# The Linear Span of Projections in AH Algebras and for Inclusions ofC*-Algebras

Authors: Dinh Trung Hoa; Toan Minh Ho; Hiroyuki Osaka
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In the first part of this paper, we show that an AH algebraA=lim→(Ai,ϕi) has the LP property if and only if every element of the centre of Ai 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 AG 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.


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## Body

## 1. Introduction

AC*-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 ofdiagonal 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 amenableC*-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 (Section2), 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⟶(Ai,ϕi) (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 Ai 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 Mn(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 aC*-algebra A is inherited to the matrix tensor product Mn(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 AG 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,Mn stands for the algebra of all n×n complex matrices, e={est}s,t=1,n¯ denotes the standard basis of Mn (for convenience, we also use this system of matrix unit for any size of matrix algebras). Let us denote by Mn(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 Xi and X=⊔i=1kXi, then
(1)Mn(C(X))=⨁i=1kMn(C(Xi)).
Hence, without lots of generality we can always assume that the spectrum of each component of a homogeneous C*-algebra is connected.Denote bydiag(a1,a2,…,an) the block diagonal matrix with entries a1,a2,…,an in some algebras.LetA be a C*-algebra. Any element a in A can be considered as an element in Mn(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 theC*-index theory.


## 2. Linear Span of Projections in AH Algebras

### 2.1. Linear Span of Projections in an Inductive Limit of Matrix Algebras overC*-Algebras

Let(2)A=lim⟶(Ai,φi),
where Ai=⊕t=1kiMnit(Bit) and Bit are C*-algebras. Let Sit be a spanning set of Bit (as a vector space) and Si be the union of Sit for t=1,…,ki. 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 Si are self-adjoint.Theorem 1.
LetA be an inductive limit C*-algebra as above. Then the following statements are equivalent.(i)
Ahas the LP property.(ii)
For any integeri, any x∈Si and any ε>0, there exists an integer j≥i such that φij(x) can be approximated by an element in L(Aj) to within ε.(iii)
For any integeri, there exist a spanning set of Ai such that the images of all elements in that spanning set under φi∞ 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 Ai.
Let us prove the implication (ii)⇒ (iii). Clearly, without lots of generality we can assume that Ai=Mni(Bi) for every i. For a fixed integer i, we put
(3)e⊗Si={est⊗x+ets⊗x*,x∈Si}.
Hence, there exists a unitary u∈Mni such that
(4)u(est⊗x+ets⊗x)u*=(0x0x00000),
where the 0 in the last column and the last row is of order ni-2.
It is evident that every element inAi is a linear combination of elements in e⊗Si∪D, where D is the set of all diagonal elements with coefficients in Si. Thus, e⊗Si∪D is the spanning set of Ai. Now, we claim that this spanning set satisfies the requirement of (iii).
Firstly, letd=diag(x1,…,xni)(xt∈Si) be an element in D. By (ii), φi∞(xt)∈L(A) for every t. Hence φi∞(d)∈L(A). Lastly, let a∈e⊗Si. By Identity (4), a can be assumed to be
(5)(0x0x00000)forsomex∈Si.
Moreover,
(6)a=u*(-x000x0000)u,whereu=(12120-12120001).
In addition, there exists an integer j≥i such that φij(x) can be approximated by an element of L(Aj) to within ε. Hence φij(a) can be approximated by an element of L(Aj) to within ε.Corollary 2.
LetA=lim⟶(Ai,φi) be an AH algebra, where Ai=⊕t=1kiMnit(C(Xit)) and Xit are connected compact Hausdorff spaces. Then the following statements are equivalent.(i)
A has the LP property.(ii)
For any integeri, any f∈∪t=1kiC(Xit) and any ε>0, there exists an integer j≥i such that φij(f) can be approximated by an element in L(Aj) to within ε.From the proof of Theorem1, we can obtain the following.Corollary 3.
LetA be a C*-algebra. If A has the LP property, then A⊗Mn 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 Mn(C(X)) to Mnm+k(C(Y)) is said to be diagonal if there exist continuous maps {λi,i=1,n¯} from Y to X such that
(7)ϕ(f)=diag(f∘λ1,f∘λ2,…,f∘λn,0),(f∘λ1,f∘λ2,…,f∘λn)f∈Mn(C(X)),
where 0 is a zero matrix of order k (k≥0). If the size k=0, the map is unital.Theλi are called the eigenvalue maps (or simply eigenvalues) of ϕ. The family {λ1,λ2,…,λm} is called the eigenvalue pattern of ϕ. In addition, let p and q be projections in Mn(C(X)) and Mnm+k(C(Y)), respectively. An *-homomorphism ψ from pMn(C(X))p to qMnm+k(C(Y))q is called diagonal if there exists a diagonal *-homomorphism ϕ from Mn(C(X)) to Mnm+k(C(Y)) such that ψ is reduced from ϕ on pMn(C(X))p and ϕ(p)=q. This definition can also be extended to a *-homomorphism
(8)ϕ:⨁i=1npiMni(C(Xi))pi⟶⨁j=1mqjMmj(C(Yj))qj
by requiring that each partial map
(9)ϕi,j:piMni(C(Xi))pi⟶qjMmj(C(Yj))qj
induced by ϕ be diagonal.


