The Linear Span of Projections in AH Algebras and for Inclusions of 𝐶^∗-Algebras

Abstract

In the first part of this paper, we show that an AH algebra 𝐴=lim_→(𝐴_𝑖,𝜙_𝑖) has the LP property if and only if every element of the centre of 𝐴_𝑖 belongs to the closure of the linear span of projections in 𝐴. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unital 𝐶^∗-algebras 𝑃⊂𝐴 with a finite Watatani index, if a faithful conditional expectation 𝐸∶𝐴→𝑃 has the Rokhlin property in the sense of Kodaka et al., then 𝑃 has the LP property under the condition that𝐴 has the LP property. As an application, let 𝐴 be a simple unital 𝐶^∗-algebra with the LP property, 𝛼 an action of a finite group 𝐺 onto Aut(𝐴). If 𝛼 has the Rokhlin property in the sense of Izumi, then the fixed point algebra 𝐴^𝐺 and the crossed product algebra 𝐴⋊_𝛼𝐺 have the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.

1. Introduction

A 𝐶^∗-algebra is said to have the LP property if the linear span of projections (i.e., the set of all linear combinations of projections in the algebra) is dense in this algebra. A picture of the problem which asks to characterize the simple 𝐶^∗-algebras to have the LP property was considered in [1]. The LP property of a 𝐶^∗-algebra is weaker than real rank zero since the latter means that every self-adjoint element can be arbitrarily closely approximated by linear combinations of orthogonal projections in this 𝐶^∗-algebra. In the class of simple AH algebras with slow dimension growth, real rank zero and small eigenvalue variation in the sense of Bratteli and Elliott are equivalent (see [2, 3]). It is not known whether the equivalence still holds when the algebras do not have slow dimension growth.

The concept of diagonal AH algebras (AH algebra which can be written as an inductive limit of homogeneous 𝐶^∗-algebras with diagonal connecting maps) was introduced in [4] or [5]. Let us denote by 𝒟 the class of diagonal AH algebras. AF-, AI-, and AT-algebras, Goodearl algebras [6], and Villadsen algebras of the first type [7] are diagonal AH algebras. The algebras constructed by Toms in [8] specially which have the same 𝐾-groups and tracial data but different Cuntz semigroups are Villadsen algebras of the first type and so belong to 𝒟. This means that the class 𝒟 contains “ugly” and interesting 𝐶^∗-algebras and has not been classified by Elliott’s program so far.

Note that the classification program of Elliott, the goal of which is to classify amenable 𝐶^∗-algebras by their 𝐾-theoretical data, has been successful for many classes of 𝐶^∗-algebras, in particular for simple AH algebras with slow dimension growth (see, e.g., [9–11]). Unfortunately, for AH algebras with higher dimension growth, very little is known.

In the first part of this paper (Section 2), we consider the LP property of inductive limits of matrix algebras over 𝐶^∗-algebras. The necessary and sufficient conditions for such an inductive limit to have the LP property will be presented in Theorem 1. In particular, we will show that an AH algebra 𝐴=lim_⟶(𝐴_𝑖,𝜙_𝑖) (which need not be diagonal nor simple) has the LP property provided that the image of every element of the centre of the building blocks 𝐴_𝑖 can be approximated by a linear combination of projections in 𝐴 (Corollary 2). In Section 2.4, using the idea of bubble sort, we can rearrange the entries on a diagonal element in 𝑀_𝑛(𝐶(𝑋)) to obtain a new diagonal element with increasing entries such that the eigenvalue variations are the same (Lemma 5) and the eigenvalue variation of the latter element is easy to evaluate. As a consequence, it will be shown that a diagonal AH algebra has the LP property if it has the small eigenvalue property (Theorem 6) without any condition on the dimension growth.

It is well known that the LP property of a 𝐶^∗-algebra 𝐴 is inherited to the matrix tensor product 𝑀_𝑛(𝐴) and the quotient 𝜋(𝐴) for any ∗-homomorphism 𝜋. But it is not stable under the hereditary subalgebra of 𝐴. In the second part of this paper (Section 3), we will present the stability of the LP property of an inclusion of a unital 𝐶^∗-algebra with certain conditions and some examples illustrated the instability of such the property. More precisely, let 1∈𝑃⊂𝐴 be an inclusion of unital 𝐶^∗-algebras with a finite Watatani index and 𝐸∶𝐴→𝑃 a faithful conditional expectation. Then the LP property of 𝑃 can be inherited from that of 𝐴 provided that 𝐸 has the Rokhlin property in the sense of Osaka and Teruya (Theorem 23). As a consequence, given a simple unital 𝐶^∗-algebra 𝐴 with the LP property if an action 𝛼 of a finite group 𝐺 to Aut(𝐴) has the Rokhlin property in the sense of Izumi, then the fixed point algebra 𝐴^𝐺 and the crossed product 𝐴⋊_𝛼𝐺 have the LP property (Theorem 24). Furthermore, we also give an example of a simple unital 𝐶^∗-algebra with the LP property, but its fixed point algebra does not have the LP property (Example 14).

Let us recall some notations. Throughout the paper, 𝑀_𝑛 stands for the algebra of all 𝑛×𝑛 complex matrices, 𝑒={𝑒_𝑠𝑡}_{𝑠,𝑡=1,𝑛} denotes the standard basis of 𝑀_𝑛 (for convenience, we also use this system of matrix unit for any size of matrix algebras). Let us denote by 𝑀_𝑛(𝐶(𝑋)) the matrix algebra with entries from the algebra 𝐶(𝑋) of all continuous functions on space 𝑋. If 𝑋 has finitely many connected components 𝑋_𝑖 and 𝑋=⊔^𝑘_𝑖=1𝑋_𝑖, then 𝑀_𝑛(𝐶(𝑋))=^𝑘_𝑖=1𝑀_𝑛𝐶𝑋_𝑖. (1) Hence, without lots of generality we can always assume that the spectrum of each component of a homogeneous 𝐶^∗-algebra is connected.

Denote by diag(𝑎₁,𝑎₂,…,𝑎_𝑛) the block diagonal matrix with entries 𝑎₁,𝑎₂,…,𝑎_𝑛 in some algebras.

Let 𝐴 be a 𝐶^∗-algebra. Any element 𝑎 in 𝐴 can be considered as an element in 𝑀_𝑛(𝐴) via the embedding 𝑎→diag(𝑎,0). We also denote by 𝐿(𝐴) the closure of the set of all linear combinations of finitely many projections in 𝐴.

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 𝐶^∗-index theory.

2. Linear Span of Projections in AH Algebras

2.1. Linear Span of Projections in an Inductive Limit of Matrix Algebras over 𝐶^∗-Algebras

Let 𝐴=lim_⟶𝐴_𝑖,𝜑_𝑖, (2) where 𝐴_𝑖=⊕^𝑘𝑖_𝑡=1𝑀_𝑛𝑖𝑡(𝐵_𝑖𝑡) and 𝐵_𝑖𝑡 are 𝐶^∗-algebras. Let 𝑆_𝑖𝑡 be a spanning set of 𝐵_𝑖𝑡 (as a vector space) and 𝑆_𝑖 be the union of 𝑆_𝑖𝑡 for 𝑡=1,…,𝑘_𝑖. Since every element of a 𝐶^∗-algebra can be written as a sum of two self-adjoint elements, we can assume that all elements of 𝑆_𝑖 are self-adjoint.

Theorem 1. Let 𝐴 be an inductive limit 𝐶^∗-algebra as above. Then the following statements are equivalent.(i)𝐴has the LP property.(ii)For any integer 𝑖, any 𝑥∈𝑆_𝑖 and any 𝜀>0, there exists an integer 𝑗≥𝑖 such that 𝜑_𝑖𝑗(𝑥) can be approximated by an element in 𝐿(𝐴_𝑗) to within 𝜀.(iii)For any integer 𝑖, there exist a spanning set of 𝐴_𝑖 such that the images of all elements in that spanning set under 𝜑_𝑖∞ belong to 𝐿(𝐴).

Proof. The implication (iii) ⇒ (i) is obvious.
To prove the implication (i) ⇒ (ii), it suffices to mention that every element (projection) in 𝐴 can be arbitrarily closely approximated by elements (projections, resp.) in 𝐴_𝑖.
Let us prove the implication (ii) ⇒ (iii). Clearly, without lots of generality we can assume that 𝐴_𝑖=𝑀_𝑛𝑖(𝐵_𝑖) for every 𝑖. For a fixed integer 𝑖, we put 𝑒⊗𝑆_𝑖=𝑒_𝑠𝑡⊗𝑥+𝑒_𝑡𝑠⊗𝑥^∗,𝑥∈𝑆_𝑖. (3) Hence, there exists a unitary 𝑢∈𝑀_𝑛𝑖 such that 𝑢𝑒_𝑠𝑡⊗𝑥+𝑒_𝑡𝑠⊗𝑥𝑢^∗=,0𝑥0𝑥00000 (4) where the 0 in the last column and the last row is of order 𝑛_𝑖−2.
It is evident that every element in 𝐴_𝑖 is a linear combination of elements in 𝑒⊗𝑆_𝑖∪𝐷, where 𝐷 is the set of all diagonal elements with coefficients in 𝑆_𝑖. Thus, 𝑒⊗𝑆_𝑖∪𝐷 is the spanning set of 𝐴_𝑖. Now, we claim that this spanning set satisfies the requirement of (iii).
Firstly, let 𝑑=diag(𝑥₁,…,𝑥_𝑛𝑖)(𝑥_𝑡∈𝑆_𝑖) be an element in 𝐷. By (ii), 𝜑_𝑖∞(𝑥_𝑡)∈𝐿(𝐴) for every 𝑡. Hence 𝜑_𝑖∞(𝑑)∈𝐿(𝐴). Lastly, let 𝑎∈𝑒⊗𝑆_𝑖. By Identity (4), 𝑎 can be assumed to be 0𝑥0𝑥00000forsome𝑥∈𝑆_𝑖. (5) Moreover, 𝑎=𝑢^∗−𝑥000𝑥0000𝑢,where1𝑢=√21√20−1√21√20.001 (6) In addition, there exists an integer 𝑗≥𝑖 such that 𝜑_𝑖𝑗(𝑥) can be approximated by an element of 𝐿(𝐴_𝑗) to within 𝜀. Hence 𝜑_𝑖𝑗(𝑎) can be approximated by an element of 𝐿(𝐴_𝑗) to within 𝜀.

