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Academic research paper on topic " The Linear Span of Projections in AH Algebras and for Inclusions of C * -Algebras "

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 204319,12 pages http://dx.doi.org/10.1155/2013/204319

Research Article

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

Dinh Trung Hoa,1 Toan Minh Ho,2 and Hiroyuki Osaka3

1 Center of Research and Development, Duy Tan University, K7/25 Quang Trung, Da Nang, Vietnam

2 Institute of Mathematics, VAST, 18 Hoang Quoc Viet, Ha Noi 10307, Vietnam

3 Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

Correspondence should be addressed to Hiroyuki Osaka; osaka@se.ritsumei.ac.jp Received 18 October 2012; Accepted 6 January 2013 Academic Editor: Ivanka Stamova

Copyright © 2013 Dinh Trung Hoa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the first part of this paper, we show that an AH algebra A = lim(A t, ) 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 c 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, a an action of a finite group G onto Aut(A). If a has the Rokhlin property in the sense of Izumi, then the fixed point algebra AG and the crossed product algebra Axa G 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; ) (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 At 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 a C*-algebra A is inherited to the matrix tensor product Mn(A) and the quotient n(A) for any * -homomorphism n. 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 e P c 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 a 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 Axa G 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 x n complex matrices, e = {est}s t=Yn 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 Xt and X = uki=l X, then

Mn (C(X))=®Mn (C(X,)). (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,.. -,an) the block diagonal matrix with entries a1,a2,... ,an in some algebras.

Let A be a C*-algebra. Any element a in A can be considered as an element in Mn(A) via the embedding a ^ diag(fl, 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 (Ai,Vi), (2)

where Aj = ®kt= xMn,t(Bit) and Bit are C*-algebras. Let Sit be a spanning set of Bit (as a vector space) and St be the union of Sit for t = 1,..., к ¡. Since every element of a С * -algebra can be written as a sum of two self-adjoint elements, we can assume that all elements of St are self-adjoint.

Theorem 1. Let A be an inductive limit C*-algebra as above. Then the following statements are equivalent.

(i) A has the LP property.

(ii) For any integer i, any x e St and any e > 0, there exists an integer j > i such that <р^(х) can be approximated by an element in L(Aj) to within e.

(iii) For any integer i, there exist a spanning set of At such that the images of all elements in that spanning set under fim 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;.

Let us prove the implication (ii) ^ (iii). Clearly, without lots of generality we can assume that A t = Mn (Bt) for every i. For a fixed integer i, we put

e®St = {est ®х + е^®х*,х eSt}. (3)

Hence, there exists a unitary и e Mn such that

/0 x 0\

и (est ®x + ets ®x)u* = ( x 0 0), (4)

\0 0 0/

where the 0 in the last column and the last row is of order щ - 2.

It is evident that every element in At is a linear combination of elements in e ® St UD, where D is the set of all diagonal elements with coefficients in St. Thus, e ® St U D is the spanning set of A¡. Now, we claim that this spanning set satisfies the requirement of (iii).

Firstly, let d = diag^,. ..,xn,) (xt e St) be an element in D. By (ii), fim(xt) e L(A) for every t. Hence <pitx>(d) e L(A). Lastly, let a e e® St. By Identity (4), a can be assumed to be

0 x 0\

x 0 0) for some x e St. (5)

0 0 0/

Moreover,

/-x 0 0\ a = u* ( 0 x 0)u, where и = \ 0 0 0/

1 1 о

ф, ф.

In addition, there exists an integer j > i such that (x) can be approximated by an element of L(Aj) to within e. Hence fij(a) can be approximated by an element of L(A j) to within e. □

Corollary 2. Let A = lim(At,ft) be an AH algebra, where

A j = ®kt= 1Mn (C(Xit)) and Xit are connected compact Hausdorffspaces. Then the following statements are equivalent.

(i) A has the LP property.

(ii) For any integer i, any f e Ukt= 1 C(Xit) and any e > 0, there exists an integer j > i such that (f) can be approximated by an element in L(A j) to within e.

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 ® 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 0 from Mn(C(X)) to Mnm+k(C(Y)) is said to be diagonal if there exist continuous maps [Xt, i = 1, n} from Y to X such that

$(f) = diag (foX1,foX 2,...,f"Xn ,0), feMn (C(X)),

where 0 is a zero matrix of order k (k > 0). If the size k = 0, the map is unital.