### 2.3. Eigenvalue Variation

Suppose thatB 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].LetB be a C*-algebra. Suppose that
(10)B=⨁i=1kC(Xi)⊗Mni,
where Xi is a connected compact Hausdorff space for every i. Set X=⊔i=1kXi. The following theorem and notations are quoted from [2, 12].Leta be any self-adjoint element in B. For any x in Xi, any positive integer m, 1≤m≤ni, let λm denote the mth lowest eigenvalue of a(x) counted with multiplicity. So λm is a function on each Xi, for i=1,2,…,k. The fact is
(11)|λm(x)-λm(y)|≤||a(x)-a(y)||.
Hence, λm is continuous, for m=1,2,…,k for a given summand of B.Thevariation of the eigenvalues of a, denoted by EV(a), is defined as the maximum of the nonnegative real numbers
(12)sup{|λm(x)-λm(y)|;x,y∈Xi},
over all i and all possible values of m.Thevariation of the normalized trace of a, denoted by TV(a), is defined as
(13)sup{|1ni∑m=1ni‍(λm(x)-λm(y))|;x,y∈Xi}=sup{|tr(a(x))-tr(a(y))|x,y∈Xi}
over all i, where tr denotes the normalized trace of Mn for any positive integer n.Theorem 4 (see [2]).
LetB be an inductive limit of homogeneous C*-algebras Bi with morphisms ϕij from Bi to Bj. Suppose that Bi has the form
(14)Bi=⨁t=1kiMnit(C(Xit)),
where ki and nit are positive integers, and Xit is a connected compact Hausdorff space for every positive integer i and 1≤t≤ki. Consider the following conditions.(1)
The projections ofB separate the traces on B.(2)
For any self-adjoint elementa in Bi and ε>0, there is a j≥i such that
(15)TV(ϕij(a))<ε.(3)
For any self-adjoint elementa in Bi and any positive number ε, there is a j≥i such that
(16)EV(ϕij(a))<ε.(4)
B has real rank zero.
(i)
The following implications hold in general:(17)(4)⟹(3)⟹(2)⟹(1).(ii)
IfB is simple, then the following equivalences hold:
(18)(3)⟺(2)⟺(1).(iii)
IfB 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 AHC*-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,…,an) in Mn(C(X)), we need to rearrange the ai so that the obtained one b=diag(b1,…,bn) with b1≤b2≤⋯≤bn 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 elementh in M2(C(S4)) which is not unitarily equivalent to diag(λ1,λ2) but the eigenvalue variations of both elements are equal, where λi is the ith lowest eigenvalue of h counted with multiplicity [13, Section 2]. In general, given a self-adjoint element h∈Mn(C(X)), for each x∈X, there is a (point-wise) unitary u(x)∈Mn such that h(x)=u(x)diag(λ1(x),λ2(x),…,λn(x))u*(x), where λi(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,…,λn)) but u(x) need not be continuous. The fact is that if u(x) is continuous for any self-adjoint h in Mn(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.Leth=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,
(19)λ1(x)={xifx≤121-xifx>12,λ2(x)={1-xifx≤12xifx>12.
Then EV(h)=EV(diag(λ1,λ2))=1/2.It is straightforward to check that∥uhu*-diag(λ1,λ2)∥≤ε.Lemma 5.
LetX be a connected compact Hausdorff space and h=diag(f1,f2,…,fn) a self-adjoint element in Mn(C(X)), where f1,f2,…,fn are continuous maps from X to ℝ. For any positive number ε, there is a unitary u∈Mn(C(X)) such that
(20)∥uhu*-diag(λ1,λ2,…,λn)∥<ε,
where the λi(x) is the ith lowest eigenvalue of h(x) counted with multiplicity for every x∈X.Proof.
Iff1≤f2≤⋯≤fn, then the unitary u is just the identity of Mn and λi=fi. Therefore, to prove the lemma, we, roughly speaking, only need to rearrange the given family {f1,f2,…,fn} 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.
LetZ=(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 thatE 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
(21)p(0)=1top(1)=(0110).
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.
Forx∈X∖(E∪F)=Z, we have
(22)|λ1(x)-λ2(x)|<ε2,|fi(x)-λ1(x)|<ε2,i=1,2.
Hence,
(23)∥diag(λ1(x),λ2(x))-diag(λ1(x),λ1(x))∥<ε2,∥h(x)-diag(λ1(x),λ1(x))∥<ε2.
On account to (23) we have
(24)∥u(x)h(x)u*(x)-diag(λ1(x),λ2(x))∥≤∥u(x)[h(x)-diag(λ1(x),λ1(x))]u*(x)∥+∥diag(λ1(x),λ1(x))-diag(λ1(x),λ2(x))∥<ε.
Therefore,
(25)∥uhu*-diag(λ1,λ2)∥<ε.The main result of this section as follows.Theorem 6.
Given an AH algebraA=lim⟶(Ai,ϕi), where the ϕi are diagonal *-homomorphisms from Ai to Ai+1, where Ai=⊕t=1kiMnit(C(Xit)) and the Xit 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 Corollary2, it suffices to show that ϕi∞(f)∈L(A) for every real-valued function f∈C(Xit). By the same argument in the proof of Theorem 1, we can assume that each At has only one component; that is, At=Mnt(C(Xt)). 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(ϕij(f))<ε. Let {μ1,…,μn} be the eigenvalue pattern of ϕij(n=nj/ni). Then,
(26)ϕij(f)=ϕij(diag(f,0))=diag(f∘μ1,0,f∘μ2,0,…,f∘μn,0)=vdiag(f1,f2,…,fn,0)v*,
where fi=f∘μi and v is the permutation matrix in Mnj moving all the zero to the bottom left-hand corner. Note that
(27)EV(ϕij(f))=EV(diag(f1,f2,…,fn,fn+1)),
where fn+1(x)=0 for all x∈Xj. By Lemma 5, there exists a unitary u∈Mn+1(C(Xj)) and eigenvalue maps λ1≤λ2≤⋯≤λn+1 of diag(f1,f2,…,fn,fn+1) such that
(28)∥udiag(f1,f2,…,fn,fn+1)u*-diag(λ1,…,λn+1)∥<ε.
Put(29)δi=12(maxx∈Xjλi(x)+minx∈Xjλi(x)),i=1,2,…,n+1.
Then for any i, we have
(30)maxx∈Xjλi(x)-minx∈Xjλi(x)≤EV(ϕij(f))<ε
and so
(31)|λi(x)-δi|<ε,∀x∈Xj.
Thus,
(32)∥udiag(f1,f2,…,fn,fn+1)u*-∑i=1n+1‍δieii∥<2ε,
where {eij} is the standard basis of Mn+1. This implies that
(33)∥ϕij(f)-b∥<2ε,
where b=v*diag(u*(∑i=1n+1δieii)u,0)v is a linear combination of projections in Aj.
Therefore,(34)ϕi∞(f)∈L(A).


### 2.5. Another Form of Theorem6

Lemma 7.
LetB 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, ifB=Mn(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 Mm(C(X)).Proof.
By assumption, there exists a partial isometryv such that p = v*v and q = vv*. Let us consider the following maps:
(35)ϕ(x)=vxv*(x∈pBp),ψ(y)=v*yv(y∈qBq).
It is straightforward to check that the compositions of ϕ and ψ are the identity maps.
In the caseB=Mn(C(X)) and q is a constant projection of rank m in B, we have qBq=Mm(C(X)). Therefore, pBp is *-isomorphic to Mm(C(X)).Theorem 8 (another form of Theorem6).
LetA=lim⟶(Ai,ϕi) be a diagonal AH algebra, where the pit are projections in Mnit(C(Xit)), Ai=⊕t=1kipitMnit(C(Xit))pit, and the ϕi are unital diagonal. Suppose that each projection p1t 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 thatAi=piMni(C(Xi))pi, for all i. It is easy to see that p1 is Murray-von Neumann equivalent to q1=e11+e22+⋯+em1, where m1 is the rank of p1. For i>1, define qi=ϕi-1(qi-1). Then qi=ϕ1i(q1) is constant, since q1 is constant and ϕ1i is diagonal. Let us denote by mi the rank of qi, then mi+1∣mi. By Lemma 7, there are *-isomorphisms Θi from piMni(C(Xi))pi to qiMni(C(Xi))qi=Mmi(C(X)) such that
(36)Θi(a)=viavi*,
where p1=v1*v1~v1v1*=q1 and vi=ϕ1i(v1). Since ϕi is diagonal, there exists its extension ϕ~i which is a diagonal *-homomorphism from Mni(C(Xi)) to Mni+1(C(Xi+1)). Let ψi be the restriction of ϕ~i on qiMni(C(Xi))qi. Then ψi(qi)=qi+1. Therefore, the map ψi can be viewed as the map from qiMni(C(Xi))qi to qi+1Mni+1(C(Xi+1))qi+1 and so lim⟶(Mmi(C(Xi)),ψi) is a diagonal AH-algebra.
On the other hand, it is straightforward to check thatΘi+1∘ϕi=ψi∘Θi and hence A=lim⟶(Mmi(C(Xi)),ψi). 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.
LetA=lim⟶(Mν(n)(C(X)),ϕn) be a Goodearl algebra [6] and ωt,1 the weighted identity ratio for ϕt,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)
limt→∞ωt,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, letA 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.
LetA 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.
LetA=lim⟶(An,ϕn) and K=lim⟶(Mn,in), where An is a homogeneous algebra, ϕn is an injective diagonal homomorphism from An to An+1, and in is the embedding from Mn to Mn+1 which associates each a∈Mn to diag(a,0)∈Mn+1 for each positive integer n. Let us consider the inductive limit  lim⟶(An⊗Mn,ϕn⊗in). For each integer n≥1, denote by in,∞ and ϕn,∞ the homomorphisms from Mn and  An to K and A in the inductive limit of K and A, respectively. Then
(37)(ϕn+1,∞⊗in+1,∞)∘(ϕn⊗in)=ϕn,∞⊗in,∞.
Hence, by the universal property of inductive limit, there exists a unique homomorphism Φ from lim⟶(An⊗Mn,ϕn⊗in) to A⊗K such that
(38)Φ∘(ϕn⊗in)=(ϕn,∞⊗in,∞).
It is straightforward to check that the image of Φ is dense in A⊗K and since all the maps ϕn and in are injective, we have lim⟶(An⊗Mn,ϕn⊗in) is A⊗K. Furthermore, for each n, we identify an element a⊗b in An⊗Mn with the matrix (aijb) in Mn(An), where a=(aij)∈Mn and b∈An. By interchanging rows and columns (independent of a⊗b) of (ϕn⊗in)(a⊗b), we obtain diag(a⊗b∘λ1,…,a⊗b∘λm,0), where λ1,…,λm are the eigenvalue maps of ϕn. This means that there is a permutation matrix un∈Mn+1(An+1) such that unϕn⊗inun* is diagonal. The fact is that the inductive limit is unchanged under unitary equivalence; that is,
(39)lim⟶(An⊗Mn,ϕn⊗in)=lim⟶(An⊗Mn,unϕn⊗inun*).
Hence, lim⟶(An⊗Mn,ϕn⊗in) is diagonal.Example 11.
LetB 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 Lemma10, 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.


