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The Linear Span of Projections in AH Algebras and for Inclusions of -Algebras
In the first part of this paper, we show that an AH algebra has the LP property if and only if every element of the centre of belongs to the closure of the linear span of projections in . 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 -algebras with a finite Watatani index, if a faithful conditional expectation has the Rokhlin property in the sense of Kodaka et al., then has the LP property under the condition that has the LP property. As an application, let be a simple unital -algebra with the LP property, an action of a finite group onto . If has the Rokhlin property in the sense of Izumi, then the fixed point algebra and the crossed product algebra 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.
A -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 -algebras to have the LP property was considered in . The LP property of a -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 -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 -algebras with diagonal connecting maps) was introduced in  or . Let us denote by the class of diagonal AH algebras. AF-, AI-, and AT-algebras, Goodearl algebras , and Villadsen algebras of the first type  are diagonal AH algebras. The algebras constructed by Toms in  specially which have the same -groups and tracial data but different Cuntz semigroups are Villadsen algebras of the first type and so belong to . This means that the class contains “ugly” and interesting -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 -algebras by their -theoretical data, has been successful for many classes of -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 -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 (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 can be approximated by a linear combination of projections in (Corollary 2). In Section 2.4, using the idea of bubble sort, we can rearrange the entries on a diagonal element in 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 -algebra is inherited to the matrix tensor product and the quotient for any -homomorphism . But it is not stable under the hereditary subalgebra of . In the second part of this paper (Section 3), we will present the stability of the LP property of an inclusion of a unital -algebra with certain conditions and some examples illustrated the instability of such the property. More precisely, let be an inclusion of unital -algebras with a finite Watatani index and a faithful conditional expectation. Then the LP property of can be inherited from that of provided that has the Rokhlin property in the sense of Osaka and Teruya (Theorem 23). As a consequence, given a simple unital -algebra with the LP property if an action of a finite group to has the Rokhlin property in the sense of Izumi, then the fixed point algebra and the crossed product have the LP property (Theorem 24). Furthermore, we also give an example of a simple unital -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, stands for the algebra of all complex matrices, denotes the standard basis of (for convenience, we also use this system of matrix unit for any size of matrix algebras). Let us denote by the matrix algebra with entries from the algebra of all continuous functions on space . If has finitely many connected components and , then Hence, without lots of generality we can always assume that the spectrum of each component of a homogeneous -algebra is connected.
Denote by the block diagonal matrix with entries in some algebras.
Let be a -algebra. Any element in can be considered as an element in via the embedding . We also denote by the closure of the set of all linear combinations of finitely many projections in .
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 -index theory.
2. Linear Span of Projections in AH Algebras
2.1. Linear Span of Projections in an Inductive Limit of Matrix Algebras over -Algebras
Let where and are -algebras. Let be a spanning set of (as a vector space) and be the union of for . Since every element of a -algebra can be written as a sum of two self-adjoint elements, we can assume that all elements of are self-adjoint.
Theorem 1. Let be an inductive limit -algebra as above. Then the following statements are equivalent.(i)has the LP property.(ii)For any integer , any and any , there exists an integer such that can be approximated by an element in to within .(iii)For any integer , there exist a spanning set of such that the images of all elements in that spanning set under belong to .
Proof. The implication (iii) (i) is obvious.
To prove the implication (i) (ii), it suffices to mention that every element (projection) in can be arbitrarily closely approximated by elements (projections, resp.) in .
Let us prove the implication (ii) (iii). Clearly, without lots of generality we can assume that for every . For a fixed integer , we put Hence, there exists a unitary such that where the in the last column and the last row is of order .
It is evident that every element in is a linear combination of elements in , where is the set of all diagonal elements with coefficients in . Thus, is the spanning set of . Now, we claim that this spanning set satisfies the requirement of (iii).
Firstly, let be an element in . By (ii), for every . Hence . Lastly, let . By Identity (4), can be assumed to be Moreover, In addition, there exists an integer such that can be approximated by an element of to within . Hence can be approximated by an element of to within .
Corollary 2. Let be an AH algebra, where and are connected compact Hausdorff spaces. Then the following statements are equivalent.(i) has the LP property.(ii)For any integer , any and any , there exists an integer such that can be approximated by an element in to within .
From the proof of Theorem 1, we can obtain the following.