The Xi are called the eigenvalue maps (or simply eigenvalues) of <p. The family [X1,X2,..., Xm} is called the eigenvalue pattern of <p. In addition, let p and q be projections in Mn(C(X)) and Mnm+k(C(Y)), respectively. An *-homomorphism y from pMn(C(X))p to qMnm+k(C(Y))q is called diagonal if there exists a diagonal *-homomorphism 0 from Mn(C(X)) to Mnm+k(C(Y)) such that f is reduced from 0 on pMn(C(X))p and <p(p) = q. This definition can also be extended to a *-homomorphism

0 :0 P,Mni (C (Xt)) pt 0 qjMmj (C (Yj)) qj (8)

¡=1 j=i

by requiring that each partial map

<i>i'j : P,Mni (c (X,)) p, qjMm) (C (Yj)) qj (9) induced by 0 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=^C(X,)®Mni,

where Xt is a connected compact Hausdorff space for every i. Set X = Uki=lXt. The following theorem and notations are quoted from [2,12].

Let a be any self-adjoint element in B. For any x in Xt, any positive integer m, 1 < m < nt, let Xm denote the mth lowest eigenvalue of a(x) counted with multiplicity. So Xm is a function on each Xt, for i = 1,2, ...,k. The fact is

I {y*)\ < \\a{x)-a{y)\\.

Hence, X m is continuous, for m = 1,2, ...,k for a given summand of B.

The variation of the eigenvalues of a, denoted by EN (a), is defined as the maximum of the nonnegative real numbers

sup {\xm (x)-Xm Ml'^y e xi}

over all i and all possible values of m.

The variation of the normalized trace of a, denoted by TV(fl), is defined as

-Y(Xm(x)-Xm(y))

= sup {\tr (a (x)) - tr (a (y))\ x,ye X¡}

over all i, where tr denotes the normalized trace of Mn for any positive integer n.

Theorem 4 (see [2]). Let B be an inductive limit of homogeneous C*-algebras Bt with morphisms ^¡j from Bt to Bj. Suppose that Bt has the form

Bi =QMn,t (C(X>t)),

where kt and nit are positive integers, and Xit is a connected compact Hausdorff space for every positive integer i and 1 < t < kt. Consider the following conditions.

(1) The projections of B separate the traces on B.

(2) For any self-adjoint element a in Bt and e > 0, there is a j > i such that

TV(0y (a))<e.

(3) For any self-adjoint element a in Bt and any positive number e, there isa j > i such that

EV(h} (a))<e.

(4) B has real rank zero.

Denote byX1 and X2 the eigenvalue maps of h; that is,

(i) The following implications hold in general:

(4)=^(3)=^(2)=^(1). (17)

(ii) If B is simple, then the following equivalences hold:

(3) ^ (2)

(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(fl1, ...,an)in Mn(C(X)), we need to rearrange the ai so that the obtained one b = diag(fc1 ,...,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 element h in M2(C(S4)) which is not unitarily equivalent to diag(A1, X2) but the eigenvalue variations of both elements are equal, where Xi is the (th lowest eigenvalue of h counted with multiplicity [13, Section 2]. In general, given a self-adjoint element h e Mn(C(X)), for each x e X, there is a (point-wise) unitary u(x) e Mn such that h(x) = u(x) diag(A1(x), X2(x),..., Xn(x))u* (x), where Xi(x) is the (th lowest eigenvalue of h(x) counted with multiplicity. Denote by EV(h) the eigenvalue variation of h, then EV(h) = EV(diag(A1 ,X2,...,Xn)) 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.

Let h = diag(x, 1-x) e M2(C[0,1]). Given any 1/2 > e> 0. By [4, Lemma 2.5], there is a unitary« e M2(C[0,1]) such that

(i) u(x) = 1 eM2, for all x e [0,1/2-t],

(ii) u(x) = (), for all x e [1/2 + t,1].

K(x) =

if x <

1-x if x >

X2 (x) =

1-x if X <

if X >

ThenEV(h) = EV(diag(A1, X2)) = 1/2.

