q x and so τ p τ q .Since τ is faithful, we have that p q, and so Thus,
and Ψ a ∈ P ↑ t, s, α .⇒ Fix t and s, and let β vary over R. The hyperplanes P t, s, β are all parallel and hence there exists a unique β 0 such that P t, s, β 0 supports B and B ⊂ P ↑ t, s, β 0 .But we have seen that α satisfies these conditions and so α β 0 .
and F Ψ p − t,s , p t,s .
Proof.Let α τ b t − s1 p ± t,s .By our previous result, P t, s, α is a supporting hyperplane for B. Hence, F : P t, s, α ∩ B is a face of B. By Theorem 2.12, there are unique projections p ≤ q in M such that Ψ −1 F ∩ M 1 p, q and Ψ p, q F. If a ∈ M 1 and Ψ a ∈ P t, s, α , then Ψ a ∈ P t, s, α ∩ B, and therefore p, q We would like to show that p p − t,s and q p t,s .Since Ψ p ± t,s ∈ F, we have 3.9 But π t P t, s, α L t s, α and π t B B t .Hence, Therefore, 3.12

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 n-tuple, g 1 , . . ., g n .Let N be the algebra generated by 1 and b i χ w b i where w ranges over the bounded Borel subsets of R and i ranges from 1 to n.
• is a bounded linear functional on M, we have the desired convergence.We now show that we only need N to generate the spectral scale for the n-tuple g 1 , . . ., g n with respect to τ.By Proposition 2.35 from 5, page 232 , there exists a faithful normal projection E : M → N, with E 1 such that τ τ • E. Hence for a ∈ M,

4.1
Since E is faithful and normal, Ψ M 1 Ψ N 1 as desired.We now introduce additional notation and change some of the old notation.Notation 5. Let U and V be finite von Neumann algebras equipped with faithful, normal, tracial states τ U and τ V , respectively.Let H U and H V be the associated Hilbert spaces obtained by the tracial Gelfand-Naimark-Segal GNS construction 2, pages 278, 279 .Let g 1 , . . ., g n ∈ U * and h 1 , . . ., h n ∈ V * be self-adjoint.Then there exist b 1 , . . ., b n closed, densely defined, self-adjoint operators affiliated with U such that τ U b i u g i u for all u ∈ U. Similarly, there exist c 1 , . . ., c n closed, densely-defined, self-adjoint operators affiliated with V such that International Journal of Mathematics and Mathematical Sciences τ V c i v h i v for all v ∈ V .Let M be the von Neumann algebra generated by 1 and b i χ ω b i , where i 1, . . ., n and ω ranges over the bounded Borel subsets of R. Similarly, let N be the von Neumann algebra generated by 1 and c i χ ω c i .Note that g 1 , . . ., g n ∈ M * and h 1 , . . ., h n ∈ N * .When we are concerned only with objects restricted to M and N, we will write • M and • N , respectively.
Let B be the spectral scale for b 1 , . . ., b n relative to τ M determined by Ψ M and C the spectral scale for c 1 , . . ., c n relative to τ N determined by Ψ N .Let π M and π N be the GNS representations of M and N. Let ξ M and ξ N be the canonical cyclic vectors that arise from this tracial GNS construction.This definition is unsatisfying since it requires uncountably many conditions.We believe that there exists a more satisfactory definition of equivalence using the g i 's and the h i 's.We have not to date been able to formulate such a definition.Proposition 4.2.Suppose that B C. Then there exists an isometry, Φ, from M * to N * such that Φ g i h i for i 1, . . ., n and Φ τ M τ N .
Proof.Let us temporarily denote τ M g n 1 and τ N h n 1 .For i 1, . . ., n 1, define Φ g i h i .We would first like to show that Φ is well defined and can be extended linearly to the span of the g i 's.Suppose one of the g i 's is a linear combination of the others.Without loss of generality, g 1 ∞ n 2 α i g i .Let v ∈ N 1 .Since B C, there exists u ∈ M 1 such that g i u h i v for every i.Thus, There exists u ∈ M 1 such that g i u h i w for every i.Therefore, Since any element in N is a finite linear combination of elements in N , it follows that if Hence, Φ is well-defined and we can therefore extend it linearly to linear combinations of the g i 's and hence to all of M * .
We now show that Φ is an isometry.For v ∈ M * let us denote Φ v Φ v .Let M 1 be the set of points in M with norm 1, and let N 1 be the set of points in N with norm 1.We need to show that v sup Consider x ≥ 0. Then x ∈ M 1 .There exists y ∈ N 1 such that g i x h i y for i 1, . . ., n 1, and y / 0. Let w y/||y||, and let v n 1 i 1 α i g i .Hence,