## 2.1. Linear Span of Projections in an Inductive Limit of Matrix Algebras overC*-Algebras

Let(2)A=lim⟶(Ai,φi),
where Ai=⊕t=1kiMnit(Bit) and Bit are C*-algebras. Let Sit be a spanning set of Bit (as a vector space) and Si be the union of Sit for t=1,…,ki. 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 Si are self-adjoint.Theorem 1.
LetA be an inductive limit C*-algebra as above. Then the following statements are equivalent.(i)
Ahas the LP property.(ii)
For any integeri, any x∈Si and any ε>0, there exists an integer j≥i such that φij(x) can be approximated by an element in L(Aj) to within ε.(iii)
For any integeri, there exist a spanning set of Ai such that the images of all elements in that spanning set under φi∞ 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 Ai.
Let us prove the implication (ii)⇒ (iii). Clearly, without lots of generality we can assume that Ai=Mni(Bi) for every i. For a fixed integer i, we put
(3)e⊗Si={est⊗x+ets⊗x*,x∈Si}.
Hence, there exists a unitary u∈Mni such that
(4)u(est⊗x+ets⊗x)u*=(0x0x00000),
where the 0 in the last column and the last row is of order ni-2.
It is evident that every element inAi is a linear combination of elements in e⊗Si∪D, where D is the set of all diagonal elements with coefficients in Si. Thus, e⊗Si∪D is the spanning set of Ai. Now, we claim that this spanning set satisfies the requirement of (iii).
Firstly, letd=diag(x1,…,xni)(xt∈Si) be an element in D. By (ii), φi∞(xt)∈L(A) for every t. Hence φi∞(d)∈L(A). Lastly, let a∈e⊗Si. By Identity (4), a can be assumed to be
(5)(0x0x00000)forsomex∈Si.
Moreover,
(6)a=u*(-x000x0000)u,whereu=(12120-12120001).
In addition, there exists an integer j≥i such that φij(x) can be approximated by an element of L(Aj) to within ε. Hence φij(a) can be approximated by an element of L(Aj) to within ε.Corollary 2.
LetA=lim⟶(Ai,φi) be an AH algebra, where Ai=⊕t=1kiMnit(C(Xit)) and Xit are connected compact Hausdorff spaces. Then the following statements are equivalent.(i)
A has the LP property.(ii)
For any integeri, any f∈∪t=1kiC(Xit) and any ε>0, there exists an integer j≥i such that φij(f) can be approximated by an element in L(Aj) to within ε.From the proof of Theorem1, we can obtain the following.Corollary 3.
LetA be a C*-algebra. If A has the LP property, then A⊗Mn 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 Mn(C(X)) to Mnm+k(C(Y)) is said to be diagonal if there exist continuous maps {λi,i=1,n¯} from Y to X such that
(7)ϕ(f)=diag(f∘λ1,f∘λ2,…,f∘λn,0),(f∘λ1,f∘λ2,…,f∘λn)f∈Mn(C(X)),
where 0 is a zero matrix of order k (k≥0). If the size k=0, the map is unital.Theλi are called the eigenvalue maps (or simply eigenvalues) of ϕ. The family {λ1,λ2,…,λm} is called the eigenvalue pattern of ϕ. In addition, let p and q be projections in Mn(C(X)) and Mnm+k(C(Y)), respectively. An *-homomorphism ψ from pMn(C(X))p to qMnm+k(C(Y))q is called diagonal if there exists a diagonal *-homomorphism ϕ from Mn(C(X)) to Mnm+k(C(Y)) such that ψ is reduced from ϕ on pMn(C(X))p and ϕ(p)=q. This definition can also be extended to a *-homomorphism
(8)ϕ:⨁i=1npiMni(C(Xi))pi⟶⨁j=1mqjMmj(C(Yj))qj
by requiring that each partial map
(9)ϕi,j:piMni(C(Xi))pi⟶qjMmj(C(Yj))qj
induced by ϕ be diagonal.