Corollary 2. Let 𝐴=lim_⟶(𝐴_𝑖,𝜑_𝑖) be an AH algebra, where 𝐴_𝑖=⊕^𝑘𝑖_𝑡=1𝑀_𝑛𝑖𝑡(𝐶(𝑋_𝑖𝑡)) and 𝑋_𝑖𝑡 are connected compact Hausdorff spaces. Then the following statements are equivalent.(i)𝐴 has the LP property.(ii)For any integer 𝑖, any 𝑓∈∪^𝑘𝑖_𝑡=1𝐶(𝑋_𝑖𝑡) and any 𝜀>0, there exists an integer 𝑗≥𝑖 such that 𝜑_𝑖𝑗(𝑓) can be approximated by an element in 𝐿(𝐴_𝑗) to within 𝜀.

From the proof of Theorem 1, we can obtain the following.

Corollary 3. Let 𝐴 be a 𝐶^∗-algebra. If 𝐴 has the LP property, then 𝐴⊗𝑀_𝑛 and 𝐴⊗𝐾 have the LP property, where 𝐾 is the algebra of compact operators on a separable Hilbert space.

2.2. Linear Span of Projections in a Diagonal AH Algebra

For convenience of the reader, let us recall the notions from [4]. Let 𝑋 and 𝑌 be compact Hausdorff spaces. A ∗-homomorphism 𝜙 from 𝑀_𝑛(𝐶(𝑋)) to 𝑀_{𝑛𝑚+𝑘}(𝐶(𝑌)) is said to be diagonal if there exist continuous maps {𝜆_𝑖,𝑖=1,𝑛} from 𝑌 to 𝑋 such that 𝜙𝑓=diag𝑓∘𝜆₁,𝑓∘𝜆₂,…,𝑓∘𝜆_𝑛,0,𝑓∈𝑀_𝑛(𝐶(𝑋)), (7) where 0 is a zero matrix of order 𝑘 (𝑘≥0). If the size 𝑘=0, the map is unital.

The 𝜆_𝑖 are called the eigenvalue maps (or simply eigenvalues) of 𝜙. The family {𝜆₁,𝜆₂,…,𝜆_𝑚} is called the eigenvalue pattern of 𝜙. In addition, let 𝑝 and 𝑞 be projections in 𝑀_𝑛(𝐶(𝑋)) and 𝑀_{𝑛𝑚+𝑘}(𝐶(𝑌)), respectively. An ∗-homomorphism 𝜓 from 𝑝𝑀_𝑛(𝐶(𝑋))𝑝 to 𝑞𝑀_{𝑛𝑚+𝑘}(𝐶(𝑌))𝑞 is called diagonal if there exists a diagonal ∗-homomorphism 𝜙 from 𝑀_𝑛(𝐶(𝑋)) to 𝑀_{𝑛𝑚+𝑘}(𝐶(𝑌)) such that 𝜓 is reduced from 𝜙 on 𝑝𝑀_𝑛(𝐶(𝑋))𝑝 and 𝜙(𝑝)=𝑞. This definition can also be extended to a ∗-homomorphism 𝜙∶^𝑛_𝑖=1𝑝_𝑖𝑀_𝑛𝑖𝐶𝑋_𝑖𝑝_𝑖⟶^𝑚_𝑗=1𝑞_𝑗𝑀_𝑚𝑗𝐶𝑌_𝑗𝑞_𝑗 (8) by requiring that each partial map 𝜙^𝑖,𝑗∶𝑝_𝑖𝑀_𝑛𝑖𝐶𝑋_𝑖𝑝_𝑖⟶𝑞_𝑗𝑀_𝑚𝑗𝐶𝑌_𝑗𝑞_𝑗 (9) induced by 𝜙 be diagonal.

2.3. Eigenvalue Variation

Suppose that 𝐵 is a simple AH algebra. Then, 𝐵 has real rank zero if and only if its projections separate the traces provided that this algebra has slow dimension growth (see [12]). This equivalence was first studied when the dimensions of the spectra of the building blocks in the inductive limit decomposition of 𝐵 are not more than two, see [2].

Let 𝐵 be a 𝐶^∗-algebra. Suppose that 𝐵=^𝑘_𝑖=1𝐶𝑋_𝑖⊗𝑀_𝑛𝑖, (10) where 𝑋_𝑖 is a connected compact Hausdorff space for every 𝑖. Set 𝑋=⊔^𝑘_𝑖=1𝑋_𝑖. The following theorem and notations are quoted from [2, 12].

Let 𝑎 be any self-adjoint element in 𝐵. For any 𝑥 in 𝑋_𝑖, any positive integer 𝑚, 1≤𝑚≤𝑛_𝑖, let 𝜆_𝑚 denote the 𝑚th lowest eigenvalue of 𝑎(𝑥) counted with multiplicity. So 𝜆_𝑚 is a function on each 𝑋_𝑖, for 𝑖=1,2,…,𝑘. The fact is 𝜆_𝑚(𝑥)−𝜆_𝑚𝑦≤𝑎(𝑥)−𝑎𝑦. (11) Hence, 𝜆_𝑚 is continuous, for 𝑚=1,2,…,𝑘 for a given summand of 𝐵.

The variation of the eigenvalues of 𝑎, denoted by EV(𝑎), is defined as the maximum of the nonnegative real numbers sup𝜆_𝑚(𝑥)−𝜆_𝑚𝑦;𝑥,𝑦∈𝑋_𝑖, (12) over all 𝑖 and all possible values of 𝑚.

The variation of the normalized trace of 𝑎, denoted by TV(𝑎), is defined as 1sup𝑛_𝑖^𝑛𝑖_𝑚=1𝜆_𝑚(𝑥)−𝜆_𝑚𝑦;𝑥,𝑦∈𝑋_𝑖=suptr(𝑎(𝑥))−tr𝑎𝑦𝑥,𝑦∈𝑋_𝑖 (13) over all 𝑖, where tr denotes the normalized trace of 𝑀_𝑛 for any positive integer 𝑛.

Theorem 4 (see [2]). Let 𝐵 be an inductive limit of homogeneous 𝐶^∗-algebras 𝐵_𝑖 with morphisms 𝜙_𝑖𝑗 from 𝐵_𝑖 to 𝐵_𝑗. Suppose that 𝐵_𝑖 has the form 𝐵_𝑖=^𝑘𝑖_𝑡=1𝑀_𝑛𝑖𝑡𝐶𝑋_𝑖𝑡, (14) where 𝑘_𝑖 and 𝑛_𝑖𝑡 are positive integers, and 𝑋_𝑖𝑡 is a connected compact Hausdorff space for every positive integer 𝑖 and 1≤𝑡≤𝑘_𝑖. Consider the following conditions.(1)The projections of 𝐵 separate the traces on 𝐵.(2)For any self-adjoint element 𝑎 in 𝐵_𝑖 and 𝜀>0, there is a 𝑗≥𝑖 such that TV𝜙_𝑖𝑗(𝑎)<𝜀. (15)(3)For any self-adjoint element 𝑎 in 𝐵_𝑖 and any positive number 𝜀, there is a 𝑗≥𝑖 such that EV𝜙_𝑖𝑗(𝑎)<𝜀. (16)(4)𝐵 has real rank zero.(i)The following implications hold in general: (4)⟹(3)⟹(2)⟹(1). (17)(ii)If 𝐵 is simple, then the following equivalences hold: (3)⟺(2)⟺(1). (18)(iii)If 𝐵 is simple and has slow dimension growth, then all the conditions (1), (2), (3), and (4) are equivalent.

Proof. The statements (i) and (ii) are proved in Theorem 1.3 of [2]. The statement (iii) is an immediate consequence of the statement (ii) and Theorem 2 of [12].

An AH 𝐶^∗-algebra 𝐵 is said to have small eigenvalue variation (in the sense of Bratteli and Elliott, [3]) if 𝐵 satisfies statement (3) of Theorem 4.

2.4. Rearrange Eigenvalue Pattern

In order to evaluate the eigenvalue variation [3] of a diagonal element 𝑎=diag(𝑎₁,…,𝑎_𝑛) in 𝑀_𝑛(𝐶(𝑋)), we need to rearrange the 𝑎_𝑖 so that the obtained one 𝑏=diag(𝑏₁,…,𝑏_𝑛) with 𝑏₁≤𝑏₂≤⋯≤𝑏_𝑛 has the same eigenvalue variation of 𝑎.