4.5
A similar calculation shows the reverse inequality and therefore sup  Define uπ M φ b ω ξ M π N φ c ω ξ N .Extend this definition by linearity to polynomials.Let φ 1 , φ 2 be two such polynomials.Then for i 1, . . ., n we have

4.12
Such polynomials are dense in H M and H N , so u extends to a unitary transformation from H M to H N with the desired properties.⇒ Suppose that the tracial representations of M and N are equivalent.Then

4.13
Suppose that M is Abelian and d, e are closed, densely defined operators affiliated with M. Then d e and de are closable, densely defined operators whose closures are affiliated with M and M , the set of operators affiliated with M is an Abelian * -algebra 2, pages 351, 352 .If in addition d and e are self-adjoint, then d e and de are also self-adjoint and hence closed 14, page 536 .Further, we have for every a ∈ M. Thus, To proceed with the theory as given in 1, Section 3, page 281 ff., it would be convenient if the following conjecture were true., x n x k 1 1 • • • x k n n denotes a monomial in the commuting variables x 1 , . . ., x n , then 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 t.

Miscellaneous Results
A natural question is to ask whether convergence of n-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 .We first establish a result for the original definition of a spectral scale, that is the spectral scale from Definition 1.4.

5.4
Theorem 5.4.Let g n g 1n , . . ., g mn , g g 10 , . . ., g m0 be self-adjoint m-tuples of bounded linear functionals on our finite von Neumann algebra M. Suppose that g jn → g j0 in the dual norm.Then B g n → B g in the Hausdorff metric.

5.8
Note that does not depend on j.Hence taking supremums over all the j's, we have that α − β ∞ < .Hence, B g ⊂ B g n 2 .Similarly, B g n ⊂ B g 2 .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 A be a finite von Neumann algebra equipped with a faithful, normal, tracial state, τ.Let b be a self-adjoint operator affiliated with A. The Spectral Scale is defined for t ∈ 0, 1 as follows: inf s | 1 − τ p s ≤ t , 5.9 where p s χ −∞,s b 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 λ t b can take for a given t.To this end, fix t and note that we can write λ t b inf s | τ p s ≥ 1 − t .

5.10
There are 3 cases to consider.
Case 1.There exists s 0 ∈ σ b such that τ p s 0 1 − t.In this case, s < s 0 ⇒ τ p s < 1 − t and hence, λ t b s 0 .
Since σ b is closed, and τ is weak- * continuous, we may choose s 0 ∈ σ b so that s 0 > s 1 ⇒ τ p s 1 < 1 − t.Hence, s 0 is the smallest real value such that τ p s 0 > 1 − t, and so s 0 λ t b .

Definition 1 . 1 .
Let H be a Hilbert space.Let M be a subalgebra of B H .If M is closed in the weak operator topology, self-adjoint, and contains 1, then M is a von Neumann algebra 2, page 308 .Let M denote the set of positive elements of M. Definition 1.2.Let τ : M → 0, ∞ be a function such that for a, b, a α ∈ M and λ, μ ∈ 0, ∞ we have: τ λa μb λτ a μτ b , τ a * a τ aa * , a α ↑ a ⇒ τ a α ↑ τ a normal ,

Theorem 2 . 4
see von Neumann in 9, page 562 .Let b be a (densely defined) self-adjoint operator in H with domain D b .Then ∃! algebraic * -homomorphism φ takes bounded Borel functions on R into B H such that the following hold.a φ is norm continuous.b Let {h n x } n∈N be a sequence of bounded Borel functions with h n x → x as n → ∞ for each x and |h n x | ≤ |x| for every x ∈ R and n ∈ N. Then for ψ ∈ D b , φ h n ψ → bψ as n → ∞.The convergence is in norm.