## 2.3. Eigenvalue Variation

Suppose thatB 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].LetB be a C*-algebra. Suppose that
(10)B=⨁i=1kC(Xi)⊗Mni,
where Xi is a connected compact Hausdorff space for every i. Set X=⊔i=1kXi. The following theorem and notations are quoted from [2, 12].Leta be any self-adjoint element in B. For any x in Xi, any positive integer m, 1≤m≤ni, let λm denote the mth lowest eigenvalue of a(x) counted with multiplicity. So λm is a function on each Xi, for i=1,2,…,k. The fact is
(11)|λm(x)-λm(y)|≤||a(x)-a(y)||.
Hence, λm is continuous, for m=1,2,…,k for a given summand of B.Thevariation of the eigenvalues of a, denoted by EV(a), is defined as the maximum of the nonnegative real numbers
(12)sup{|λm(x)-λm(y)|;x,y∈Xi},
over all i and all possible values of m.Thevariation of the normalized trace of a, denoted by TV(a), is defined as
(13)sup{|1ni∑m=1ni‍(λm(x)-λm(y))|;x,y∈Xi}=sup{|tr(a(x))-tr(a(y))|x,y∈Xi}
over all i, where tr denotes the normalized trace of Mn for any positive integer n.Theorem 4 (see [2]).
LetB be an inductive limit of homogeneous C*-algebras Bi with morphisms ϕij from Bi to Bj. Suppose that Bi has the form
(14)Bi=⨁t=1kiMnit(C(Xit)),
where ki and nit are positive integers, and Xit is a connected compact Hausdorff space for every positive integer i and 1≤t≤ki. Consider the following conditions.(1)
The projections ofB separate the traces on B.(2)
For any self-adjoint elementa in Bi and ε>0, there is a j≥i such that
(15)TV(ϕij(a))<ε.(3)
For any self-adjoint elementa in Bi and any positive number ε, there is a j≥i such that
(16)EV(ϕij(a))<ε.(4)
B has real rank zero.
(i)
The following implications hold in general:(17)(4)⟹(3)⟹(2)⟹(1).(ii)
IfB is simple, then the following equivalences hold:
(18)(3)⟺(2)⟺(1).(iii)
IfB 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 AHC*-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,…,an) in Mn(C(X)), we need to rearrange the ai so that the obtained one b=diag(b1,…,bn) with b1≤b2≤⋯≤bn 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 elementh in M2(C(S4)) which is not unitarily equivalent to diag(λ1,λ2) but the eigenvalue variations of both elements are equal, where λi is the ith lowest eigenvalue of h counted with multiplicity [13, Section 2]. In general, given a self-adjoint element h∈Mn(C(X)), for each x∈X, there is a (point-wise) unitary u(x)∈Mn such that h(x)=u(x)diag(λ1(x),λ2(x),…,λn(x))u*(x), where λi(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,…,λn)) but u(x) need not be continuous. The fact is that if u(x) is continuous for any self-adjoint h in Mn(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.Leth=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,
(19)λ1(x)={xifx≤121-xifx>12,λ2(x)={1-xifx≤12xifx>12.
Then EV(h)=EV(diag(λ1,λ2))=1/2.It is straightforward to check that∥uhu*-diag(λ1,λ2)∥≤ε.Lemma 5.
LetX be a connected compact Hausdorff space and h=diag(f1,f2,…,fn) a self-adjoint element in Mn(C(X)), where f1,f2,…,fn are continuous maps from X to ℝ. For any positive number ε, there is a unitary u∈Mn(C(X)) such that
(20)∥uhu*-diag(λ1,λ2,…,λn)∥<ε,
where the λi(x) is the ith lowest eigenvalue of h(x) counted with multiplicity for every x∈X.Proof.
Iff1≤f2≤⋯≤fn, then the unitary u is just the identity of Mn and λi=fi. Therefore, to prove the lemma, we, roughly speaking, only need to rearrange the given family {f1,f2,…,fn} 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.
LetZ=(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 thatE 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
(21)p(0)=1top(1)=(0110).
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.
Forx∈X∖(E∪F)=Z, we have
(22)|λ1(x)-λ2(x)|<ε2,|fi(x)-λ1(x)|<ε2,i=1,2.
Hence,
(23)∥diag(λ1(x),λ2(x))-diag(λ1(x),λ1(x))∥<ε2,∥h(x)-diag(λ1(x),λ1(x))∥<ε2.
On account to (23) we have
(24)∥u(x)h(x)u*(x)-diag(λ1(x),λ2(x))∥≤∥u(x)[h(x)-diag(λ1(x),λ1(x))]u*(x)∥+∥diag(λ1(x),λ1(x))-diag(λ1(x),λ2(x))∥<ε.
Therefore,
(25)∥uhu*-diag(λ1,λ2)∥<ε.The main result of this section as follows.Theorem 6.
Given an AH algebraA=lim⟶(Ai,ϕi), where the ϕi are diagonal *-homomorphisms from Ai to Ai+1, where Ai=⊕t=1kiMnit(C(Xit)) and the Xit 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 Corollary2, it suffices to show that ϕi∞(f)∈L(A) for every real-valued function f∈C(Xit). By the same argument in the proof of Theorem 1, we can assume that each At has only one component; that is, At=Mnt(C(Xt)). 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(ϕij(f))<ε. Let {μ1,…,μn} be the eigenvalue pattern of ϕij(n=nj/ni). Then,
(26)ϕij(f)=ϕij(diag(f,0))=diag(f∘μ1,0,f∘μ2,0,…,f∘μn,0)=vdiag(f1,f2,…,fn,0)v*,
where fi=f∘μi and v is the permutation matrix in Mnj moving all the zero to the bottom left-hand corner. Note that
(27)EV(ϕij(f))=EV(diag(f1,f2,…,fn,fn+1)),
where fn+1(x)=0 for all x∈Xj. By Lemma 5, there exists a unitary u∈Mn+1(C(Xj)) and eigenvalue maps λ1≤λ2≤⋯≤λn+1 of diag(f1,f2,…,fn,fn+1) such that
(28)∥udiag(f1,f2,…,fn,fn+1)u*-diag(λ1,…,λn+1)∥<ε.
Put(29)δi=12(maxx∈Xjλi(x)+minx∈Xjλi(x)),i=1,2,…,n+1.
Then for any i, we have
(30)maxx∈Xjλi(x)-minx∈Xjλi(x)≤EV(ϕij(f))<ε
and so
(31)|λi(x)-δi|<ε,∀x∈Xj.
Thus,
(32)∥udiag(f1,f2,…,fn,fn+1)u*-∑i=1n+1‍δieii∥<2ε,
where {eij} is the standard basis of Mn+1. This implies that
(33)∥ϕij(f)-b∥<2ε,
where b=v*diag(u*(∑i=1n+1δieii)u,0)v is a linear combination of projections in Aj.
Therefore,(34)ϕi∞(f)∈L(A).


## 2.5. Another Form of Theorem6

Lemma 7.
LetB 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, ifB=Mn(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 Mm(C(X)).Proof.
By assumption, there exists a partial isometryv such that p = v*v and q = vv*. Let us consider the following maps:
(35)ϕ(x)=vxv*(x∈pBp),ψ(y)=v*yv(y∈qBq).
It is straightforward to check that the compositions of ϕ and ψ are the identity maps.
In the caseB=Mn(C(X)) and q is a constant projection of rank m in B, we have qBq=Mm(C(X)). Therefore, pBp is *-isomorphic to Mm(C(X)).Theorem 8 (another form of Theorem6).
LetA=lim⟶(Ai,ϕi) be a diagonal AH algebra, where the pit are projections in Mnit(C(Xit)), Ai=⊕t=1kipitMnit(C(Xit))pit, and the ϕi are unital diagonal. Suppose that each projection p1t 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 thatAi=piMni(C(Xi))pi, for all i. It is easy to see that p1 is Murray-von Neumann equivalent to q1=e11+e22+⋯+em1, where m1 is the rank of p1. For i>1, define qi=ϕi-1(qi-1). Then qi=ϕ1i(q1) is constant, since q1 is constant and ϕ1i is diagonal. Let us denote by mi the rank of qi, then mi+1∣mi. By Lemma 7, there are *-isomorphisms Θi from piMni(C(Xi))pi to qiMni(C(Xi))qi=Mmi(C(X)) such that
(36)Θi(a)=viavi*,
where p1=v1*v1~v1v1*=q1 and vi=ϕ1i(v1). Since ϕi is diagonal, there exists its extension ϕ~i which is a diagonal *-homomorphism from Mni(C(Xi)) to Mni+1(C(Xi+1)). Let ψi be the restriction of ϕ~i on qiMni(C(Xi))qi. Then ψi(qi)=qi+1. Therefore, the map ψi can be viewed as the map from qiMni(C(Xi))qi to qi+1Mni+1(C(Xi+1))qi+1 and so lim⟶(Mmi(C(Xi)),ψi) is a diagonal AH-algebra.
On the other hand, it is straightforward to check thatΘi+1∘ϕi=ψi∘Θi and hence A=lim⟶(Mmi(C(Xi)),ψi). 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.
LetA=lim⟶(Mν(n)(C(X)),ϕn) be a Goodearl algebra [6] and ωt,1 the weighted identity ratio for ϕt,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)
limt→∞ωt,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, letA 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.
LetA 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.
LetA=lim⟶(An,ϕn) and K=lim⟶(Mn,in), where An is a homogeneous algebra, ϕn is an injective diagonal homomorphism from An to An+1, and in is the embedding from Mn to Mn+1 which associates each a∈Mn to diag(a,0)∈Mn+1 for each positive integer n. Let us consider the inductive limit  lim⟶(An⊗Mn,ϕn⊗in). For each integer n≥1, denote by in,∞ and ϕn,∞ the homomorphisms from Mn and  An to K and A in the inductive limit of K and A, respectively. Then
(37)(ϕn+1,∞⊗in+1,∞)∘(ϕn⊗in)=ϕn,∞⊗in,∞.
Hence, by the universal property of inductive limit, there exists a unique homomorphism Φ from lim⟶(An⊗Mn,ϕn⊗in) to A⊗K such that
(38)Φ∘(ϕn⊗in)=(ϕn,∞⊗in,∞).
It is straightforward to check that the image of Φ is dense in A⊗K and since all the maps ϕn and in are injective, we have lim⟶(An⊗Mn,ϕn⊗in) is A⊗K. Furthermore, for each n, we identify an element a⊗b in An⊗Mn with the matrix (aijb) in Mn(An), where a=(aij)∈Mn and b∈An. By interchanging rows and columns (independent of a⊗b) of (ϕn⊗in)(a⊗b), we obtain diag(a⊗b∘λ1,…,a⊗b∘λm,0), where λ1,…,λm are the eigenvalue maps of ϕn. This means that there is a permutation matrix un∈Mn+1(An+1) such that unϕn⊗inun* is diagonal. The fact is that the inductive limit is unchanged under unitary equivalence; that is,
(39)lim⟶(An⊗Mn,ϕn⊗in)=lim⟶(An⊗Mn,unϕn⊗inun*).
Hence, lim⟶(An⊗Mn,ϕn⊗in) is diagonal.Example 11.
LetB 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 Lemma10, 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 UnitalC*-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.
LetA be a projectionless simple unital C*-algebra with a unique tracial state. Then for any n∈ℕ with n>1, Mn(A) has the LP property.Proof.
Note thatMn(A) has also a unique tracial state.
SinceA is unital, Mn(A) has a nontrivial projection. Then by [1, Corollary 5], Mn(A) has the LP property.Remark 13.
LetA be the Jiang-Su algebra. Then we know that RR(A)=1 [14]. Since Mn(A) is an AH algebra without real rank zero, RR(Mn(A))=1. But from Lemma 12, Mn(A) has the LP property.Using this observation, we can construct aC*-algebra with the LP property such that the fixed point algebra does not have the LP property.Example 14.
A simple unital AI algebraA 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 Bβ does not have the LP property.