It is straightforward to check that \\uhu* - diag(A1 ,X2)\\ <

Lemma 5. Let X 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 e, there is a unitary u e Mn(C(X)) such that

\\uhu* - diag (X1,X2,..., < e,

where the Xi (x) is the ith lowest eigenvalue ofh(x) counted with multiplicity for every x e X.

Proof. If f1 < f2 < ••• < fn, then the unitary u is just the identity of Mn and Xi = 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 ofbubble sort, we can reduce to the case n = 2.

Let Z= (X1 -X2)-1 (-e/2,e/2). Set E= {x e X : f1(x) < f2(x)} n(X\Z) and F={x eX: f1(x) > f2(x)} n (X \ Z).

It is clear that E and F are disjoint closed sets and X = Eu F U Z. We have X1(x) = min{f1(x),f2(x)} and X2(x) = max{f1(x),f2(x)} for all x e X. If E(F) is empty, then the unitary u can be chosen as (00) (1 e M2, resp.). Thus, we can assume both E and F are nonempty. By Urysohn's Lemma, there is a continuous map ^ : X ^ [0,1] such that ^ 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) = 1 to P(1) = (JJ). (21)

Consequently, u = p ° ^ is a unitary in M2(C(X)) and

u(x)h(x)u* (x) = diag(X1(x), X2(x)) for all x e EuF.

For x e X \ (E U F) = Z, we have

£ £ \\1 (x)-X2 (x)\<2> \fi (x)-X 1 (x)\<2> i=1,2.

Hence,

||diag (\1 (x), X2 (x)) - diag (\1 (x), X1 (%))|| <

\\h(x)- diag(X! (x),X! (x))|| < 2. On account to (23) we have

Ilu (x) h (x) u* (x) - diag (Xx (x), X2 (%))||

< (x) [h (x) - diag (Xx (x), X1 (%))] u* (x)||

+ Il diag (Xi (x), Xi (x)) - diag (Xi (x), X2 (%))||

Therefore,

\\uhu* - diag(X1,X2)\| < £■

(25) □

The main result of this section as follows.

Theorem 6. Given an AH algebra A = lim(Ai,0i), where the are diagonal *-homomorphisms from At to Ai+1, where Aj = ®kt= 1 MUt(C(Xit)) and the Xit are connected compact Hausdorff spaces. If A has small eigenvalue variation in the sense ofBratteli and Elliott, then A has the LP property.

Proof. By Corollary 2, it suffices to show that (f) e L(A) for every real-valued function f e 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 = Mn (C(Xt)). Let e > 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 (^¡j(f)) < e. Let {ft,..., ftn} be the eigenvalue pattern of <p{j (n = Ujlnt). Then,

0y (f) = $tj (diag (f,0))

= diag (f ° ft,0, f ° ft,0,...,f°pn> 0) (26)

= V diag (fl,f2,...,fn, 0)V*>

where ft = f ° ft and v is the permutation matrix in Mn moving all the zero to the bottom left-hand corner. Note that

EV ($tj (f)) = EV (diag (f1,f2, ...,fn, fn+1)), (27)

where fn+1 (x) = 0 for all x e Xj. By Lemma 5, there exists a unitary u e Mn+1 (C(Xj)) and eigenvalue maps

\1 <^2 <■■ < A n+1 of diag(/1 J2'...'fn' fn+1 ) such that

\\U diag (f1,f2'...'fn 'fn+1)u* - diag n+1 )ll < £.

Sj = - ( max Xj (x) + min X ¡ (%)), i = 1,2,... ,n + I.

2 y xeXj xeXj J

Then for any i, we have

m Xx Xj (x) - nun Xj (x) < EV (faj (f)) < e (30)

and so

\Xt (x)-5j\<£, VxeXj. (31)

idiag (fl,f2,...,fn,fn+1 )u* -

where {et } is the standard basis of Mn+1. This implies that

(f)-b\\<2e,

where b = v* diag(M*(^"=t11 Sieii)u,0)v is a linear combination of projections in Aj. Therefore,

(f)eL(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 = 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 isometry v such that p = v*v and q = vv*. Let us consider the following maps:

0 (x) = vxv* (x e pBp), f (y) = v*yv (y e qBq).

It is straightforward to check that the compositions of 0 and f are the identity maps.