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 ℎ in 𝑀₂(𝐶(𝑆⁴)) which is not unitarily equivalent to diag(𝜆₁,𝜆₂) but the eigenvalue variations of both elements are equal, where 𝜆_𝑖 is the 𝑖th lowest eigenvalue of ℎ counted with multiplicity [13, Section 2]. In general, given a self-adjoint element ℎ∈𝑀_𝑛(𝐶(𝑋)), for each 𝑥∈𝑋, there is a (point-wise) unitary 𝑢(𝑥)∈𝑀_𝑛 such that ℎ(𝑥)=𝑢(𝑥)diag(𝜆₁(𝑥),𝜆₂(𝑥),…,𝜆_𝑛(𝑥))𝑢^∗(𝑥), where 𝜆_𝑖(𝑥) is the 𝑖th lowest eigenvalue of ℎ(𝑥) counted with multiplicity. Denote by EV(ℎ) the eigenvalue variation of ℎ, then EV(ℎ)=EV(diag(𝜆₁,𝜆₂,…,𝜆_𝑛)) but 𝑢(𝑥) need not be continuous. The fact is that if 𝑢(𝑥) is continuous for any self-adjoint ℎ in 𝑀_𝑛(𝐶(𝑋)), then dim(𝑋) is less than 3 [13]. However, when replacing the equality “=” by some approximation “≈” and in some spacial cases (diagonal elements) discussed below, we can get such a continuous unitary without any hypothesis on dimension. Let us see the idea via the following example.

Let ℎ=diag(𝑥,1−𝑥)∈𝑀₂(𝐶[0,1]). Given any 1/2>𝜀>0. By [4, Lemma 2.5], there is a unitary 𝑢∈𝑀₂(𝐶[0,1]) such that(i)𝑢(𝑥)=1∈𝑀₂, for all 𝑥∈[0,1/2−𝜀],(ii)𝑢(𝑥)=⁰¹₁₀, for all 𝑥∈[1/2+𝜀,1].Denote by 𝜆₁ and 𝜆₂ the eigenvalue maps of ℎ; that is, 𝜆₁(𝑥)=𝑥if1𝑥≤21−𝑥if1𝑥>2,𝜆₂(𝑥)=1−𝑥if1𝑥≤2𝑥if1𝑥>2. (19) Then EV(ℎ)=EV(diag(𝜆₁,𝜆₂))=1/2.

It is straightforward to check that ‖𝑢ℎ𝑢^∗−diag(𝜆₁,𝜆₂)‖≤𝜀.

Lemma 5. Let 𝑋 be a connected compact Hausdorff space and ℎ=diag(𝑓₁,𝑓₂,…,𝑓_𝑛) a self-adjoint element in 𝑀_𝑛(𝐶(𝑋)), where 𝑓₁,𝑓₂,…,𝑓_𝑛 are continuous maps from 𝑋 to ℝ. For any positive number 𝜀, there is a unitary 𝑢∈𝑀_𝑛(𝐶(𝑋)) such that 𝑢ℎ𝑢^∗−diag𝜆₁,𝜆₂,…,𝜆_𝑛<𝜀, (20) where the 𝜆_𝑖(𝑥) is the 𝑖th lowest eigenvalue of ℎ(𝑥) counted with multiplicity for every 𝑥∈𝑋.

Proof. If 𝑓₁≤𝑓₂≤⋯≤𝑓_𝑛, then the unitary 𝑢 is just the identity of 𝑀_𝑛 and 𝜆_𝑖=𝑓_𝑖. Therefore, to prove the lemma, we, roughly speaking, only need to rearrange the given family {𝑓₁,𝑓₂,…,𝑓_𝑛} to obtain an increasing ordered family. For 𝑛=1, the lemma is obvious. Otherwise, using the idea of bubble sort, we can reduce to the case 𝑛=2.
Let 𝑍=(𝜆₁−𝜆₂)⁻¹(−𝜀/2,𝜀/2). Set 𝐸={𝑥∈𝑋∶𝑓₁(𝑥)≤𝑓₂(𝑥)}∩(𝑋⧵𝑍) and 𝐹={𝑥∈𝑋∶𝑓₁(𝑥)≥𝑓₂(𝑥)}∩(𝑋⧵𝑍).
It is clear that 𝐸 and 𝐹 are disjoint closed sets and 𝑋=𝐸∪𝐹∪𝑍. We have 𝜆₁(𝑥)=min{𝑓₁(𝑥),𝑓₂(𝑥)} and 𝜆₂(𝑥)=max{𝑓₁(𝑥),𝑓₂(𝑥)} for all 𝑥∈𝑋. If 𝐸(𝐹) is empty, then the unitary 𝑢 can be chosen as 0110 (1∈𝑀₂, resp.). Thus, we can assume both 𝐸 and 𝐹 are nonempty. By Urysohn’s Lemma, there is a continuous map 𝜇∶𝑋→[0,1] such that 𝜇 is equal to 0 on 𝐸 and 1 on 𝐹. Since the space of unitary matrices of 𝑀₂ is path connected, there is a unitary path 𝑝 linking 𝑝(0)=1to𝑝(1)=.0110 (21) Consequently, 𝑢=𝑝∘𝜇 is a unitary in 𝑀₂(𝐶(𝑋)) and 𝑢(𝑥)ℎ(𝑥)𝑢^∗(𝑥)=diag(𝜆₁(𝑥),𝜆₂(𝑥)) for all 𝑥∈𝐸∪𝐹.
For 𝑥∈𝑋⧵(𝐸∪𝐹)=𝑍, we have 𝜆₁(𝑥)−𝜆₂(𝑥<𝜀)2,𝑓_𝑖(𝑥)−𝜆₁(𝑥<𝜀)2,𝑖=1,2. (22) Hence, diag𝜆₁(𝑥),𝜆₂(𝑥)−diag𝜆₁(𝑥),𝜆₁(𝑥)<𝜀2,ℎ(𝑥)−diag𝜆₁(𝑥),𝜆₁(𝑥)<𝜀2. (23) On account to (23) we have 𝑢(𝑥)ℎ(𝑥)𝑢^∗(𝑥)−diag𝜆₁(𝑥),𝜆₂(𝑥)≤𝑢(𝑥)ℎ(𝑥)−diag𝜆₁(𝑥),𝜆₁(𝑥)𝑢^∗(𝑥+𝜆)diag₁(𝑥),𝜆₁(𝑥)−diag𝜆₁(𝑥),𝜆₂(𝑥)<𝜀. (24) Therefore, 𝑢ℎ𝑢^∗−diag𝜆₁,𝜆₂<𝜀. (25)

The main result of this section as follows.

Theorem 6. Given an AH algebra 𝐴=lim_⟶(𝐴_𝑖,𝜙_𝑖), where the 𝜙_𝑖 are diagonal ∗-homomorphisms from 𝐴_𝑖 to 𝐴_𝑖+1, where 𝐴_𝑖=⊕^𝑘𝑖_𝑡=1𝑀_𝑛𝑖𝑡(𝐶(𝑋_𝑖𝑡)) and the 𝑋_𝑖𝑡 are connected compact Hausdorff spaces. If 𝐴 has small eigenvalue variation in the sense of Bratteli and Elliott, then 𝐴 has the LP property.

Proof. By Corollary 2, it suffices to show that 𝜙_𝑖∞(𝑓)∈𝐿(𝐴) for every real-valued function 𝑓∈𝐶(𝑋_𝑖𝑡). By the same argument in the proof of Theorem 1, we can assume that each 𝐴_𝑡 has only one component; that is, 𝐴_𝑡=𝑀_𝑛𝑡(𝐶(𝑋_𝑡)). Let 𝜀>0 be arbitrary. Since 𝐴 has small eigenvalue variation in the sense of Bratteli and Elliott, there is an integer 𝑗≥𝑖 such that EV(𝜙_𝑖𝑗(𝑓))<𝜀. Let {𝜇₁,…,𝜇_𝑛} be the eigenvalue pattern of 𝜙_𝑖𝑗(𝑛=𝑛_𝑗/𝑛_𝑖). Then, 𝜙_𝑖𝑗𝑓=𝜙_𝑖𝑗diag𝑓,0=diag𝑓∘𝜇₁,0,𝑓∘𝜇₂,0,…,𝑓∘𝜇_𝑛𝑓,0=𝑣diag₁,𝑓₂,…,𝑓_𝑛,0𝑣^∗, (26) where 𝑓_𝑖=𝑓∘𝜇_𝑖 and 𝑣 is the permutation matrix in 𝑀_𝑛𝑗 moving all the zero to the bottom left-hand corner. Note that EV𝜙_𝑖𝑗𝑓=EVdiag𝑓₁,𝑓₂,…,𝑓_𝑛,𝑓_𝑛+1, (27) where 𝑓_𝑛+1(𝑥)=0 for all 𝑥∈𝑋_𝑗. By Lemma 5, there exists a unitary 𝑢∈𝑀_𝑛+1(𝐶(𝑋_𝑗)) and eigenvalue maps 𝜆₁≤𝜆₂≤⋯≤𝜆_𝑛+1 of diag(𝑓₁,𝑓₂,…,𝑓_𝑛,𝑓_𝑛+1) such that 𝑢diag𝑓₁,𝑓₂,…,𝑓_𝑛,𝑓_𝑛+1𝑢^∗−diag𝜆₁,…,𝜆_𝑛+1<𝜀. (28)
Put 𝛿_𝑖=12max_{𝑥∈𝑋𝑗}𝜆_𝑖(𝑥)+min_{𝑥∈𝑋𝑗}𝜆_𝑖(𝑥),𝑖=1,2,…,𝑛+1. (29) Then for any 𝑖, we have max_{𝑥∈𝑋𝑗}𝜆_𝑖(𝑥)−min_{𝑥∈𝑋𝑗}𝜆_𝑖(𝑥)≤EV𝜙_𝑖𝑗𝑓<𝜀 (30) and so 𝜆_𝑖(𝑥)−𝛿_𝑖<𝜀,∀𝑥∈𝑋_𝑗. (31) Thus, 𝑢diag𝑓₁,𝑓₂,…,𝑓_𝑛,𝑓_𝑛+1𝑢^∗−^𝑛+1_𝑖=1𝛿_𝑖𝑒_𝑖𝑖<2𝜀, (32) where {𝑒_𝑖𝑗} is the standard basis of 𝑀_𝑛+1. This implies that 𝜙_𝑖𝑗𝑓−𝑏<2𝜀, (33) where 𝑏=𝑣^∗diag(𝑢^∗(∑^𝑛+1_𝑖=1𝛿_𝑖𝑒_𝑖𝑖)𝑢,0)𝑣 is a linear combination of projections in 𝐴_𝑗.
Therefore, 𝜙_𝑖∞𝑓∈𝐿(𝐴). (34)