### 3.2.C*-Index Theory

According to Example14, there is a faithful conditional expectation E:B→Bβ. We extend this observation to an inclusion of unital C*-algebras with a finite Watatani index as follows.In this section we recall theC*-basic construction defined by Watatani.Definition 15.
LetA⊃P be an inclusion of unital C*-algebras with a conditional expectation E from A to P.(1)
Aquasi-basis for E is a finite set {(ui,vi)}i=1n⊂A×A such that for every a∈A,
(40)a=∑i=1n‍uiE(via)=∑i=1n‍E(aui)vi.(2)
When{(ui,vi)}i=1n is a quasi-basis for E, we define indexE by
(41)IndexE=∑i=1n‍uivi.
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{(wi,wi*)}i=1m, which shows that IndexE is a positive element [17, Lemma 2.1.6].(3)
By the above statements, ifA is a simple C*-algebra, then IndexE is a positive scalar.(4)
IfIndexE<∞, then E is faithful; that is, E(x*x)=0 implies x=0 for x∈A.Next we recall theC*-basic construction defined by Watatani.LetE:A→P be a faithful conditional expectation. Then AP(=A) is a pre-Hilbert module over P with a P valued inner product
(42)〈x,y〉P=E(x*y),x,y∈AP.
We denote by ℰE and ηE 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 ℰE. Then ℰE is a Hilbert C*-module over P. Since E is faithful, the inclusion map ηE from A to ℰE is injective. Let LP(ℰE) be the set of all (right) P module homomorphisms T:ℰE→ℰE with an adjoint right P module homomorphism T*:ℰE→ℰE such that
(43)〈Tξ,ζ〉=〈ξ,T*ζ〉,ξ,ζ∈ℰE.
Then LP(ℰE) is a C*-algebra with the operator norm ∥T∥=sup{∥Tξ∥:∥ξ∥=1}. There is an injective *-homomorphism λ:A→LP(ℰE) defined by
(44)λ(a)ηE(x)=ηE(ax)
for x∈AP and a∈A, so that A can be viewed as a C*-subalgebra of LP(ℰE). Note that the map eP:AP→AP defined by
(45)ePηE(x)=ηE(E(x)),x∈AP
is bounded and thus it can be extended to a bounded linear operator, denoted by eP again, on ℰE. Then eP∈LP(ℰE) and eP=eP2=eP*; that is, eP is a projection in LP(ℰE). A projection eP is called the Jones projection of E.The(reduced)C*-basic construction is a C*-subalgebra of LP(ℰE), defined as
(46)Cr*〈A,eP〉=span{λ(x)ePλ(y)∈LP(ℰE):x,y∈A}¯∥·∥.Remark 17.
Watatani proved the following in [17].(1)
IndexE is finite if and only if Cr*〈A,eP〉 has the identity (equivalently Cr*〈A,eP〉=LP(ℰE)) and there exists a constant c>0 such that E(x*x)≥cx*x for x∈A; that is, ∥x∥P2≥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 ℰE=A.(2)
IfindexE is finite, then each element z in Cr*〈A,eP〉 has a form
(47)z=∑i=1n‍λ(xi)ePλ(yi)
for some xi and yi in A.(3)
LetCmax*〈A,eP〉 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 Cr*〈A,eP〉 to Cmax*〈A,eP〉 [17, Proposition 2.2.9]. Therefore, we can identify Cr*〈A,eP〉 with Cmax*〈A,eP〉. So we call Cr*〈A,eP〉 the C*-basic construction and denote it by C*〈A,eP〉. Moreover, we identify λ(A) with A in C*〈A,ep〉(=Cr*〈A,eP〉), and we define it as
(48)C*〈A,ep〉={∑i=1n‍xiePyi:xi,yi∈A,n∈ℕ}.(4)
TheC*-basic construction C*〈A,ep〉 is isomorphic to qMn(P)q for some n∈ℕ and projection q∈Mn(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,eP〉 to A such that
(49)E^(xePy)=(IndexE)-1xyforx,y∈A
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 thatindexE 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,eP〉 such that π(e)=eP and π(a)=a for a∈A [17, Proposition 2.2.11].