In the case B = 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 Theorem 6). Let A = lim(A;, be a diagonal AH algebra, where the pit are projections in Mntt(C(Xit)), A; = pitMntt(C(Xit))pa, and the fa are unital diagonal. Suppose that each projection p1t is Murray-von Neumann equivalent to some constant projection in A P Then A has the LP property provided that A has small eigenvalue variation in the sense ofBratteli and Elliott.

Proof. We can assume that Ai = piMn,(C(Xi))pi, for all i. It is easy to see that p1 is Murray-von Neumann equivalent to

q1 = e11 +e22 +-----+emi, where m1 is the rank of p1. For i > 1,

define q; = (q—1 ). Then q; = (q1) is constant, since q1 is constant and is diagonal. Let us denote by m; the rank of q;, then mi+1 | m;. By Lemma 7, there are *-isomorphisms from p,Mni(C(X,))p, to q,Mn (C(X,))q, = Mm,(C(X)) such that

0 j (a) = vjav*,

where p1 = v*v1 ~ v1v* = q1 and Vj = (w1). Since fa is diagonal, there exists its extension fa which is a diagonal *-homomorphism from Mn (C(Xi)) to Mn+ (C(Xi+1)).

Let be the restriction of <f>i on qiMn (C(Xi))qi. Then Vi(li) = . Therefore, the map can be viewed as the map from qiMn.(C(Xi))q{ to Mn,+i (C(Xi+1 ))qi+1 and so lim(Mm (C(Xj)), is a diagonal AH-algebra.

On the other hand, it is straightforward to check that ®i+i oCPi = Yi i and hence A = lim(Mm.(C(Xi)),fi). 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(Mv(n)(C(X)),<pn) be a Goodearl algebra [6] and &t1 the weighted identity ratio for . 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^m^ti = 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(An, $n) and K = lim(Mn, in), where An is a homogeneous algebra, <pn is an injective diagonal homomor-phism from An to An+1, and in is the embedding from Mn to Mn+1 which associates each a e Mn to diag(a, 0) e Mn+1 for each positive integer n. Let us consider the inductive limit lim(An ® Mn, (pn ® in). For each integer n> 1, denote by in tx>

and <pnm the homomorphisms from Mn and An to K and A in the inductive limit of K and A, respectively. Then

)o($n ® in) = ® ■ (37)

Hence, by the universal property of inductive limit, there exists a unique homomorphism O from lim(An®Mn, <pn®in)

to A®K such that

°o($n ®in) = ($n,m )■ (38)

It is straightforward to check that the image of O is dense in A® K and since all the maps <pn and in are injective, we have lim(An ® Mn, <fin ® in) is A® K. Furthermore, for each n, we identify an element a® b in An ® Mn with the matrix (a^b) in M„(An), where a = (a{j) e Mn and b e An. By interchanging rows and columns (independent of a ® b) of (4*n ® in) (a ® b),we obtain diag(a ®b o X1,.. ,,a®b o Xm, 0), where X1,...,Xm are the eigenvalue maps of <pn. This means that there is a permutation matrix un e Mn+1 (An+1) such that un$n ® inK is diagonal. The fact is that the inductive limit is unchanged under unitary equivalence; that is,

hm (An ® Mn, ® in) = lim {An ® Mn, Un$n ® inK) ■

Hence, lim(An ® Mn, cpn ® in) is diagonal. □

Example 11. Let Bbeasimpleunital diagonalAHalgebrawith 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 isa 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 e N with n > 1, Mn(A) has the LP property.

Proof. Note that Mn(A) has also a unique tracial state.

Since A is unital, Mn(A) has a nontrivial projection. Then by [1, Corollary 5], Mn(A) has the LP property. □

Remark 13. Let A 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 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 a on A constructed by Elliott such that AxaZ/2Z is a UHF algebra. Since the fixed point algebra (AxaZ/2Zf = A, where p is the dual action of a. This shows that there is a simple unital C* -algebra B with the LP property such that the fixed point algebra B^ does not have the LP property.

3.2. C*-Index Theory. According to Example 14, there is a faithful conditional expectation E : B ^ . 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 Ad 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 {(ut, vi)}1=1 c Ax A such that for every a e A,

a = ^UjE (vja) = ^E (aUj) vt.