Notation 2 .
Recall that we are assuming that M ⊂ B H is a finite von Neumann algebra equipped with a faithful, normal, tracial state τ.The operator b is unbounded and self-adjoint on H affiliated with M obtained from a linear functional g ∈ M * i.e., g a τ ba for each a ∈ M .B : { τ a , τ ba | a ∈ M 1 } is the spectral scale of b.The lower boundary of B is given by Fix s ∈ R. If s is an eigenvalue, then b p s − p − s s p s − p − s .

Figure 1 :
Figure 1: Graph of b for Example 2.18.

Example 2 .
18. Define b x 1/x almost everywhere.Then b ∈ M * is densely defined and self-adjoint on H.It turns out that the equation of the lower boundary function is w f y 2 − 2 1 − y .This was obtained by integrating b multiplied by appropriate characteristic functions.Observe that 1 ≤ f y ≤ ∞ and τ b

Figure 2 :
Figure 2: Spectral scale for b in Example 2.18.

Figure 3 :
Figure 3: Graph of b for Example 2.19.

Figure 4 :
Figure 4: Spectral scale for b in Example 2.19.

Definition 4 . 1 .
Suppose that there exists a surjective unitary transformation u :H M → H N such that uξ M ξ N and uπ M b i χ ω b iπ N c i χ ω c i u for i 1, . . ., n and all bounded Borel subsets ω of R. Then the tracial representations of M and N are said to be equivalent.

. 15 for
every a ∈ M. Let b be the positive part of b t − n i 1 t i b i and b − the negative part.Choose a b χ ω b , where ω is any bounded Borel set.Then a is positive since M is commutative and b t − n i 1 t i b i a b 2 χ ω b ≥ 0. 4.16 International Journal of Mathematics and Mathematical Sciences But τ b 2 χ ω b 0, since τ is faithful b 2 χ ω b 0. Thus b 0 on its domain.Similarly b − 0, and so b

Conjecture 4 . 5 .
If M and N are Abelian and B C, then the tracial representations of M and N are equivalent.By Lemma 4.4, it is enough to show that, if φ x 1 , . . .

Definition 5 . 1 .
Let X, d be a metric space, with A and B being nonempty subsets of X. Defined A, B inf{d a, b | a ∈ A, b ∈ B}.For γ > 0, let us define A γ x ∈ X | d {x}, A < γ , B γ x ∈ X | d {x}, B < γ .inf γ > 0 | A ⊂ B γ , B ⊂ A γ .5.2Then d H A, B is the Hausdorff distance between A and B.

Theorem 5 . 2 .≤ τ a 2 Let
Let m ∈ N. Suppose that b n → b strongly in each coordinate for b n , b ∈ M m where b n and b are self-adjoint (in each coordinate).If B b n and B b are the corresponding spectral scales, then B b n → B b in the Hausdorff metric induced by the usual topology on R m 1 .Proof.Write b jn for the jth coordinate of b n and b j0 for the jth coordinate of b.Let > 0. Now B b n { τ a , τ b 1n a , . . ., τ b mn a | a ∈ M 1 } and B b { τ a , τ b 10 a , . . ., τ b m0 a | a ∈ M 1 }.Fix a j.Define c jn b jn − b j0 and note that c jn is self-adjoint.Further, c jn → 0 strongly, and so c 2 jn → 0 strongly as well.Since τ is normal, τ c 2 jn → 0. Therefore, there exists N j ∈ N such that n ≥ N j ⇒ |τ c 2 jn | < 2 .Since τ is a weight defined on all of M 4, page 486 , the map a, b → τ b * a a, b ∈ M is a positive-definite inner product on M 4, page 489 .The map is definite because τ is faithful.Hence, the Cauchy-Schwarz inequality applies.For a ∈ M 1 and n ≥ N j , we have τ c jn a a, c jn N max j 1,...,m N j , and fix n ≥ N.For a ∈ M 1 , let α a τ a , τ b 1n a , . . ., τ b mn a , β a τ a , τ b 10 a , . . ., τ b m0 a .