Corollary 3. Let be a -algebra. If has the LP property, then and have the LP property, where 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 . Let and be compact Hausdorff spaces. A -homomorphism from to is said to be diagonal if there exist continuous maps from to such that where is a zero matrix of order (). If the size , the map is unital.
The are called the eigenvalue maps (or simply eigenvalues) of . The family is called the eigenvalue pattern of . In addition, let and be projections in and , respectively. An -homomorphism from to is called diagonal if there exists a diagonal -homomorphism from to such that is reduced from on and . This definition can also be extended to a -homomorphism by requiring that each partial map induced by be diagonal.
2.3. Eigenvalue Variation
Suppose that is a simple AH algebra. Then, has real rank zero if and only if its projections separate the traces provided that this algebra has slow dimension growth (see ). This equivalence was first studied when the dimensions of the spectra of the building blocks in the inductive limit decomposition of are not more than two, see .
Let be any self-adjoint element in . For any in , any positive integer , , let denote the th lowest eigenvalue of counted with multiplicity. So is a function on each , for . The fact is Hence, is continuous, for for a given summand of .
The variation of the eigenvalues of , denoted by , is defined as the maximum of the nonnegative real numbers over all and all possible values of .
The variation of the normalized trace of , denoted by , is defined as over all , where denotes the normalized trace of for any positive integer .
Theorem 4 (see ). Let be an inductive limit of homogeneous -algebras with morphisms from to . Suppose that has the form where and are positive integers, and is a connected compact Hausdorff space for every positive integer and . Consider the following conditions.(1)The projections of separate the traces on .(2)For any self-adjoint element in and , there is a such that (3)For any self-adjoint element in and any positive number , there is a such that (4) has real rank zero.(i)The following implications hold in general: (ii)If is simple, then the following equivalences hold: (iii)If is simple and has slow dimension growth, then all the conditions , , , and are equivalent.
2.4. Rearrange Eigenvalue Pattern
In order to evaluate the eigenvalue variation  of a diagonal element in , we need to rearrange the so that the obtained one with has the same eigenvalue variation of .
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 in which is not unitarily equivalent to but the eigenvalue variations of both elements are equal, where is the th lowest eigenvalue of counted with multiplicity [13, Section 2]. In general, given a self-adjoint element , for each , there is a (point-wise) unitary such that , where is the th lowest eigenvalue of counted with multiplicity. Denote by the eigenvalue variation of , then but need not be continuous. The fact is that if is continuous for any self-adjoint in , then is less than 3 . 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 . Given any . By [4, Lemma 2.5], there is a unitary such that(i), for all ,(ii), for all .Denote by and the eigenvalue maps of ; that is, Then .
It is straightforward to check that .
Lemma 5. Let be a connected compact Hausdorff space and a self-adjoint element in , where are continuous maps from to . For any positive number , there is a unitary such that where the is the th lowest eigenvalue of counted with multiplicity for every .
Proof. If , then the unitary is just the identity of and . Therefore, to prove the lemma, we, roughly speaking, only need to rearrange the given family to obtain an increasing ordered family. For , the lemma is obvious. Otherwise, using the idea of bubble sort, we can reduce to the case .
Let . Set and .
It is clear that and are disjoint closed sets and . We have and for all . If is empty, then the unitary can be chosen as (, resp.). Thus, we can assume both and are nonempty. By Urysohn’s Lemma, there is a continuous map such that is equal to on and on . Since the space of unitary matrices of is path connected, there is a unitary path linking Consequently, is a unitary in and for all .
For , we have Hence, On account to (23) we have Therefore,
The main result of this section as follows.
Theorem 6. Given an AH algebra , where the are diagonal -homomorphisms from to , where and the are connected compact Hausdorff spaces. If has small eigenvalue variation in the sense of Bratteli and Elliott, then has the LP property.
Proof. By Corollary 2, it suffices to show that for every real-valued function . By the same argument in the proof of Theorem 1, we can assume that each has only one component; that is, . Let be arbitrary. Since has small eigenvalue variation in the sense of Bratteli and Elliott, there is an integer such that . Let be the eigenvalue pattern of . Then,
where and is the permutation matrix in moving all the zero to the bottom left-hand corner. Note that
where for all . By Lemma 5, there exists a unitary and eigenvalue maps of such that
Put Then for any , we have and so Thus, where is the standard basis of . This implies that where is a linear combination of projections in .