(2) When {(ui,vi)}1=1 is a quasi-basis for E, we define index E by

Index E =

When there is no quasi-basis, we write Index E = >x>. index E is called the Watatani index of E.

Remark 16. We give several remarks about the above definitions.

(1) Index E does not depend on the choice of the quasibasis 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 {(wt, w* )}'=1 ,which showsthat 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 Index E < >x>, then E is faithful; that is, E(x* x) = 0 implies x = 0 for x e A.

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

Let E : A ^ P be a faithful conditional expectation. Then AP(= A) is a pre-Hilbert module over P with a P valued inner product

(x,y)p = E(x*y), x,yeAt

We denote by Ee and 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 qE from A to E E is injective. Let Lp(Ee) be the set of all (right) P module homomorphisms T : Ee ^ Ee with an adjoint

^ E E such that

right P module homomorphism T* : EE

= «e EE.

Then Lp(Ee) is a C*-algebra with the operator norm ||T|| = sup{||T£|| : ||£|| = 1}. Thereis aninjective *-homomorphism X: A ^ Lp(Ee) defined by

X (а) цЕ (x) = tjE (ax)

for x e AP and a e A, so that A can be viewed as a C*-subalgebra of Lp(Ee). Note that the map eP : AP ^ AP defined by

eplE (x) = 1e (e(x))> xeaf

is bounded and thus it can be extended to a bounded linear operator, denoted by eP again, on Ee. Then eP e Lp(Ee)

= 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 L p(Ee), defined as

C* (A, eP) = span [X (x) ePX (y) e LP (E E): x,y e A}

Remark 17. Watatani proved the following in [17].

(1) Index E is finite if and only if C**{A,eP) has the identity (equivalently C** (A, eP) = Lp(Ee)) and there exists a constant c > 0 such that E(x*x) > cx*x for x e A; that is, \\x\\p > c\\x\\2 for x in A by [17, Proposition 2.1.5]. Since \\x\\ > \\x\\P for x in A, if index E is finite, then E e = A.

(2) If index E is finite, then each element z in C** (A, eP) has a form

= (xi) ePX (yi)

for some xt and yi in A.

(3) Let C**ax(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 index E is finite, then there exists an isomorphism from C** (A, eP) to Cmax(A,eP) [17, Proposition 2.2.9]. Therefore, we can identify C**(A,eP) with C"mzx(A,eP). So we call C**(A,eP) the C*-basic construction and denote it by C* (A, eP). Moreover, we identify X(A) with A in C* (A, ep)(= C** (A, eP)), and we define it as

^x¡ePy¡ : x¡,yj e A, n e N

(4) The C* -basic construction C* {A, ep) is isomorphic to qMn(P)q for some ne N and projection q e Mn(P) [17, Lemma 3.3.4]. If index £ is finite, then index £ is a central invertible element of A and there is the dual conditional expectation E from C* {A, eP) to A such that

E(xePy) = (IndexE)-1xy for x,ye A (49)

by [17, Proposition 2.3.2]. Moreover, E has a finite index and faithfulness. If A is simple unital C* -algebra, index E e 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 e A. If a map P 5 x ^ xe e B(H) is injective, then there exists an isomorphism n from the norm closure of a linear span of AeA to C* {A, eP) such that n(e) = eP and n(a) = a for a e A [17, Proposition 2.2.11].

3.3. Rokhlin Property for an Inclusion of Unital C* -Algebras. For a C*-algebra A, we set

Co (A) = {(an) eT (N, A) Mim \\an\ = o},

A<x = riNA

Co (A) •

We identify A with the C*-subalgebra of Am consisting of the equivalence classes of constant sequences and set

Aœ = ATO nA1. (51)

For an automorphism a e Aut(A), we denote by am and am the automorphisms of A00 and A œ induced by a,respectively.

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

Definition 18. Let a be an action of a finite group G on a unital C*-algebra A. a is said to have the Rokhlin property if there exists a partition of unity [eg}^eG c Aœ consisting of projections satisfying

{ag)œ (eh) = egh for g,heG. (52)

We call {eg}geG the Rokhlin projections.