### 3.3. Rokhlin Property for an Inclusion of UnitalC*-Algebras

For aC*-algebra A, we set
(50)c0(A)={(an)∈l∞(ℕ,A):limn→∞∥an∥=0},A∞=l∞(ℕ,A)c0(A).We identifyA with the C*-subalgebra of A∞ consisting of the equivalence classes of constant sequences and set
(51)A∞=A∞∩A′.
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.
Letα 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 {eg}g∈G⊂A∞ consisting of projections satisfying
(52)(αg)∞(eh)=eghforg,h∈G.
We call {eg}g∈G the Rokhlin projections.
LetA⊃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 {(ui,vi)}i=1n, then E∞ also has a finite index with a quasi-basis {(ui,vi)}i=1n and Index(E∞)=IndexE.Motivated by Definition18, Kodaka et al. introduced the Rokhlin property for an inclusion of unital C*-algebras with a finite index [19].Definition 19.
A conditional expectationE 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
(53)E∞(e)=(IndexE)-1·1
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 simpleC*-algebra to its fixed point algebra has the Rokhlin property in the sense of Definition 19.Proposition 20 (see [19]).
Letα 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=Aα defined by
(54)E(x)=1#G∑g∈G‍αg(x),forx∈A,
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]).
LetP⊂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.
LetP⊂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
(55)P⊂A⊂C*〈A,eP〉(:=B)⊂C*〈B,eA〉(:=B1).
IfE:A→P has the Rokhlin property with a Rokhlin projection e∈A∞, then the double dual conditional expectation E^^(:=EB):C*〈B,eA〉→B has the Rokhlin property.Proof.
Note that from Remark17(4) and [19, Corollary 3.8], C*-algebras C*〈A,eP〉 and C*〈B,eA〉 are simple.
SinceePeeP=E∞(e)eP=(IndexE)-1ep, (IndexE)eepe≤e, and
(56)E^∞(e-(IndexE)eepe)=e-(IndexE)eE^∞(eP)e=e-e=0,
we have e=(IndexE)eePe. Then, for any x,y∈A(57)e(xePy)e=exeePeye=(IndexE)-1exye=E^(xePy)e.
Hence, from Remark 17(3), we have eze=E^(z)e for any z∈C*〈A,eP〉.
Let{(wi,wi*)}⊂B×B be a quasi-basis for E^(=EA) and eA be the Jones projection of E^. Set g=∑i‍wieeAwi*∈B1∞. Then g is a projection and g∈B1′. Indeed, since
(58)g2=∑i,j‍wieeAwi*wjeeAwj*=∑i,j‍wieeAE^(wi*wj)wj*=∑i‍wieeA(∑j‍E^(wi*wj)wj*)=∑i‍wieeAwi*=g,g is a projection.
Consider the following:(59)geA=∑i‍wieeAwi*eA=∑i‍wiE^(wi*)eeA=eeA,eAg=eA∑i‍wieeAwi*=∑i‍E^(wi)eAewi*=eAe∑i‍E^(wi)wi*=eeA=geA.
Moreover, for any z∈C*〈A,eP〉, we have
(60)gz=∑i‍wieeAwi*z=∑i‍wieeA(∑j‍E^(wi*zwj)wj*)=∑i‍wi∑j‍E^(wi*xzwj)eeAwj*=∑j‍(∑i‍wiE^(wi*zwj))eeAwj*=∑j‍zwjeeAwj*=z∑j‍wjeeAwj*=zg.
Since B1=C*〈C*〈A,eP〉,eA〉, g∈B1′∩B1∞.
To prove that the double dual conditional expectationE^^ 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,wP〉, by Remark 17(5), there exists an isomorphism π:C*〈C*〈A,eP〉,eA〉→C*〈C*〈A,eP〉,e〉 such that π(eA)=e and π(z)=z for z∈C*〈A,eP〉. Then
(61)E^^∞(g)=∑i‍wieE^^∞(eA)wi*=∑i‍wi(IndexE)-1ewi*=(IndexE)-1∑i‍wiπ(eA)wi*=(IndexE)-1π(∑i‍wieAwi*)=(IndexE)-11=(IndexE^^)-11,
hence E^^ has the Rokhlin property.


### 3.4. Main Results

Theorem 23.
Let1∈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.
Letx∈P and ε>0. Since A has the LP property, x can be approximated by a line sum of projection ∑i=1nλipi(pi∈A) such that ∥x-∑i=1nλipi∥<ε.
Sinceβ:A∞→P∞ is an injective *-homomorphism by Proposition 21, we have
(62)∥β(x-∑i=1n‍λip)∥=∥β(x)-∑i=1n‍λiβ(pi)∥<ε.
Since β|P=id, we have ∥x-∑i=1nλiβ(pi)∥<ε. Each projection in P∞ can be lifted to a projection in ℓ∞(ℕ,P), so we can find a set of projections {qi}i=1n⊂P such that
(63)∥x-∑i=1n‍λiqi∥<ε.
Therefore, P has the LP property.Theorem 24.
Letα 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=Aα defined by
(64)E(x)=1#G∑g∈G‍αg(x),forx∈A,
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 Theorem24 consider the following two basic constructions:
(65)Aα⊂A⊂C*〈A,eP〉⊂C*〈B,eA〉(B=C*〈A,eP〉)(Aα)⊂A⊂A⋊αG⊂C*〈A⋊αG,eF〉,
where F:A⋊αG→A is a canonical conditional expectation. Then there is an isomorphism π:C*〈A,eP〉→A⋊αG and π~:C*〈B,eA〉→C*〈A⋊αG,eF〉 such that(1)
π(a)=a for all a∈A,(2)
π(ep)=q, where q=(1/|G|)∑g∈Gug,(3)
A⋊αG=C*〈A,q〉,(4)
π~(b)=π(b) for all b∈B,(5)
π~(eA)=eF.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,
(66)A⋊αG=linearspanof{∑g∈G‍(αg(x)ug)*(∑g∈G‍αg(y)ug)∣x,y∈A}¯=linearspanof{1|G|∑g∈G‍x*αg(y)ug∣x,y∈A}¯.
On the contrary, for anyx,y∈A, we have
(67)xqy=x1|G|∑g∈G‍ugy=1|G|x∑g∈G‍ugyug*ug=1|G|∑g∈G‍xαg(y)ug,
hence A⋊αG=C*〈A,q〉.
Since for anya∈A(68)qaq=1|G|2∑g∈G‍uga∑h∈G‍uh=1|G|2∑g,h∈G‍ugaug*ugh=1|G|2∑g,h∈G‍αg(a)ugh=1|G|∑g∈G‍αg(a)1|G|∑h∈G‍ugh=E(a)q,
by Remark 17(5) there is an isomorphism π:C*〈A,eP〉→C*〈A,q〉=A⋊αG such that π(a)=a for any a∈A and π(eP)=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∈A(69)(eFπ(xePy)eF)(aqb)=(eFxqy)F(1|G|∑g∈G‍aαg(b)ug)=1|G|(eFxqyab)=1|G|2xyab.
On the contrary,
(70)π(E^(xePy))eF(aqb)=π(1|G|xy)1|G|ab=1|G|2xyab.
Hence, we have eFπ(xepy)eF=π(E^(xePy)). By Remark 17(5), there is an isomorphism π~:C*〈B,eA〉→C*〈A⋊αG,eF〉 such that π~(b)=π(b) for any b∈B and π~(eA)=eF.
The condition(6) comes from the direct computation.Lemma 26.
Under the same conditions in Lemma25  C*〈A⋊αG,eF〉 is isomorphic to M|G|(A).Proof.
Note that{(ug*,ug)}g∈G is a quasi-basis for F. By [17, Lemma 3.3.4], there is an isomorphism from C*〈A⋊αG,eF〉 to rM|G|(A)r, where r=[E(ug*uh)]g,h∈G=I|G|. Hence C*〈A⋊αG,eF〉 is isomorphic to M|G|(A).Proof of Theorem24.
Let{eg}g∈G be the Rokhlin projection of E. From Proposition 20,  E:A→AG 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
(71)AG⊂A⊂C*〈A,eP〉⊂C*〈B,eA〉(B=C*〈A,eP〉).
SinceA is simple, the map A∋x↦xe is injective, hence we know that E has the Rokhlin property. Therefore, AG has the LP property by Theorem 23.
SinceC*〈A⋊αG,eF〉 is isomorphic to M|G|(A) by Lemma 26 and A has the LP property, C*〈A⋊αG,eF〉 has the LP property. Hence, C*〈B,eA〉 has the LP property, because that C*〈A⋊αG,eF〉 is isomorphic to C*〈B,eA〉 from Lemma 25. From Proposition 22,  E^^:C*〈B,eA〉→C*〈A,eP〉 has the Rokhlin property, hence we conclude that C*〈A,eP〉 has the LP property by Theorem 23. Since C*〈A,eP〉 is isomorphic to A⋊αG by Lemma 25, we conclude that A⋊αG has the LP property.Remark 27.
(1)
When an actionα 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 AG 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 groupG 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.LetA 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 eg∈A for g∈G such that(1)
∥αg(eh)-egh∥<ε for all g,h∈G and all a∈F,(2)
∥ega-aeg∥<ε for all g∈G and all a∈F,(3)
withe=∑g∈Geg, the projection 1-e is the Murray-von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x,(4)
withe as in (3), we have ∥exe∥>1-ε.It is obvious that the tracial Rokhlin property is weaker than the Rokhlin property.Proposition 28.
Letα 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.
Sinceα is a pointwise outer (i.e., for any g∈G∖{0}αg 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 actionsα 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].