2.5. Another Form of Theorem 6

Lemma 7. Let 𝐵 be a 𝐶^∗-algebra, and 𝑝 and 𝑞 projections in 𝐵. If 𝑝 and 𝑞 are Murray-von Neumann equivalent, then 𝑝𝐵𝑝 is isomorphic to 𝑞𝐵𝑞.
In particular, if 𝐵=𝑀_𝑛(𝐶(𝑋)) (where 𝑋 is a connected compact Hausdorff space) and 𝑞 is a constant projection of rank 𝑚 in 𝐵, then 𝑞𝐵𝑞 is ∗-isomorphic to 𝑀_𝑚(𝐶(𝑋)).

Proof. By assumption, there exists a partial isometry 𝑣 such that 𝑝 = 𝑣^∗𝑣 and 𝑞 = 𝑣𝑣^∗. Let us consider the following maps: 𝜙(𝑥)=𝑣𝑥𝑣^∗,𝜓𝑦𝑥∈𝑝𝐵𝑝=𝑣^∗𝑦𝑣𝑦∈𝑞𝐵𝑞. (35) It is straightforward to check that the compositions of 𝜙 and 𝜓 are the identity maps.
In the case 𝐵=𝑀_𝑛(𝐶(𝑋)) and 𝑞 is a constant projection of rank 𝑚 in 𝐵, we have 𝑞𝐵𝑞=𝑀_𝑚(𝐶(𝑋)). Therefore, 𝑝𝐵𝑝 is ∗-isomorphic to 𝑀_𝑚(𝐶(𝑋)).

Theorem 8 (another form of Theorem 6). Let 𝐴=lim_⟶(𝐴_𝑖,𝜙_𝑖) be a diagonal AH algebra, where the 𝑝_𝑖𝑡 are projections in 𝑀_𝑛𝑖𝑡(𝐶(𝑋_𝑖𝑡)), 𝐴_𝑖=⊕^𝑘𝑖_𝑡=1𝑝_𝑖𝑡𝑀_𝑛𝑖𝑡(𝐶(𝑋_𝑖𝑡))𝑝_𝑖𝑡, and the 𝜙_𝑖 are unital diagonal. Suppose that each projection 𝑝_1𝑡 is Murray-von Neumann equivalent to some constant projection in 𝐴₁. Then 𝐴 has the LP property provided that 𝐴 has small eigenvalue variation in the sense of Bratteli and Elliott.

Proof. We can assume that 𝐴_𝑖=𝑝_𝑖𝑀_𝑛𝑖(𝐶(𝑋_𝑖))𝑝_𝑖, for all 𝑖. It is easy to see that 𝑝₁ is Murray-von Neumann equivalent to 𝑞₁=𝑒₁₁+𝑒₂₂+⋯+𝑒_𝑚1, where 𝑚₁ is the rank of 𝑝₁. For 𝑖>1, define 𝑞_𝑖=𝜙_𝑖−1(𝑞_𝑖−1). Then 𝑞_𝑖=𝜙_1𝑖(𝑞₁) is constant, since 𝑞₁ is constant and 𝜙_1𝑖 is diagonal. Let us denote by 𝑚_𝑖 the rank of 𝑞_𝑖, then 𝑚_𝑖+1∣𝑚_𝑖. By Lemma 7, there are ∗-isomorphisms Θ_𝑖 from 𝑝_𝑖𝑀_𝑛𝑖(𝐶(𝑋_𝑖))𝑝_𝑖 to 𝑞_𝑖𝑀_𝑛𝑖(𝐶(𝑋_𝑖))𝑞_𝑖=𝑀_𝑚𝑖(𝐶(𝑋)) such that Θ_𝑖(𝑎)=𝑣_𝑖𝑎𝑣^∗_𝑖, (36) where 𝑝₁=𝑣^∗₁𝑣₁∼𝑣₁𝑣^∗₁=𝑞₁ and 𝑣_𝑖=𝜙_1𝑖(𝑣₁). Since 𝜙_𝑖 is diagonal, there exists its extension 𝜙^∼_𝑖 which is a diagonal ∗-homomorphism from 𝑀_𝑛𝑖(𝐶(𝑋_𝑖)) to 𝑀_𝑛𝑖+1(𝐶(𝑋_𝑖+1)). Let 𝜓_𝑖 be the restriction of 𝜙^∼_𝑖 on 𝑞_𝑖𝑀_𝑛𝑖(𝐶(𝑋_𝑖))𝑞_𝑖. Then 𝜓_𝑖(𝑞_𝑖)=𝑞_𝑖+1. Therefore, the map 𝜓_𝑖 can be viewed as the map from 𝑞_𝑖𝑀_𝑛𝑖(𝐶(𝑋_𝑖))𝑞_𝑖 to 𝑞_𝑖+1𝑀_𝑛𝑖+1(𝐶(𝑋_𝑖+1))𝑞_𝑖+1 and so lim_⟶(𝑀_𝑚𝑖(𝐶(𝑋_𝑖)),𝜓_𝑖) is a diagonal AH-algebra.
On the other hand, it is straightforward to check that Θ_𝑖+1∘𝜙_𝑖=𝜓_𝑖∘Θ_𝑖 and hence 𝐴=lim_⟶(𝑀_𝑚𝑖(𝐶(𝑋_𝑖)),𝜓_𝑖). By Theorem 6, 𝐴 has the LP property.

2.6. Examples

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

Example 9. Let 𝐴=lim_⟶(𝑀_𝜈(𝑛)(𝐶(𝑋)),𝜙_𝑛) be a Goodearl algebra [6] and 𝜔_𝑡,1 the weighted identity ratio for 𝜙_𝑡,1. Suppose that 𝑋 is not totally disconnected and has finitely many connected components, then the following statements are equivalent.(i)𝐴 has real rank zero.(ii)lim_𝑡→∞𝜔_𝑡,1=0.(iii)𝐴 has small eigenvalue variation in the sense of Bratteli and Elliott.(iv)𝐴 has the LP property.

Proof. Indeed, (i) and (ii) are equivalent by [6, Theorem 9]. The implication (i) ⇒ (iii) follows from [2, Theorem 1.3]. By [14, Theorem 2.6], (i) implies (iv). Using Theorem 6 we get the implication (iii) ⇒ (iv). Finally, (iv) implies (ii) by [6, Theorem 6].

In general, the LP property cannot imply small eigenvalue variation in the sense of Bratteli and Elliott nor real rank zero. For example, let 𝐴 be a simple AH algebra with slow dimension growth and 𝐻 be a simple hereditary 𝐶^∗-subalgebra of 𝐴. By [5, Theorem 3.5], 𝐻 has nontrivial projections. Hence, 𝐻⊗𝐾 has the LP property by [1, Corollary 5]. However, 𝐴 has real rank zero if and only if it has small eigenvalue variation in the sense of Bratteli and Elliott [3]. This means that we can choose 𝐻 with real rank nonzero such that 𝐻⊗𝐾 has the LP property and does not have small eigenvalue variation in the sense of Bratteli and Elliott.

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

Lemma 10. Let 𝐴 be a diagonal AH algebra and 𝐾 be the 𝐶^∗-algebra of compact operators on an infinite dimensional Hilbert space. Then the tensor product 𝐴⊗𝐾 is again diagonal.

