We extend the notion of a spectral scale to

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

In Section

In Section

In Section

Section

Finally, in Section

Let us start with some preliminary definitions.

Let

Let

Let

Then

Let

Two projections

Further, we will assume that there exists a faithful, finite, normal trace

A crucial property of

Let

Now

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

In 2004, Akemann and David Sherman conjectured that, if we replace

Some results on “noncommutative integration” will prove useful in our exposition. We will use Nelson's 1972 [

In his paper, Nelson defines

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

Let

From the theory of noncommutative integration, we see that

We will often write

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

If

Our main goal in this section is to show that

Let

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.

Let

Let

If

If

If

For a given

For

For the most part, we will only need spectral projections obtained from intervals. Note that, for

Let

Using the decomposition

Hence,

For every

Therefore

Let

Let

The following set relations hold:

The range projection of

Let

Taking the limit as

We have shown that

Now

Hence,

The second statement in the lemma follows from an analogous proof.

Let

If

Write

Note that

We remark that in the original paper on spectral scales [

Fix

If

If

Note that

(

(

Since

Suppose that

We next state a theorem proved by Akemann and Pedersen [

If

The following results are generalizations of the main theorems for the

Recall that we are assuming that

Let

Our next result describes the faces of the lower boundary of

We have the following steps.

We show that

Fix

But

We now show that

For

Fix

Write

Let

We show that we have accounted for all of the graph of

Fix a point

By definition,

Suppose that

This result is a direct consequence of applying Proposition

Let

If

Since

If

Choose

The corners of

Let

For each

Fix

In this situation,

Note that

In this case

Since

The case for

Conversely, for fixed

For the last statement, consider

If

Similarly, if

From the above results, if

Conversely, if one of

Here the bounded and unbounded spectral scale theories do not coincide, since, in the bounded situation,

In both situations, we can read spectral data of the lower boundary curve as follows

1-dimensional faces correspond to eigenvalues of

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

Define

Graph of

Spectral scale for

Define

Graph of

Spectral scale for

We now examine a question posed to the author by Crandall. We start by stating the necessary properties that

A

Further, if

The set

Let

Since

Since

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

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

Let

From the symmetry required for

Let us denote

This section is devoted to further generalizations of results from the original paper on spectral scales by Akemann et al. [

In Section

Let

Essentially the motivation for the introduction of

Define

The equality

For

As a consequence of this calculation,

We next introduce some additional notation.

Let

Let

Let

Let

Let

The following results discuss the geometrical properties of

If

Fix an extreme point

Next, suppose that

Suppose that

Let

Therefore

Fix

If

Let

We would like to show that

Since

Therefore,

Let

If

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

Observe that

We now show that we only need

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

Let

Let

Suppose that there exists a surjective unitary transformation

This definition is unsatisfying since it requires uncountably many conditions. We believe that there exists a more satisfactory definition of equivalence using the

Suppose that

Let us temporarily denote

We would first like to show that

We now show that

Consider

Recall that

The following are equivalent:

(

(

Suppose that

(

Fix

(

Given (

(

Since characteristic functions on intervals are bounded and Borel, this is immediate.

(

Take

(

Define

(

Define

The tracial representations of

We begin by making the notation a little less cumbersome. Let

Suppose that

Define

Suppose that the tracial representations of

Suppose that

To proceed with the theory as given in [

If

By Lemma

A natural question is to ask whether convergence of

Let

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

Let

Write

Theorem

Let

Let

Spectral scales for Example

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

Let

Let

Since

The spectral scale as given in Definition

Let

We shall call this spectral scale the

We now show how the two notions are related. To this end, we first find what values

There exists

In this case,

For every

Since

For every

Note that, in this case,

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

Let

By the rotational symmetry of the AAW spectral scale,

At

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

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

Let

Then

We can write

In the unbounded situation, we start with

Finally, we ask whether Theorem

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.