Let A d 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 Am to Pm induced by E. If E has a finite index with a quasi-basis {(ut, vi)}'n=1, then Em also has a finite index with a quasi-basis {(ut,vi)}''=l and Index (Em) = 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 e ATO satisfying

Em (e) = (IndexE)-1 -1 (53)

and a map A 5 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 a 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 = Aa defined

E(x) = (x)' forxeA> (54)

where #G is the order ofG. Then a has the Rokhlin property if and only if there is a projection e e ATO such that E™(e) = (1/#G) ■ 1, where Em is the conditional expectation from A00 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 c A be an inclusion of unital C*-algebras and E a conditional expectation from A to P with a finite index. IfE has the Rokhlin property with a Rokhlin projection e e Athen there is a unital linear map p : Am ^ Pm such that for any x e Am there exists the unique element y ofPm such that xe = ye = p(x)e and p(A' nA™) c P' nPm. Inparticular, ¡3\ is a unital injective *-homomorphism and p(x) = x for all x e P.

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

Proposition 22. Let P c 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

PcAcC* (A, ep) (:= B) c C* (B, eA) (:= . (55)

If E : A ^ P has the Rokhlin property with a Rokhlin projection e e Athen the double dual conditional

expectation E(:= EB) : C*{B,eA) ^ B has the Rokhlin property.

Proof. Note that from Remark 17(4) and [19, Corollary 3.8], C*-algebras C* {A, ep) and C* (B, eA) are simple.

Since ePeeP = Em(e)eP = (IndexE) 1 ep, (IndexE)eepe < Moreover, for any z e C* (A, eP),wehave e, and

È (e - (Index E) eepe) = e - (Index E) eÈ (eP) e

= e - e = 0,

we have e = (Index E)eePe. Then, for any x,yeA

e (xePy) e = exeePeye

= (Index E) 1 exye = E (xePy) e.

Hence, from Remark 17(3), we have eze = E(z)e for any z e C*(A,eP).

Let {(w^w*)} с BxBbe a quasi-basis for E(= EA) andeA be the Jones projection of E. Set g = ^ e Б™. Then

g is a projection and g e B[. Indeed, since

wieeAwi WjeeAWj

в2 = Y

= XWieeAÊ (W*Wj) W

= JwteeA ( XÊ{wîwj)w*j

= XwieeAwj = 9>

g is a projection.

Consider the following:

geA = YwieeAwjeA =

gz = X

w;eeAw: z

= Y.WieeA ^^È (wjjzwj) Wj ^

= TWiTP {wjxzwj) eeAWj

= T WiÈ (wjzwj))eeAwj

= TzwjeeAwj = zTwieeAW

Since B1 = C*(C*(A,eP),eA), g e B[ n£™.

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 e C* (A, wP),by Remark 17(5), there exists an isomorphism n : C* (C* (A,eP),eA) ^ C* (C* (A,eP),e) such that n(eA) = e and n(z) = z for z e C* (A, eP). Then

{g) = Ywie^° (ëa) wj i

= Y^Index E)-1 ewjj = (Index E)-1 Twin (eA)wj

= (Index E) 1 п(т wieAwj)

= (Index E)-11

= ( Index È) 1,

eAg = eAYwieeAwj = Yß { i

= eAeYÊ{wi)wj

= eeA = geA.

hence E has the Rokhlin property.

3.4. Main Results

Theorem 23. Let 1 e P c A be an inclusion of unital C*-algebras with a finite Watatani index and E : A ^ P afaithful conditional expectation. Suppose that A has the LP property and E has the Rokhlin property. Then P has the LP property.

Proof. Let x e P and e > 0. Since A has the LP property, x can be approximated by a line sum of projection ^i=l Xipi (pi e A) such that \\x - Y^=l hipi\\ < e.

Since ß : Aœ ^ Pœ is an injective *-homomorphism by Proposition 21, we have

ß( x-^XiP

ß(x)-fjXiß(pi)

Since = id,wehave\\x-^li=1\fi(pi )|| < e. Each projection in Pm can be lifted to a projection in lm(N, P), so we can find a set of projections {qiTi=1 c P such that

Therefore, P has the LP property.