Proof. Let 𝐴=lim_⟶(𝐴_𝑛,𝜙_𝑛) and 𝐾=lim_⟶(𝑀_𝑛,𝑖_𝑛), where 𝐴_𝑛 is a homogeneous algebra, 𝜙_𝑛 is an injective diagonal homomorphism from 𝐴_𝑛 to 𝐴_𝑛+1, and 𝑖_𝑛 is the embedding from 𝑀_𝑛 to 𝑀_𝑛+1 which associates each 𝑎∈𝑀_𝑛 to diag(𝑎,0)∈𝑀_𝑛+1 for each positive integer 𝑛. Let us consider the inductive limit lim_⟶(𝐴_𝑛⊗𝑀_𝑛,𝜙_𝑛⊗𝑖_𝑛). For each integer 𝑛≥1, denote by 𝑖_𝑛,∞ and 𝜙_𝑛,∞ the homomorphisms from 𝑀_𝑛 and 𝐴_𝑛 to 𝐾 and 𝐴 in the inductive limit of 𝐾 and 𝐴, respectively. Then 𝜙_𝑛+1,∞⊗𝑖_𝑛+1,∞∘𝜙_𝑛⊗𝑖_𝑛=𝜙_𝑛,∞⊗𝑖_𝑛,∞. (37) Hence, by the universal property of inductive limit, there exists a unique homomorphism Φ from lim_⟶(𝐴_𝑛⊗𝑀_𝑛,𝜙_𝑛⊗𝑖_𝑛) to 𝐴⊗𝐾 such that Φ∘𝜙_𝑛⊗𝑖_𝑛=𝜙_𝑛,∞⊗𝑖_𝑛,∞. (38) It is straightforward to check that the image of Φ is dense in 𝐴⊗𝐾 and since all the maps 𝜙_𝑛 and 𝑖_𝑛 are injective, we have lim_⟶(𝐴_𝑛⊗𝑀_𝑛,𝜙_𝑛⊗𝑖_𝑛) is 𝐴⊗𝐾. Furthermore, for each 𝑛, we identify an element 𝑎⊗𝑏 in 𝐴_𝑛⊗𝑀_𝑛 with the matrix (𝑎_𝑖𝑗𝑏) in 𝑀_𝑛(𝐴_𝑛), where 𝑎=(𝑎_𝑖𝑗)∈𝑀_𝑛 and 𝑏∈𝐴_𝑛. By interchanging rows and columns (independent of 𝑎⊗𝑏) of (𝜙_𝑛⊗𝑖_𝑛)(𝑎⊗𝑏), we obtain diag(𝑎⊗𝑏∘𝜆₁,…,𝑎⊗𝑏∘𝜆_𝑚,0), where 𝜆₁,…,𝜆_𝑚 are the eigenvalue maps of 𝜙_𝑛. This means that there is a permutation matrix 𝑢_𝑛∈𝑀_𝑛+1(𝐴_𝑛+1) such that 𝑢_𝑛𝜙_𝑛⊗𝑖_𝑛𝑢^∗_𝑛 is diagonal. The fact is that the inductive limit is unchanged under unitary equivalence; that is, lim_⟶𝐴_𝑛⊗𝑀_𝑛,𝜙_𝑛⊗𝑖_𝑛=lim_⟶𝐴_𝑛⊗𝑀_𝑛,𝑢_𝑛𝜙_𝑛⊗𝑖_𝑛𝑢^∗_𝑛. (39) Hence, lim_⟶(𝐴_𝑛⊗𝑀_𝑛,𝜙_𝑛⊗𝑖_𝑛) is diagonal.

Example 11. Let 𝐵 be a simple unital diagonal AH algebra with real rank one without the LP property (e.g., take a Goodearl algebra, see Example 9), then 𝐵⊗𝐾 is a diagonal AH algebra of real rank one with the LP property.

Proof. By Lemma 10, 𝐵⊗𝐾 is a diagonal AH algebra. The real rank of 𝐵⊗𝐾 is one since that of 𝐵 is nonzero. Since 𝐵 is unital, 𝐵⊗𝐾 has a nontrivial projection. By [1, Corollary 5], 𝐵⊗𝐾 has the LP property.

3. The LP Property for an Inclusion of Unital 𝐶^∗-Algebras

3.1. Examples

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.

Lemma 12. Let 𝐴 be a projectionless simple unital 𝐶^∗-algebra with a unique tracial state. Then for any 𝑛∈ℕ with 𝑛>1, 𝑀_𝑛(𝐴) has the LP property.

Proof. Note that 𝑀_𝑛(𝐴) has also a unique tracial state.
Since 𝐴 is unital, 𝑀_𝑛(𝐴) has a nontrivial projection. Then by [1, Corollary 5], 𝑀_𝑛(𝐴) has the LP property.

Remark 13. Let 𝐴 be the Jiang-Su algebra. Then we know that 𝑅𝑅(𝐴)=1 [14]. Since 𝑀_𝑛(𝐴) is an AH algebra without real rank zero, 𝑅𝑅(𝑀_𝑛(𝐴))=1. But from Lemma 12, 𝑀_𝑛(𝐴) has the LP property.

Using this observation, we can construct a 𝐶^∗-algebra with the LP property such that the fixed point algebra does not have the LP property.

Example 14. A simple unital AI algebra 𝐴 in [15, Example 9], which comes from Thomsen’s construction, has two extremal tracial states; so by [16, Theorem 4.4], 𝐴 does not have the LP property. There is a symmetry 𝛼 on 𝐴 constructed by Elliott such that 𝐴⋊_𝛼ℤ/2ℤ is a UHF algebra. Since the fixed point algebra (𝐴⋊_𝛼ℤ/2ℤ)^𝛽=𝐴, where 𝛽 is the dual action of 𝛼. This shows that there is a simple unital 𝐶^∗-algebra 𝐵 with the LP property such that the fixed point algebra 𝐵^𝛽 does not have the LP property.

3.2. 𝐶^∗-Index Theory

According to Example 14, there is a faithful conditional expectation 𝐸∶𝐵→𝐵^𝛽. We extend this observation to an inclusion of unital 𝐶^∗-algebras with a finite Watatani index as follows.

In this section we recall the 𝐶^∗-basic construction defined by Watatani.

Definition 15. Let 𝐴⊃𝑃 be an inclusion of unital 𝐶^∗-algebras with a conditional expectation 𝐸 from 𝐴 to 𝑃.(1)A quasi-basis for 𝐸 is a finite set {(𝑢_𝑖,𝑣_𝑖)}^𝑛_𝑖=1⊂𝐴×𝐴 such that for every 𝑎∈𝐴, 𝑎=^𝑛_𝑖=1𝑢_𝑖𝐸𝑣_𝑖𝑎=^𝑛_𝑖=1𝐸𝑎𝑢_𝑖𝑣_𝑖. (40)(2)When {(𝑢_𝑖,𝑣_𝑖)}^𝑛_𝑖=1 is a quasi-basis for 𝐸, we define index𝐸 by Index𝐸=^𝑛_𝑖=1𝑢_𝑖𝑣_𝑖. (41) When there is no quasi-basis, we write Index𝐸=∞. index𝐸 is called the Watatani index of 𝐸.

Remark 16. We give several remarks about the above definitions.(1)Index𝐸 does not depend on the choice of the quasi-basis in the above formula, and it is a central element of 𝐴 [17, Proposition 1.2.8].(2)Once we know that there exists a quasi-basis, we can choose one of the form {(𝑤_𝑖,𝑤^∗_𝑖)}^𝑚_𝑖=1, which shows that Index𝐸 is a positive element [17, Lemma 2.1.6].(3)By the above statements, if 𝐴 is a simple 𝐶^∗-algebra, then Index𝐸 is a positive scalar.(4)If Index𝐸<∞, then 𝐸 is faithful; that is, 𝐸(𝑥^∗𝑥)=0 implies 𝑥=0 for 𝑥∈𝐴.

Next we recall the 𝐶^∗-basic construction defined by Watatani.

Let 𝐸∶𝐴→𝑃 be a faithful conditional expectation. Then 𝐴_𝑃(=𝐴) is a pre-Hilbert module over 𝑃 with a 𝑃 valued inner product 𝑥,𝑦_𝑃=𝐸𝑥^∗𝑦,𝑥,𝑦∈𝐴_𝑃. (42) We denote by ℰ_𝐸 and 𝜂_𝐸 the Hilbert 𝑃 module completion of 𝐴 by the norm ‖𝑥‖_𝑃=‖⟨𝑥,𝑥⟩_𝑃‖^1/2 for 𝑥 in 𝐴 and the natural inclusion map from 𝐴 to ℰ_𝐸. Then ℰ_𝐸 is a Hilbert 𝐶^∗-module over 𝑃. Since 𝐸 is faithful, the inclusion map 𝜂_𝐸 from 𝐴 to ℰ_𝐸 is injective. Let 𝐿_𝑃(ℰ_𝐸) be the set of all (right) 𝑃 module homomorphisms 𝑇∶ℰ_𝐸→ℰ_𝐸 with an adjoint right 𝑃 module homomorphism 𝑇^∗∶ℰ_𝐸→ℰ_𝐸 such that 𝑇𝜉,𝜁=𝜉,𝑇^∗𝜁,𝜉,𝜁∈ℰ_𝐸. (43) Then 𝐿_𝑃(ℰ_𝐸) is a 𝐶^∗-algebra with the operator norm ‖𝑇‖=sup{‖𝑇𝜉‖∶‖𝜉‖=1}. There is an injective ∗-homomorphism 𝜆∶𝐴→𝐿_𝑃(ℰ_𝐸) defined by 𝜆(𝑎)𝜂_𝐸(𝑥)=𝜂_𝐸(𝑎𝑥) (44) for 𝑥∈𝐴_𝑃 and 𝑎∈𝐴, so that 𝐴 can be viewed as a 𝐶^∗-subalgebra of 𝐿_𝑃(ℰ_𝐸). Note that the map 𝑒_𝑃∶𝐴_𝑃→𝐴_𝑃 defined by 𝑒_𝑃𝜂_𝐸(𝑥)=𝜂_𝐸(𝐸(𝑥)),𝑥∈𝐴_𝑃 (45) is bounded and thus it can be extended to a bounded linear operator, denoted by 𝑒_𝑃 again, on ℰ_𝐸. Then 𝑒_𝑃∈𝐿_𝑃(ℰ_𝐸) and 𝑒_𝑃=𝑒²_𝑃=𝑒^∗_𝑃; that is, 𝑒_𝑃 is a projection in 𝐿_𝑃(ℰ_𝐸). A projection 𝑒_𝑃 is called the Jones projection of 𝐸.