2.5. Another Form of Theorem 6
Lemma 7. Let be a -algebra, and and projections in . If and are Murray-von Neumann equivalent, then is isomorphic to .
In particular, if (where is a connected compact Hausdorff space) and is a constant projection of rank in , then is -isomorphic to .
Proof. By assumption, there exists a partial isometry such that = and = . Let us consider the following maps:
It is straightforward to check that the compositions of and are the identity maps.
In the case and is a constant projection of rank in , we have . Therefore, is -isomorphic to .
Theorem 8 (another form of Theorem 6). Let be a diagonal AH algebra, where the are projections in , , and the are unital diagonal. Suppose that each projection is Murray-von Neumann equivalent to some constant projection in . Then has the LP property provided that has small eigenvalue variation in the sense of Bratteli and Elliott.
Proof. We can assume that , for all . It is easy to see that is Murray-von Neumann equivalent to , where is the rank of . For , define . Then is constant, since is constant and is diagonal. Let us denote by the rank of , then . By Lemma 7, there are -isomorphisms from to such that
where and . Since is diagonal, there exists its extension which is a diagonal -homomorphism from to . Let be the restriction of on . Then . Therefore, the map can be viewed as the map from to and so is a diagonal AH-algebra.
On the other hand, it is straightforward to check that and hence . By Theorem 6, has the LP property.
In some special cases, small eigenvalue variation in the sense of Bratteli and Elliott and the LP property are equivalent.
Example 9. Let be a Goodearl algebra  and the weighted identity ratio for . Suppose that is not totally disconnected and has finitely many connected components, then the following statements are equivalent.(i) has real rank zero.(ii).(iii) has small eigenvalue variation in the sense of Bratteli and Elliott.(iv) 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 be a simple AH algebra with slow dimension growth and be a simple hereditary -subalgebra of . By [5, Theorem 3.5], has nontrivial projections. Hence, has the LP property by [1, Corollary 5]. However, has real rank zero if and only if it has small eigenvalue variation in the sense of Bratteli and Elliott . This means that we can choose with real rank nonzero such that 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 be a diagonal AH algebra and be the -algebra of compact operators on an infinite dimensional Hilbert space. Then the tensor product is again diagonal.
Proof. Let and , where is a homogeneous algebra, is an injective diagonal homomorphism from to , and is the embedding from to which associates each to for each positive integer . Let us consider the inductive limit . For each integer , denote by and the homomorphisms from and to and in the inductive limit of and , respectively. Then Hence, by the universal property of inductive limit, there exists a unique homomorphism from to such that It is straightforward to check that the image of is dense in and since all the maps and are injective, we have is . Furthermore, for each , we identify an element in with the matrix in , where and . By interchanging rows and columns (independent of ) of , we obtain , where are the eigenvalue maps of . This means that there is a permutation matrix such that is diagonal. The fact is that the inductive limit is unchanged under unitary equivalence; that is, Hence, is diagonal.
Example 11. Let 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 is a diagonal AH algebra of real rank one with the LP property.
3. The LP Property for an Inclusion of Unital -Algebras
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 be a projectionless simple unital -algebra with a unique tracial state. Then for any with , has the LP property.
Proof. Note that has also a unique tracial state.
Since is unital, has a nontrivial projection. Then by [1, Corollary 5], has the LP property.
Using this observation, we can construct a -algebra with the LP property such that the fixed point algebra does not have the LP property.
Example 14. A simple unital AI algebra in [15, Example 9], which comes from Thomsen’s construction, has two extremal tracial states; so by [16, Theorem 4.4], does not have the LP property. There is a symmetry on constructed by Elliott such that is a UHF algebra. Since the fixed point algebra , where is the dual action of . This shows that there is a simple unital -algebra with the LP property such that the fixed point algebra does not have the LP property.
3.2. -Index Theory
According to Example 14, there is a faithful conditional expectation . We extend this observation to an inclusion of unital -algebras with a finite Watatani index as follows.
In this section we recall the -basic construction defined by Watatani.
Definition 15. Let be an inclusion of unital -algebras with a conditional expectation from to .(1)A quasi-basis for is a finite set such that for every , (2)When is a quasi-basis for , we define by When there is no quasi-basis, we write . is called the Watatani index of .