Lemma 2.7. Let
b 1 − p s ≥ s 1 − p s and hence b − s1 1 − p s ≥ 0. The other statement in this lemma follows via a similar argument.hbeacharacteristic function of a bounded Borel subset of R. Then Im φ h ⊂ D b .International Journal of Mathematics and Mathematical SciencesProof.Let h n be a sequence of bounded Borel functions such that lim n → ∞ h n xx for all x ∈ R and |h n x | ≤ |x|.By the Spectral theorem, 2. For s ∈ σ p b , Ψ p − s , p s are faces of B. − s , p s , and Ψ p λ is a typical point on the line segment connecting Ψ p − s and Ψ p s .Hence graph f contains this line segment.The slope is s 1 − λ p s for all λ ∈ 0, 1 .Then p λ ∈ p k i, t there is an associated self-adjoint, densely defined operator in H, b k , and for every a ∈ M we have g k a τ b k a .Then B : Ψ M 1 is the spectral scale of b 1 , . . ., b n with respect to τ and B t : Ψ t M 1 is the spectral scale of g t with respect to τ.
1, Section 2, pages 276-280 .Often, with some modifications, the proofs are the same as in the original paper.Recall that H is a Hilbert space, M ⊂ B H is a finite von Neumann algebra equipped with faithful, normal, tracial state, τ.Notation 3. In Section 2 of this paper, we considered g ∈ M * .We now consider an n-tuple of self-adjoint linear functionals, g 1 , . . ., g n ∈ M n * .Let t t 1 , . . ., t n ∈ R n \ {0}.Let g t n i 1 t i g i .Then g t is also self-adjoint since each g i is self-adjoint and each t i is real.For each g k Definition 3.1.Let Ψ a τ a , g 1 a , . . ., g n a for every a ∈ M. Let Ψ t a τ a , g t a for each a ∈ M.
As a consequence of this calculation, B t π t B .Let p t,s be the spectral projection of b t determined by −∞, s .Let p − t,s be the spectral projection of b t determined by −∞, s .

. 6
Notation 6.Recall that p t,s is the spectral projection of b t corresponding to −∞, s and p − t,s is the spectral projection of b t corresponding to −∞, s .Let q ± t,s denote the spectral projections of c t on the same intervals.Then relabeling if necessary there exists a vector x x 0 , x 1 , . . ., x n ∈ B \ C. Since C is compact and convex and x / ∈ C, there exists a hyperplane that strictly separates C from x.Thus, there exists t t 0 , t 1 , . . ., t n ∈ R n 1 , and β ∈ R such that for y y 0 , y 1 , . . ., y n ∈ C, we have There exists unique α such that P t, s, α is a hyperplane of support for B and B ⊂ P ↑ t, s, α .Since B C, then α has the same properties with respect to C. Hence, P t, s, α ∩ B P t, s, α ∩ C. Therefore, Given 3 , 4 holds when f is a characteristic function of an interval −∞, s or −∞, s .These intervals generate the Borel structure of R. Now τ M and τ N are normal and linear.Since any bounded Borel function, f, is uniformly approximated by linear combinations of characteristic functions, 4 holds for all such f.By assumption, τ M h b t τ N h c t , for every ω, a bounded Borel subset of R, and every k ∈ N.But the h b t 's are weakly dense in M, and the h c t 's are weakly dense in N. Also, τ M and τ N are normal, and so τ M f b t τ N f c t , for f being a bounded Borel function on R.
k .Therefore, π t x / π t y for every y ∈ C, and soπ t x ∈ B t \ C t . 1 ⇒ 3 .Fix s, t. ± on R by g ± a af ± a for a ∈ R. g ± is a bounded Borel function, andg ± b t b t p ± t,s .Hence, τ M b t p ± t,s τ N c t q ± t,s for every nonzero t ∈ R n , s ∈ R. We have τ M p ± t,s , τ M b t p ± t,s τ N q ± t,s , τ N c t q ± t,s .4.10These are the extreme points of the lower boundaries of B t and C t , and so the lower boundaries coincide; hence the upper boundaries coincide by Proposition 2.3, and so B t C t .k for s ∈ R.
c n χ ω c n 4.17 for every ω, a bounded Borel subset of R. By part 5 of Proposition 4.3, we know that τ M b t χ ω b t k τ N c t χ ω c t k for every k ∈ N, t ∈ R n \ {0},with ω being a bounded Borel subset of R. Let us fix k and ω, and let P be the set of all monomials φ in n commuting variables such that n i 1 k i k.Routine computations show that τ M b t χ ω b t