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In the first part of this paper, we show that an AH algebra

A

The concept of

Note that the classification program of Elliott, the goal of which is to classify amenable

In the first part of this paper (Section

It is well known that the LP property of a

Let us recall some notations. Throughout the paper,

Denote by

Let

The last two authors appreciate Duy Tan University for the warm hospitality during our visit in September 2012 and the third author would also like to thank Teruya Tamotsu for fruitful discussions about the

Let

Let

For any integer

For any integer

The implication (iii)

To prove the implication (i)

Let us prove the implication (ii)

It is evident that every element in

Firstly, let

Let

For any integer

From the proof of Theorem

Let

For convenience of the reader, let us recall the notions from [

The

Suppose that

Let

Let

The

The

Let

The projections of

For any self-adjoint element

For any self-adjoint element

The following implications hold in general:

If

If

The statements (i) and (ii) are proved in Theorem 1.3 of [

An AH

In order to evaluate the eigenvalue variation [

The eigenvalue variations of two unitary equivalent self-adjoint elements are equal since their eigenvalues are the same. However, the converse need not be true in general. More precisely, there is a self-adjoint element

Let

It is straightforward to check that

Let

If

Let

It is clear that

For

The main result of this section as follows.

Given an AH algebra

By Corollary

Put

Therefore,

Let

In particular, if

By assumption, there exists a partial isometry

In the case

Let

We can assume that

On the other hand, it is straightforward to check that

In some special cases, small eigenvalue variation in the sense of Bratteli and Elliott and the LP property are equivalent.

Let

Indeed, (i) and (ii) are equivalent by [

In general, the LP property cannot imply small eigenvalue variation in the sense of Bratteli and Elliott nor real rank zero. For example, let

Looking for examples in the class of diagonal AH algebras, we need the following lemma.

Let

Let

Let

By Lemma

In this subsection, we will show that the LP property is not stable under the fixed point operation via the given examples. Firstly, we could observe the following example which shows that the LP property is not stable under the hereditary subalgebra.

Let

Note that

Since

Let

Using this observation, we can construct a

A simple unital AI algebra

According to Example

In this section we recall the

Let

A

When

We give several remarks about the above definitions.

Once we know that there exists a quasi-basis, we can choose one of the form

By the above statements, if

If

Next we recall the

Let

The

Watatani proved the following in [

If

Let

The

Suppose that

For a

We identify

Izumi defined the Rokhlin property for a finite group action in [

Let

Let

Motivated by Definition

A conditional expectation

The following result states that the Rokhlin property of an action in the sense of Izumi implies that the canonical conditional expectation from a given simple

Let

The following is the key one in the next section.

Let

The following is contained in [

Let

If

Note that from Remark

Since

Let

Consider the following:

To prove that the double dual conditional expectation

Let

Let

Since

Let

Before giving the proof, we need the following two lemmas, which must be well known.

Under the same conditions in Theorem

At first we prove condition

On the contrary, for any

Since for any

By the similar steps we will show conditions

The condition

Under the same conditions in Lemma

Note that

Let

Since

Since

When an action

When an action of a finite group

We could also have many examples which shows that the LP property is preserved under the formulation of crossed products from the following observation.

Let

with

with

It is obvious that the tracial Rokhlin property is weaker than the Rokhlin property.

Let

From [

Since

Therefore, by [

There are many examples of actions

The research of H. Osaka was partially supported by the JSPS grant for Scientific Research no. 23540256.