## 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.
LetA be a projectionless simple unital C*-algebra with a unique tracial state. Then for any n∈ℕ with n>1, Mn(A) has the LP property.Proof.
Note thatMn(A) has also a unique tracial state.
SinceA is unital, Mn(A) has a nontrivial projection. Then by [1, Corollary 5], Mn(A) has the LP property.Remark 13.
LetA be the Jiang-Su algebra. Then we know that RR(A)=1 [14]. Since Mn(A) is an AH algebra without real rank zero, RR(Mn(A))=1. But from Lemma 12, Mn(A) has the LP property.Using this observation, we can construct aC*-algebra with the LP property such that the fixed point algebra does not have the LP property.Example 14.
A simple unital AI algebraA 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 Bβ does not have the LP property.


## 3.2.C*-Index Theory

According to Example14, there is a faithful conditional expectation E:B→Bβ. We extend this observation to an inclusion of unital C*-algebras with a finite Watatani index as follows.In this section we recall theC*-basic construction defined by Watatani.Definition 15.
LetA⊃P be an inclusion of unital C*-algebras with a conditional expectation E from A to P.(1)
Aquasi-basis for E is a finite set {(ui,vi)}i=1n⊂A×A such that for every a∈A,
(40)a=∑i=1n‍uiE(via)=∑i=1n‍E(aui)vi.(2)
When{(ui,vi)}i=1n is a quasi-basis for E, we define indexE by
(41)IndexE=∑i=1n‍uivi.
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{(wi,wi*)}i=1m, which shows that IndexE is a positive element [17, Lemma 2.1.6].(3)
By the above statements, ifA is a simple C*-algebra, then IndexE is a positive scalar.(4)
IfIndexE<∞, then E is faithful; that is, E(x*x)=0 implies x=0 for x∈A.Next we recall theC*-basic construction defined by Watatani.LetE:A→P be a faithful conditional expectation. Then AP(=A) is a pre-Hilbert module over P with a P valued inner product
(42)〈x,y〉P=E(x*y),x,y∈AP.
We denote by ℰE and ηE 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 ℰE. Then ℰE is a Hilbert C*-module over P. Since E is faithful, the inclusion map ηE from A to ℰE is injective. Let LP(ℰE) be the set of all (right) P module homomorphisms T:ℰE→ℰE with an adjoint right P module homomorphism T*:ℰE→ℰE such that
(43)〈Tξ,ζ〉=〈ξ,T*ζ〉,ξ,ζ∈ℰE.
Then LP(ℰE) is a C*-algebra with the operator norm ∥T∥=sup{∥Tξ∥:∥ξ∥=1}. There is an injective *-homomorphism λ:A→LP(ℰE) defined by
(44)λ(a)ηE(x)=ηE(ax)
for x∈AP and a∈A, so that A can be viewed as a C*-subalgebra of LP(ℰE). Note that the map eP:AP→AP defined by
(45)ePηE(x)=ηE(E(x)),x∈AP
is bounded and thus it can be extended to a bounded linear operator, denoted by eP again, on ℰE. Then eP∈LP(ℰE) and eP=eP2=eP*; that is, eP is a projection in LP(ℰE). A projection eP is called the Jones projection of E.The(reduced)C*-basic construction is a C*-subalgebra of LP(ℰE), defined as
(46)Cr*〈A,eP〉=span{λ(x)ePλ(y)∈LP(ℰE):x,y∈A}¯∥·∥.Remark 17.
Watatani proved the following in [17].(1)
IndexE is finite if and only if Cr*〈A,eP〉 has the identity (equivalently Cr*〈A,eP〉=LP(ℰE)) and there exists a constant c>0 such that E(x*x)≥cx*x for x∈A; that is, ∥x∥P2≥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 ℰE=A.(2)
IfindexE is finite, then each element z in Cr*〈A,eP〉 has a form
(47)z=∑i=1n‍λ(xi)ePλ(yi)
for some xi and yi in A.(3)
LetCmax*〈A,eP〉 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 Cr*〈A,eP〉 to Cmax*〈A,eP〉 [17, Proposition 2.2.9]. Therefore, we can identify Cr*〈A,eP〉 with Cmax*〈A,eP〉. So we call Cr*〈A,eP〉 the C*-basic construction and denote it by C*〈A,eP〉. Moreover, we identify λ(A) with A in C*〈A,ep〉(=Cr*〈A,eP〉), and we define it as
(48)C*〈A,ep〉={∑i=1n‍xiePyi:xi,yi∈A,n∈ℕ}.(4)
TheC*-basic construction C*〈A,ep〉 is isomorphic to qMn(P)q for some n∈ℕ and projection q∈Mn(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,eP〉 to A such that
(49)E^(xePy)=(IndexE)-1xyforx,y∈A
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 thatindexE 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,eP〉 such that π(e)=eP and π(a)=a for a∈A [17, Proposition 2.2.11].