The (reduced) 𝐶^∗-basic construction is a 𝐶^∗-subalgebra of 𝐿_𝑃(ℰ_𝐸), defined as 𝐶^∗_𝑟𝐴,𝑒_𝑃=span𝜆(𝑥)𝑒_𝑃𝜆𝑦∈𝐿_𝑃ℰ_𝐸∶𝑥,𝑦∈𝐴^‖⋅‖. (46)

Remark 17. Watatani proved the following in [17].(1)Index𝐸 is finite if and only if 𝐶^∗_𝑟⟨𝐴,𝑒_𝑃⟩ has the identity (equivalently 𝐶^∗_𝑟⟨𝐴,𝑒_𝑃⟩=𝐿_𝑃(ℰ_𝐸)) and there exists a constant 𝑐>0 such that 𝐸(𝑥^∗𝑥)≥𝑐𝑥^∗𝑥 for 𝑥∈𝐴; that is, ‖𝑥‖²_𝑃≥𝑐‖𝑥‖² for 𝑥 in 𝐴 by [17, Proposition 2.1.5]. Since ‖𝑥‖≥‖𝑥‖_𝑃 for 𝑥 in 𝐴, if index𝐸 is finite, then ℰ_𝐸=𝐴.(2)If index𝐸 is finite, then each element 𝑧 in 𝐶^∗_𝑟⟨𝐴,𝑒_𝑃⟩ has a form 𝑧=^𝑛_𝑖=1𝜆𝑥_𝑖𝑒_𝑃𝜆𝑦_𝑖 (47) for some 𝑥_𝑖 and 𝑦_𝑖 in 𝐴.(3)Let 𝐶^∗_max⟨𝐴,𝑒_𝑃⟩ be the unreduced 𝐶^∗-basic construction defined in Definition 2.2.5 of [17], which has the certain universality (cf.(5) below). If index𝐸 is finite, then there exists an isomorphism from 𝐶^∗_𝑟⟨𝐴,𝑒_𝑃⟩ to 𝐶^∗_max⟨𝐴,𝑒_𝑃⟩ [17, Proposition 2.2.9]. Therefore, we can identify 𝐶^∗_𝑟⟨𝐴,𝑒_𝑃⟩ with 𝐶^∗_max⟨𝐴,𝑒_𝑃⟩. So we call 𝐶^∗_𝑟⟨𝐴,𝑒_𝑃⟩ the 𝐶^∗-basic construction and denote it by 𝐶^∗⟨𝐴,𝑒_𝑃⟩. Moreover, we identify 𝜆(𝐴) with 𝐴 in 𝐶^∗⟨𝐴,𝑒_𝑝⟩(=𝐶^∗_𝑟⟨𝐴,𝑒_𝑃⟩), and we define it as 𝐶^∗𝐴,𝑒_𝑝=^𝑛_𝑖=1𝑥_𝑖𝑒_𝑃𝑦_𝑖∶𝑥_𝑖,𝑦_𝑖∈𝐴,𝑛∈ℕ. (48)(4)The 𝐶^∗-basic construction 𝐶^∗⟨𝐴,𝑒_𝑝⟩ is isomorphic to 𝑞𝑀_𝑛(𝑃)𝑞 for some 𝑛∈ℕ and projection 𝑞∈𝑀_𝑛(𝑃) [17, Lemma 3.3.4]. If index𝐸 is finite, then index𝐸 is a central invertible element of 𝐴 and there is the dual conditional expectation 𝐸 from 𝐶^∗⟨𝐴,𝑒_𝑃⟩ to 𝐴 such that 𝐸𝑥𝑒_𝑃𝑦=(Index𝐸)⁻¹𝑥𝑦for𝑥,𝑦∈𝐴 (49) by [17, Proposition 2.3.2]. Moreover, 𝐸 has a finite index and faithfulness. If 𝐴 is simple unital 𝐶^∗-algebra, index𝐸∈𝐴 by Remark 16(4). Hence index𝐸=index𝐸 by [17, Proposition 2.3.4].(5)Suppose that index𝐸 is finite and 𝐴 acts on a Hilbert space ℋ faithfully and 𝑒 is a projection on ℋ such that 𝑒𝑎𝑒=𝐸(𝑎)𝑒 for 𝑎∈𝐴. If a map 𝑃∋𝑥↦𝑥𝑒∈𝐵(ℋ) is injective, then there exists an isomorphism 𝜋 from the norm closure of a linear span of 𝐴𝑒𝐴 to 𝐶^∗⟨𝐴,𝑒_𝑃⟩ such that 𝜋(𝑒)=𝑒_𝑃 and 𝜋(𝑎)=𝑎 for 𝑎∈𝐴 [17, Proposition 2.2.11].

3.3. Rokhlin Property for an Inclusion of Unital 𝐶^∗-Algebras

For a 𝐶^∗-algebra 𝐴, we set 𝑐₀(𝐴)=𝑎_𝑛∈𝑙^∞(ℕ,𝐴)∶lim_𝑛→∞𝑎_𝑛,𝐴=0_∞=𝑙^∞(ℕ,𝐴)𝑐₀(𝐴). (50)

We identify 𝐴 with the 𝐶^∗-subalgebra of 𝐴^∞ consisting of the equivalence classes of constant sequences and set 𝐴_∞=𝐴^∞∩𝐴^′. (51) For an automorphism 𝛼∈Aut(𝐴), we denote by 𝛼^∞ and 𝛼_∞ the automorphisms of 𝐴^∞ and 𝐴_∞ induced by 𝛼, respectively.

Izumi defined the Rokhlin property for a finite group action in [18, Definition 3.1] as follows.

Definition 18. Let 𝛼 be an action of a finite group 𝐺 on a unital 𝐶^∗-algebra 𝐴. 𝛼 is said to have the Rokhlin property if there exists a partition of unity {𝑒_𝑔}_𝑔∈𝐺⊂𝐴_∞ consisting of projections satisfying 𝛼_𝑔∞𝑒_ℎ=𝑒_𝑔ℎfor𝑔,ℎ∈𝐺. (52) We call {𝑒_𝑔}_𝑔∈𝐺 the Rokhlin projections.
Let 𝐴⊃𝑃 be an inclusion of unital 𝐶^∗-algebras. For a conditional expectation 𝐸 from 𝐴 to 𝑃, we denote by 𝐸^∞ the natural conditional expectation from 𝐴^∞ to 𝑃^∞ induced by 𝐸. If 𝐸 has a finite index with a quasi-basis {(𝑢_𝑖,𝑣_𝑖)}^𝑛_𝑖=1, then 𝐸^∞ also has a finite index with a quasi-basis {(𝑢_𝑖,𝑣_𝑖)}^𝑛_𝑖=1 and Index(𝐸^∞)=Index𝐸.

Motivated by Definition 18, Kodaka et al. introduced the Rokhlin property for an inclusion of unital 𝐶^∗-algebras with a finite index [19].

Definition 19. A conditional expectation 𝐸 of a unital 𝐶^∗-algebra 𝐴 with a finite index is said to have the Rokhlin property if there exists a projection 𝑒∈𝐴_∞ satisfying 𝐸^∞(𝑒)=(Index𝐸)⁻¹⋅1 (53) and a map 𝐴∋𝑥↦𝑥𝑒 is injective. We call 𝑒 a Rokhlin projection.

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 𝐶^∗-algebra to its fixed point algebra has the Rokhlin property in the sense of Definition 19.

Proposition 20 (see [19]). Let 𝛼 be an action of a finite group 𝐺 on a unital 𝐶^∗-algebra 𝐴 and 𝐸 the canonical conditional expectation from 𝐴 to the fixed point algebra 𝑃=𝐴^𝛼 defined by 𝐸(𝑥)=1#𝐺_𝑔∈𝐺𝛼_𝑔(𝑥),for𝑥∈𝐴, (54) where #𝐺 is the order of 𝐺. Then 𝛼 has the Rokhlin property if and only if there is a projection 𝑒∈𝐴_∞ such that 𝐸^∞(𝑒)=(1/#𝐺)⋅1, where 𝐸^∞ is the conditional expectation from 𝐴^∞ to 𝑃^∞ induced by 𝐸.

The following is the key one in the next section.

Proposition 21 (see [19] and [20, Lemma 2.5]). Let 𝑃⊂𝐴 be an inclusion of unital 𝐶^∗-algebras and 𝐸 a conditional expectation from 𝐴 to 𝑃 with a finite index. If 𝐸 has the Rokhlin property with a Rokhlin projection 𝑒∈𝐴_∞, then there is a unital linear map 𝛽∶𝐴^∞→𝑃^∞ such that for any 𝑥∈𝐴^∞ there exists the unique element 𝑦 of 𝑃^∞ such that 𝑥𝑒=𝑦𝑒=𝛽(𝑥)𝑒 and 𝛽(𝐴^′∩𝐴^∞)⊂𝑃^′∩𝑃^∞. In particular, 𝛽_|𝐴 is a unital injective ∗-homomorphism and 𝛽(𝑥)=𝑥 for all 𝑥∈𝑃.

The following is contained in [19, Proposition 3.4]. But we give it for self-contained.

Proposition 22. Let 𝑃⊂𝐴 be an inclusion of unital 𝐶^∗algebras and 𝐸 conditional expectation from 𝐴 to 𝑃 with a finite index. Suppose that 𝐴 is simple. Consider the basic construction 𝑃⊂𝐴⊂𝐶^∗𝐴,𝑒_𝑃(∶=𝐵)⊂𝐶^∗𝐵,𝑒_𝐴∶=𝐵₁. (55)
If 𝐸∶𝐴→𝑃 has the Rokhlin property with a Rokhlin projection 𝑒∈𝐴_∞, then the double dual conditional expectation 𝐸(∶=𝐸_𝐵)∶𝐶^∗⟨𝐵,𝑒_𝐴⟩→𝐵 has the Rokhlin property.