Theorem 24. Let a 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 = Aa defined

E(x) = (x), for xe A,

where #G is the order of G. Suppose that a has the Rokhlin property. We have, then, that if A has the LP property, the fixedpoint algebra and the crossedproduct AxaG 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 cAcC* (A, eP) c C* (B, eA) (B = C* (A, eP)) (Aa)cAcAxaGcC* (AxaG,eP),

where F : AxaG ^ A is a canonical conditional expectation. Then there is an isomorphism n : C* {A, eP) ^ AxaG and n : C* (B, e A) ^ C* (AxaG, eF) such that

(1) n(a) = a for all a e A,

(2) n(ep) = q, where q= (1/|G|) "LgiGug,

(3) AxaG = C*(A,q),

(4) n(b) = n(b) for all b e B,

(5) n(eA) = eF. Moreover, we have

(6) F o n = E and n o E = F o n.

Proof. At first we prove condition (3). Since a is outer, a is saturated by [21, Proposition 4.9]; that is,

Ax„G

= linear span of { £ (ag (x) ug) (у)ид ) I x, y e A}

[geG \geG ) J

linear span of {—Ix*ag (y)ug \x,yeA}.

1\G\ h

On the contrary, for any x,y e A, we have

xqy=xG\Iu*y

= —x I u„yu*u„

\G\ k ^ g 3

= \G\Ixag (y)ug'

\G\ geG

hence AxaG = C* {A, q). Since for any a e A

qaq=Thî lugaluh \G\ geG heG

= 7^2 ! u3au3ugh

\G\ g,heG

= 7^2 1 ag (a)i

\G\ e \G\ he

= E (a) q,

by Remark 17(5) there is an isomorphism n : C* (A, eP) ^ C*(A,q) = AxaG such that n(a) = a for any a e A and n(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 e A

(eFn (xePy) eF) (aqb) = (eFxqy) F X aa9 (b) u9 ^

= G (eFxqyab)

;xyab.

On the contrary,

n (È {xePy)) eF (aqb) = n(

= —Txvab.

Hence, we have epn(xepy)ep = n(E(xePy)).ByRemark17(5), there is an isomorphism ix : C* (B, eA) ^ C* (A G, eF) such that fc(b) = n(b) for any b e B and fc(eA) = eF.

The condition (6) comes from the direct computation. □

Lemma 26. Under the same conditions in Lemma 25 C* (A G, eF) is isomorphic to M^ (A).

Proof. Note that {(u*g,ug)}geG is a quasi-basis for F. By [17, Lemma 3.3.4], there is an isomorphism from C* (Axa G, ep) to rMlGl(A)r, where r = [E(u*guh)]gMG = I]G]. Hence C* (A G, ep) is isomorphic to M^(A). □

Proof of Theorem 24. Let {eg}geG be the Rokhlin projection of E. From Proposition 20, E : of index finite and

has a projection e e A' n Am such that Em(e) = (1/|G|)1. Note that indexE = |G| and e = e1. Consider the basic construction

Ag cAcC* {A, eP) c C* {B, eA) (B = C* {A, eP)).

Since A is simple, the map A 5 x ^ xe is injective, hence we know that E has the Rokhlin property. Therefore, A has the LP property by Theorem 23.

Since C* (Axa G,ep) is isomorphic to M\G\(A) by Lemma 26 and A has the LP property, C* (Axa G,ep) has the LP property. Hence, C* (B, eA) has the LP property, because that C* (A G, ep) 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 AxaG by Lemma 25, we conclude that Axa G has the LP property. □

Remark 27.

(1) When an action a 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 a 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 a be an action from a finite group G on Aut(A). Recall that a has the tracial Rokhlin property if for every finite set F c A, every e > 0, and every positive element x e A with \\x\\ = 1, there are mutually orthogonal projections eg e A for g e G such that

(1) \\ag(eh) - egh\\ < £ for all g,h e G and all a e F,

(2) \\ega - aeg\\ < e for all g e G and all a e F,

(3) with e = XgeG eg, 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 - £.

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

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

Proof. From [26, Proposition 5.7], the restriction map from tracial states on the crossed product AxaG to a-invariant tracial states on A is isomorphism. Hence, Axa G has a unique tracial state.

Since a is a pointwise outer (i.e., for any g e G \ {0} ag is outer) by [23, Lemma 1.5], Axa G is simple.

Therefore, by [1, Corollary4], Axa G has the LP property.

Remark 29. There are many examples of actions a of finite groups on simple unital C*-algebras with real rank zero and a unique tracial state such that a 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.

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