## 3.3. Rokhlin Property for an Inclusion of UnitalC*-Algebras

For aC*-algebra A, we set
(50)c0(A)={(an)∈l∞(ℕ,A):limn→∞∥an∥=0},A∞=l∞(ℕ,A)c0(A).We identifyA with the C*-subalgebra of A∞ consisting of the equivalence classes of constant sequences and set
(51)A∞=A∞∩A′.
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.
Letα 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 {eg}g∈G⊂A∞ consisting of projections satisfying
(52)(αg)∞(eh)=eghforg,h∈G.
We call {eg}g∈G the Rokhlin projections.
LetA⊃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 {(ui,vi)}i=1n, then E∞ also has a finite index with a quasi-basis {(ui,vi)}i=1n and Index(E∞)=IndexE.Motivated by Definition18, Kodaka et al. introduced the Rokhlin property for an inclusion of unital C*-algebras with a finite index [19].Definition 19.
A conditional expectationE 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
(53)E∞(e)=(IndexE)-1·1
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 simpleC*-algebra to its fixed point algebra has the Rokhlin property in the sense of Definition 19.Proposition 20 (see [19]).
Letα 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=Aα defined by
(54)E(x)=1#G∑g∈G‍αg(x),forx∈A,
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]).
LetP⊂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.
LetP⊂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
(55)P⊂A⊂C*〈A,eP〉(:=B)⊂C*〈B,eA〉(:=B1).
IfE:A→P has the Rokhlin property with a Rokhlin projection e∈A∞, then the double dual conditional expectation E^^(:=EB):C*〈B,eA〉→B has the Rokhlin property.Proof.
Note that from Remark17(4) and [19, Corollary 3.8], C*-algebras C*〈A,eP〉 and C*〈B,eA〉 are simple.
SinceePeeP=E∞(e)eP=(IndexE)-1ep, (IndexE)eepe≤e, and
(56)E^∞(e-(IndexE)eepe)=e-(IndexE)eE^∞(eP)e=e-e=0,
we have e=(IndexE)eePe. Then, for any x,y∈A(57)e(xePy)e=exeePeye=(IndexE)-1exye=E^(xePy)e.
Hence, from Remark 17(3), we have eze=E^(z)e for any z∈C*〈A,eP〉.
Let{(wi,wi*)}⊂B×B be a quasi-basis for E^(=EA) and eA be the Jones projection of E^. Set g=∑i‍wieeAwi*∈B1∞. Then g is a projection and g∈B1′. Indeed, since
(58)g2=∑i,j‍wieeAwi*wjeeAwj*=∑i,j‍wieeAE^(wi*wj)wj*=∑i‍wieeA(∑j‍E^(wi*wj)wj*)=∑i‍wieeAwi*=g,g is a projection.
Consider the following:(59)geA=∑i‍wieeAwi*eA=∑i‍wiE^(wi*)eeA=eeA,eAg=eA∑i‍wieeAwi*=∑i‍E^(wi)eAewi*=eAe∑i‍E^(wi)wi*=eeA=geA.
Moreover, for any z∈C*〈A,eP〉, we have
(60)gz=∑i‍wieeAwi*z=∑i‍wieeA(∑j‍E^(wi*zwj)wj*)=∑i‍wi∑j‍E^(wi*xzwj)eeAwj*=∑j‍(∑i‍wiE^(wi*zwj))eeAwj*=∑j‍zwjeeAwj*=z∑j‍wjeeAwj*=zg.
Since B1=C*〈C*〈A,eP〉,eA〉, g∈B1′∩B1∞.
To prove that the double dual conditional expectationE^^ 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,wP〉, by Remark 17(5), there exists an isomorphism π:C*〈C*〈A,eP〉,eA〉→C*〈C*〈A,eP〉,e〉 such that π(eA)=e and π(z)=z for z∈C*〈A,eP〉. Then
(61)E^^∞(g)=∑i‍wieE^^∞(eA)wi*=∑i‍wi(IndexE)-1ewi*=(IndexE)-1∑i‍wiπ(eA)wi*=(IndexE)-1π(∑i‍wieAwi*)=(IndexE)-11=(IndexE^^)-11,
hence E^^ has the Rokhlin property.


## 3.4. Main Results

Theorem 23.
Let1∈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.
Letx∈P and ε>0. Since A has the LP property, x can be approximated by a line sum of projection ∑i=1nλipi(pi∈A) such that ∥x-∑i=1nλipi∥<ε.
Sinceβ:A∞→P∞ is an injective *-homomorphism by Proposition 21, we have
(62)∥β(x-∑i=1n‍λip)∥=∥β(x)-∑i=1n‍λiβ(pi)∥<ε.
Since β|P=id, we have ∥x-∑i=1nλiβ(pi)∥<ε. Each projection in P∞ can be lifted to a projection in ℓ∞(ℕ,P), so we can find a set of projections {qi}i=1n⊂P such that
(63)∥x-∑i=1n‍λiqi∥<ε.
Therefore, P has the LP property.Theorem 24.
Letα 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=Aα defined by
(64)E(x)=1#G∑g∈G‍αg(x),forx∈A,
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 Theorem24 consider the following two basic constructions:
(65)Aα⊂A⊂C*〈A,eP〉⊂C*〈B,eA〉(B=C*〈A,eP〉)(Aα)⊂A⊂A⋊αG⊂C*〈A⋊αG,eF〉,
where F:A⋊αG→A is a canonical conditional expectation. Then there is an isomorphism π:C*〈A,eP〉→A⋊αG and π~:C*〈B,eA〉→C*〈A⋊αG,eF〉 such that(1)
π(a)=a for all a∈A,(2)
π(ep)=q, where q=(1/|G|)∑g∈Gug,(3)
A⋊αG=C*〈A,q〉,(4)
π~(b)=π(b) for all b∈B,(5)
π~(eA)=eF.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,
(66)A⋊αG=linearspanof{∑g∈G‍(αg(x)ug)*(∑g∈G‍αg(y)ug)∣x,y∈A}¯=linearspanof{1|G|∑g∈G‍x*αg(y)ug∣x,y∈A}¯.
On the contrary, for anyx,y∈A, we have
(67)xqy=x1|G|∑g∈G‍ugy=1|G|x∑g∈G‍ugyug*ug=1|G|∑g∈G‍xαg(y)ug,
hence A⋊αG=C*〈A,q〉.
Since for anya∈A(68)qaq=1|G|2∑g∈G‍uga∑h∈G‍uh=1|G|2∑g,h∈G‍ugaug*ugh=1|G|2∑g,h∈G‍αg(a)ugh=1|G|∑g∈G‍αg(a)1|G|∑h∈G‍ugh=E(a)q,
by Remark 17(5) there is an isomorphism π:C*〈A,eP〉→C*〈A,q〉=A⋊αG such that π(a)=a for any a∈A and π(eP)=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∈A(69)(eFπ(xePy)eF)(aqb)=(eFxqy)F(1|G|∑g∈G‍aαg(b)ug)=1|G|(eFxqyab)=1|G|2xyab.
On the contrary,
(70)π(E^(xePy))eF(aqb)=π(1|G|xy)1|G|ab=1|G|2xyab.
Hence, we have eFπ(xepy)eF=π(E^(xePy)). By Remark 17(5), there is an isomorphism π~:C*〈B,eA〉→C*〈A⋊αG,eF〉 such that π~(b)=π(b) for any b∈B and π~(eA)=eF.
The condition(6) comes from the direct computation.Lemma 26.
Under the same conditions in Lemma25  C*〈A⋊αG,eF〉 is isomorphic to M|G|(A).Proof.
Note that{(ug*,ug)}g∈G is a quasi-basis for F. By [17, Lemma 3.3.4], there is an isomorphism from C*〈A⋊αG,eF〉 to rM|G|(A)r, where r=[E(ug*uh)]g,h∈G=I|G|. Hence C*〈A⋊αG,eF〉 is isomorphic to M|G|(A).Proof of Theorem24.
Let{eg}g∈G be the Rokhlin projection of E. From Proposition 20,  E:A→AG 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
(71)AG⊂A⊂C*〈A,eP〉⊂C*〈B,eA〉(B=C*〈A,eP〉).
SinceA is simple, the map A∋x↦xe is injective, hence we know that E has the Rokhlin property. Therefore, AG has the LP property by Theorem 23.
SinceC*〈A⋊αG,eF〉 is isomorphic to M|G|(A) by Lemma 26 and A has the LP property, C*〈A⋊αG,eF〉 has the LP property. Hence, C*〈B,eA〉 has the LP property, because that C*〈A⋊αG,eF〉 is isomorphic to C*〈B,eA〉 from Lemma 25. From Proposition 22,  E^^:C*〈B,eA〉→C*〈A,eP〉 has the Rokhlin property, hence we conclude that C*〈A,eP〉 has the LP property by Theorem 23. Since C*〈A,eP〉 is isomorphic to A⋊αG by Lemma 25, we conclude that A⋊αG has the LP property.Remark 27.
(1)
When an actionα 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 AG 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 groupG 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.LetA 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 eg∈A for g∈G such that(1)
∥αg(eh)-egh∥<ε for all g,h∈G and all a∈F,(2)
∥ega-aeg∥<ε for all g∈G and all a∈F,(3)
withe=∑g∈Geg, the projection 1-e is the Murray-von Neumann equivalent to a projection in the hereditary subalgebra of A generated by x,(4)
withe as in (3), we have ∥exe∥>1-ε.It is obvious that the tracial Rokhlin property is weaker than the Rokhlin property.Proposition 28.
Letα 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.
Sinceα is a pointwise outer (i.e., for any g∈G∖{0}αg 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 actionsα 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].



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