Proof. Note that from Remark 17(4) and [19, Corollary 3.8], 𝐶^∗-algebras 𝐶^∗⟨𝐴,𝑒_𝑃⟩ and 𝐶^∗⟨𝐵,𝑒_𝐴⟩ are simple.
Since 𝑒_𝑃𝑒𝑒_𝑃=𝐸^∞(𝑒)𝑒_𝑃=(Index𝐸)⁻¹𝑒_𝑝, (Index𝐸)𝑒𝑒_𝑝𝑒≤𝑒, and 𝐸^∞𝑒−(Index𝐸)𝑒𝑒_𝑝𝑒=𝑒−(Index𝐸)𝑒𝐸^∞𝑒_𝑃𝑒=𝑒−𝑒=0, (56) we have 𝑒=(Index𝐸)𝑒𝑒_𝑃𝑒. Then, for any 𝑥,𝑦∈𝐴𝑒𝑥𝑒_𝑃𝑦𝑒=𝑒𝑥𝑒𝑒_𝑃=(𝑒𝑦𝑒Index𝐸)⁻¹=𝐸𝑒𝑥𝑦𝑒𝑥𝑒_𝑃𝑦𝑒. (57) Hence, from Remark 17(3), we have 𝑒𝑧𝑒=𝐸(𝑧)𝑒 for any 𝑧∈𝐶^∗⟨𝐴,𝑒_𝑃⟩.
Let {(𝑤_𝑖,𝑤^∗_𝑖)}⊂𝐵×𝐵 be a quasi-basis for 𝐸(=𝐸_𝐴) and 𝑒_𝐴 be the Jones projection of 𝐸. Set 𝑔=∑_𝑖𝑤_𝑖𝑒𝑒_𝐴𝑤^∗_𝑖∈𝐵^∞₁. Then 𝑔 is a projection and 𝑔∈𝐵^′₁. Indeed, since 𝑔²=_𝑖,𝑗𝑤_𝑖𝑒𝑒_𝐴𝑤^∗_𝑖𝑤_𝑗𝑒𝑒_𝐴𝑤^∗_𝑗=_𝑖,𝑗𝑤_𝑖𝑒𝑒_𝐴𝐸𝑤^∗_𝑖𝑤_𝑗𝑤^∗_𝑗=_𝑖𝑤_𝑖𝑒𝑒_𝐴𝑗𝐸𝑤^∗_𝑖𝑤_𝑗𝑤^∗_𝑗=_𝑖𝑤_𝑖𝑒𝑒_𝐴𝑤^∗_𝑖=𝑔, (58)𝑔 is a projection.
Consider the following:𝑔𝑒_𝐴=_𝑖𝑤_𝑖𝑒𝑒_𝐴𝑤^∗_𝑖𝑒_𝐴=_𝑖𝑤_𝑖𝐸𝑤^∗_𝑖𝑒𝑒_𝐴=𝑒𝑒_𝐴,𝑒_𝐴𝑔=𝑒_𝐴𝑖𝑤_𝑖𝑒𝑒_𝐴𝑤^∗_𝑖=_𝑖𝐸𝑤_𝑖𝑒_𝐴𝑒𝑤^∗_𝑖=𝑒_𝐴𝑒_𝑖𝐸𝑤_𝑖𝑤^∗_𝑖=𝑒𝑒_𝐴=𝑔𝑒_𝐴. (59) Moreover, for any 𝑧∈𝐶^∗⟨𝐴,𝑒_𝑃⟩, we have 𝑔𝑧=_𝑖𝑤_𝑖𝑒𝑒_𝐴𝑤^∗_𝑖𝑧=_𝑖𝑤_𝑖𝑒𝑒_𝐴𝑗𝐸𝑤^∗_𝑖𝑧𝑤_𝑗𝑤^∗_𝑗=_𝑖𝑤_𝑖𝑗𝐸𝑤^∗_𝑖𝑥𝑧𝑤_𝑗𝑒𝑒_𝐴𝑤^∗_𝑗=_𝑗𝑖𝑤_𝑖𝐸𝑤^∗_𝑖𝑧𝑤_𝑗𝑒𝑒_𝐴𝑤^∗_𝑗=_𝑗𝑧𝑤_𝑗𝑒𝑒_𝐴𝑤^∗_𝑗=𝑧_𝑗𝑤_𝑗𝑒𝑒_𝐴𝑤^∗_𝑗=𝑧𝑔. (60) Since 𝐵₁=𝐶^∗⟨𝐶^∗⟨𝐴,𝑒_𝑃⟩,𝑒_𝐴⟩, 𝑔∈𝐵^′₁∩𝐵^∞₁.
To prove that the double dual conditional expectation 𝐸 has the Rokhlin property, we will show that 𝑔 is the Rokhlin projection of 𝐸. Since 𝑒𝑧𝑒=𝐸(𝑧)𝑒 for any 𝑧∈𝐶^∗⟨𝐴,𝑤_𝑃⟩, by Remark 17(5), there exists an isomorphism 𝜋∶𝐶^∗⟨𝐶^∗⟨𝐴,𝑒_𝑃⟩,𝑒_𝐴⟩→𝐶^∗⟨𝐶^∗⟨𝐴,𝑒_𝑃⟩,𝑒⟩ such that 𝜋(𝑒_𝐴)=𝑒 and 𝜋(𝑧)=𝑧 for 𝑧∈𝐶^∗⟨𝐴,𝑒_𝑃⟩. Then 𝐸^∞𝑔=_𝑖𝑤_𝑖𝑒𝐸^∞𝑒_𝐴𝑤^∗_𝑖=_𝑖𝑤_𝑖(Index𝐸)⁻¹𝑒𝑤^∗_𝑖=(Index𝐸)⁻¹_𝑖𝑤_𝑖𝜋𝑒_𝐴𝑤^∗_𝑖=(Index𝐸)⁻¹𝜋_𝑖𝑤_𝑖𝑒_𝐴𝑤^∗_𝑖=(Index𝐸)⁻¹1=Index𝐸⁻¹1, (61) hence 𝐸 has the Rokhlin property.

3.4. Main Results

Theorem 23. Let 1∈𝑃⊂𝐴 be an inclusion of unital 𝐶^∗-algebras with a finite Watatani index and 𝐸∶𝐴→𝑃 a faithful conditional expectation. Suppose that 𝐴 has the LP property and 𝐸 has the Rokhlin property. Then 𝑃 has the LP property.

Proof. Let 𝑥∈𝑃 and 𝜀>0. Since 𝐴 has the LP property, 𝑥 can be approximated by a line sum of projection ∑^𝑛_𝑖=1𝜆_𝑖𝑝_𝑖(𝑝_𝑖∈𝐴) such that ‖𝑥−∑^𝑛_𝑖=1𝜆_𝑖𝑝_𝑖‖<𝜀.
Since 𝛽∶𝐴^∞→𝑃^∞ is an injective ∗-homomorphism by Proposition 21, we have 𝛽𝑥−^𝑛_𝑖=1𝜆_𝑖𝑝=𝛽(𝑥)−^𝑛_𝑖=1𝜆_𝑖𝛽𝑝_𝑖<𝜀. (62) Since 𝛽_|𝑃=𝑖𝑑, we have ‖𝑥−∑^𝑛_𝑖=1𝜆_𝑖𝛽(𝑝_𝑖)‖<𝜀. Each projection in 𝑃^∞ can be lifted to a projection in ℓ^∞(ℕ,𝑃), so we can find a set of projections {𝑞_𝑖}^𝑛_𝑖=1⊂𝑃 such that 𝑥−^𝑛_𝑖=1𝜆_𝑖𝑞_𝑖<𝜀. (63) Therefore, 𝑃 has the LP property.

Theorem 24. Let 𝛼 be an action of a finite group 𝐺 on a simple unital 𝐶^∗-algebra 𝐴 and 𝐸 be canonical conditional expectation from 𝐴 to the fixed point algebra 𝑃=𝐴^𝛼 defined by 𝐸(𝑥)=1#𝐺_𝑔∈𝐺𝛼_𝑔(𝑥),for𝑥∈𝐴, (64) where #𝐺 is the order of 𝐺. Suppose that 𝛼 has the Rokhlin property. We have, then, that if 𝐴 has the LP property, the fixed point algebra and the crossed product 𝐴⋊_𝛼𝐺 have the LP property.

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

Lemma 25. Under the same conditions in Theorem 24 consider the following two basic constructions: 𝐴^𝛼⊂𝐴⊂𝐶^∗𝐴,𝑒_𝑃⊂𝐶^∗𝐵,𝑒_𝐴𝐵=𝐶^∗𝐴,𝑒_𝑃𝐴^𝛼⊂𝐴⊂𝐴⋊_𝛼𝐺⊂𝐶^∗𝐴⋊_𝛼𝐺,𝑒_𝐹, (65) where 𝐹∶𝐴⋊_𝛼𝐺→𝐴 is a canonical conditional expectation. Then there is an isomorphism 𝜋∶𝐶^∗⟨𝐴,𝑒_𝑃⟩→𝐴⋊_𝛼𝐺 and 𝜋^∼∶𝐶^∗⟨𝐵,𝑒_𝐴⟩→𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ such that(1)𝜋(𝑎)=𝑎 for all 𝑎∈𝐴,(2)𝜋(𝑒_𝑝)=𝑞, where 𝑞=(1/|𝐺|)∑_𝑔∈𝐺𝑢_𝑔,(3)𝐴⋊_𝛼𝐺=𝐶^∗⟨𝐴,𝑞⟩,(4)𝜋^∼(𝑏)=𝜋(𝑏) for all 𝑏∈𝐵,(5)𝜋^∼(𝑒_𝐴)=𝑒_𝐹.Moreover, we have(6)𝐹∘𝜋=𝐸 and 𝜋∘𝐸=𝐹∘𝜋^∼.

Proof. At first we prove condition (3). Since 𝛼 is outer, 𝛼 is saturated by [21, Proposition 4.9]; that is, 𝐴⋊_𝛼𝐺=linearspanof∑_𝑔∈𝐺𝛼_𝑔(𝑥)𝑢_𝑔^∗∑_𝑔∈𝐺𝛼_𝑔𝑦𝑢_𝑔=∣𝑥,𝑦∈𝐴linearspanof1|𝐺|_𝑔∑_∈𝐺𝑥^∗𝛼_𝑔𝑦𝑢_𝑔∣𝑥,𝑦∈𝐴. (66)
On the contrary, for any 𝑥,𝑦∈𝐴, we have 1𝑥𝑞𝑦=𝑥|𝐺|_𝑔∈𝐺𝑢_𝑔𝑦=1|𝐺|𝑥_𝑔∈𝐺𝑢_𝑔𝑦𝑢^∗_𝑔𝑢_𝑔=1|𝐺|_𝑔∈𝐺𝑥𝛼_𝑔𝑦𝑢_𝑔, (67) hence 𝐴⋊_𝛼𝐺=𝐶^∗⟨𝐴,𝑞⟩.
Since for any 𝑎∈𝐴1𝑞𝑎𝑞=|𝐺|²_𝑔∈𝐺𝑢_𝑔𝑎_ℎ∈𝐺𝑢_ℎ=1|𝐺|²_{𝑔,ℎ∈𝐺}𝑢_𝑔𝑎𝑢^∗_𝑔𝑢_𝑔ℎ=1|𝐺|²_{𝑔,ℎ∈𝐺}𝛼_𝑔(𝑎)𝑢_𝑔ℎ=1|𝐺|_𝑔∈𝐺𝛼_𝑔(𝑎)1|𝐺|_ℎ∈𝐺𝑢_𝑔ℎ=𝐸(𝑎)𝑞, (68) by Remark 17(5) there is an isomorphism 𝜋∶𝐶^∗⟨𝐴,𝑒_𝑃⟩→𝐶^∗⟨𝐴,𝑞⟩=𝐴⋊_𝛼𝐺 such that 𝜋(𝑎)=𝑎 for any 𝑎∈𝐴 and 𝜋(𝑒_𝑃)=𝑞. Hence conditions (1) and (2) are proved.
By the similar steps we will show conditions (4) and (5). Since for any 𝑥,𝑦,𝑎,𝑏∈𝐴𝑒_𝐹𝜋𝑥𝑒_𝑃𝑦𝑒_𝐹𝑎𝑞𝑏=𝑒_𝐹𝐹1𝑥𝑞𝑦|𝐺|_𝑔∈𝐺𝑎𝛼_𝑔(𝑏)𝑢_𝑔=1|𝐺|𝑒_𝐹=1𝑥𝑞𝑦𝑎𝑏|𝐺|²𝑥𝑦𝑎𝑏. (69) On the contrary, 𝜋𝐸𝑥𝑒_𝑃𝑦𝑒_𝐹1𝑎𝑞𝑏=𝜋|𝐺|1𝑥𝑦|𝐺|=1𝑎𝑏|𝐺|²𝑥𝑦𝑎𝑏. (70) Hence, we have 𝑒_𝐹𝜋(𝑥𝑒_𝑝𝑦)𝑒_𝐹=𝜋(𝐸(𝑥𝑒_𝑃𝑦)). By Remark 17(5), there is an isomorphism 𝜋^∼∶𝐶^∗⟨𝐵,𝑒_𝐴⟩→𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ such that 𝜋^∼(𝑏)=𝜋(𝑏) for any 𝑏∈𝐵 and 𝜋^∼(𝑒_𝐴)=𝑒_𝐹.
The condition (6) comes from the direct computation.

Lemma 26. Under the same conditions in Lemma 25  𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ is isomorphic to 𝑀_|𝐺|(𝐴).

Proof. Note that {(𝑢^∗_𝑔,𝑢_𝑔)}_𝑔∈𝐺 is a quasi-basis for 𝐹. By [17, Lemma 3.3.4], there is an isomorphism from 𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ to 𝑟𝑀_|𝐺|(𝐴)𝑟, where 𝑟=[𝐸(𝑢^∗_𝑔𝑢_ℎ)]_{𝑔,ℎ∈𝐺}=𝐼_|𝐺|. Hence 𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ is isomorphic to 𝑀_|𝐺|(𝐴).

Proof of Theorem 24. Let {𝑒_𝑔}_𝑔∈𝐺 be the Rokhlin projection of 𝐸. From Proposition 20,  𝐸∶𝐴→𝐴^𝐺 is of index finite and has a projection 𝑒∈𝐴^′∩𝐴^∞ such that 𝐸^∞(𝑒)=(1/|𝐺|)1. Note that index𝐸=|𝐺| and 𝑒=𝑒₁. Consider the basic construction 𝐴^𝐺⊂𝐴⊂𝐶^∗𝐴,𝑒_𝑃⊂𝐶^∗𝐵,𝑒_𝐴𝐵=𝐶^∗𝐴,𝑒_𝑃. (71)
Since 𝐴 is simple, the map 𝐴∋𝑥↦𝑥𝑒 is injective, hence we know that 𝐸 has the Rokhlin property. Therefore, 𝐴^𝐺 has the LP property by Theorem 23.
Since 𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ is isomorphic to 𝑀_|𝐺|(𝐴) by Lemma 26 and 𝐴 has the LP property, 𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ has the LP property. Hence, 𝐶^∗⟨𝐵,𝑒_𝐴⟩ has the LP property, because that 𝐶^∗⟨𝐴⋊_𝛼𝐺,𝑒_𝐹⟩ is isomorphic to 𝐶^∗⟨𝐵,𝑒_𝐴⟩ from Lemma 25. From Proposition 22,  𝐸∶𝐶^∗⟨𝐵,𝑒_𝐴⟩→𝐶^∗⟨𝐴,𝑒_𝑃⟩ has the Rokhlin property, hence we conclude that 𝐶^∗⟨𝐴,𝑒_𝑃⟩ has the LP property by Theorem 23. Since 𝐶^∗⟨𝐴,𝑒_𝑃⟩ is isomorphic to 𝐴⋊_𝛼𝐺 by Lemma 25, we conclude that 𝐴⋊_𝛼𝐺 has the LP property.

Remark 27. (1)When an action 𝛼 of a finite group 𝐺 does not have the Rokhlin property, we have an example of simple unital 𝐶^∗-algebra with the LP property such that the fixed point algebra 𝐴^𝐺 does not have the LP property by Example 14. Note that the action 𝛼 does not have the Rokhlin property.(2)When an action of a finite group 𝐺 on a unital 𝐶^∗-algebra 𝐴 has the Rokhlin property, the crossed product can be locally approximated by the class of matrix algebras over corners of 𝐴 [22, Theorem 3.2]. Many kinds of properties are preserved by this method such as AF algebras [23], AI algebras, AT algebras, simple AH algebras with slow dimension growth and real rank zero [22], D-absorbing separable unital 𝐶^∗-algebras for a strongly self-absorbing 𝐶^∗-algebras D [24], simple unital separable strongly self-absorbing 𝐶^∗-algebras [20], and unital Kirchberg 𝐶^∗-algebras [22]. Like the ideal property [25], however, since the LP property is not preserved by passing to corners by Lemma 12, we cannot apply this method to determine the LP property of the crossed products.

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 𝐴 be an infinite dimensional simple 𝐶^∗-algebra and let 𝛼 be an action from a finite group 𝐺 on Aut(𝐴). Recall that 𝛼 has the tracial Rokhlin property if for every finite set 𝐹⊂𝐴, every 𝜀>0, and every positive element 𝑥∈𝐴 with ‖𝑥‖=1, there are mutually orthogonal projections 𝑒_𝑔∈𝐴 for 𝑔∈𝐺 such that(1)‖𝛼_𝑔(𝑒_ℎ)−𝑒_𝑔ℎ‖<𝜀 for all 𝑔,ℎ∈𝐺 and all 𝑎∈𝐹,(2)‖𝑒_𝑔𝑎−𝑎𝑒_𝑔‖<𝜀 for all 𝑔∈𝐺 and all 𝑎∈𝐹,(3)with 𝑒=∑_𝑔∈𝐺𝑒_𝑔, the projection 1−𝑒 is the Murray-von Neumann equivalent to a projection in the hereditary subalgebra of 𝐴 generated by 𝑥,(4)with 𝑒 as in (3), we have ‖𝑒𝑥𝑒‖>1−𝜀.

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

Proposition 28. Let 𝛼 be an action of a finite group 𝐺 on a simple unital 𝐶^∗-algebra 𝐴 with a unique tracial state. Suppose that 𝛼 has the tracial Rokhlin property. If 𝐴 has the LP property, then the crossed product 𝐴⋊_𝛼𝐺 has the LP property.

Proof. From [26, Proposition 5.7], the restriction map from tracial states on the crossed product 𝐴⋊_𝛼𝐺 to 𝛼-invariant tracial states on 𝐴 is isomorphism. Hence, 𝐴⋊_𝛼𝐺 has a unique tracial state.
Since 𝛼 is a pointwise outer (i.e., for any 𝑔∈𝐺⧵{0}𝛼_𝑔 is outer) by [23, Lemma 1.5], 𝐴⋊_𝛼𝐺 is simple.
Therefore, by [1, Corollary 4], 𝐴⋊_𝛼𝐺 has the LP property.

Remark 29. There are many examples of actions 𝛼 of finite groups on simple unital 𝐶^∗-algebras with real rank zero and a unique tracial state such that 𝛼 has the tracial Rokhlin property, see [23, 26].

Acknowledgment

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