Remark 16. We give several remarks about the above definitions.(1) does not depend on the choice of the quasi-basis in the above formula, and it is a central element of [17, Proposition ].(2)Once we know that there exists a quasi-basis, we can choose one of the form , which shows that is a positive element [17, Lemma ].(3)By the above statements, if is a simple -algebra, then is a positive scalar.(4)If , then is faithful; that is, implies for .
Next we recall the -basic construction defined by Watatani.
Let be a faithful conditional expectation. Then is a pre-Hilbert module over with a valued inner product We denote by and the Hilbert module completion of by the norm for in and the natural inclusion map from to . Then is a Hilbert -module over . Since is faithful, the inclusion map from to is injective. Let be the set of all (right) module homomorphisms with an adjoint right module homomorphism such that Then is a -algebra with the operator norm . There is an injective -homomorphism defined by for and , so that can be viewed as a -subalgebra of . Note that the map defined by is bounded and thus it can be extended to a bounded linear operator, denoted by again, on . Then and ; that is, is a projection in . A projection is called the Jones projection of .
The (reduced) -basic construction is a -subalgebra of , defined as
Remark 17. Watatani proved the following in .(1) is finite if and only if has the identity (equivalently ) and there exists a constant such that for ; that is, for in by [17, Proposition ]. Since for in , if is finite, then .(2)If is finite, then each element in has a form for some and in .(3)Let be the unreduced -basic construction defined in Definition of , which has the certain universality (cf.(5) below). If is finite, then there exists an isomorphism from to [17, Proposition ]. Therefore, we can identify with . So we call the -basic construction and denote it by . Moreover, we identify with in , and we define it as (4)The -basic construction is isomorphic to for some and projection [17, Lemma ]. If is finite, then is a central invertible element of and there is the dual conditional expectation from to such that by [17, Proposition ]. Moreover, has a finite index and faithfulness. If is simple unital -algebra, by Remark 16. Hence by [17, Proposition ].(5)Suppose that is finite and acts on a Hilbert space faithfully and is a projection on such that for . If a map is injective, then there exists an isomorphism from the norm closure of a linear span of to such that and for [17, Proposition ].
3.3. Rokhlin Property for an Inclusion of Unital -Algebras
For a -algebra , we set
We identify with the -subalgebra of consisting of the equivalence classes of constant sequences and set For an automorphism , we denote by and the automorphisms of and induced by , respectively.
Izumi defined the Rokhlin property for a finite group action in [18, Definition 3.1] as follows.
Definition 18. Let be an action of a finite group on a unital -algebra . is said to have the Rokhlin property if there exists a partition of unity consisting of projections satisfying
We call the Rokhlin projections.
Let be an inclusion of unital -algebras. For a conditional expectation from to , we denote by the natural conditional expectation from to induced by . If has a finite index with a quasi-basis , then also has a finite index with a quasi-basis and .
Definition 19. A conditional expectation of a unital -algebra with a finite index is said to have the Rokhlin property if there exists a projection satisfying and a map is injective. We call 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 -algebra to its fixed point algebra has the Rokhlin property in the sense of Definition 19.
Proposition 20 (see ). Let be an action of a finite group on a unital -algebra and the canonical conditional expectation from to the fixed point algebra defined by where is the order of . Then has the Rokhlin property if and only if there is a projection such that , where is the conditional expectation from to induced by .
The following is the key one in the next section.
Proposition 21 (see  and [20, Lemma 2.5]). Let be an inclusion of unital -algebras and a conditional expectation from to with a finite index. If has the Rokhlin property with a Rokhlin projection , then there is a unital linear map such that for any there exists the unique element of such that and . In particular, is a unital injective -homomorphism and for all .
The following is contained in [19, Proposition 3.4]. But we give it for self-contained.
Proposition 22. Let be an inclusion of unital algebras and conditional expectation from to with a finite index. Suppose that is simple. Consider the basic construction
If has the Rokhlin property with a Rokhlin projection , then the double dual conditional expectation has the Rokhlin property.
Proof. Note that from Remark 17 and [19, Corollary 3.8], -algebras and are simple.
Since , , and we have . Then, for any Hence, from Remark 17, we have for any .
Let be a quasi-basis for and be the Jones projection of . Set . Then is a projection and . Indeed, since is a projection.
Consider the following: Moreover, for any , we have Since , .
To prove that the double dual conditional expectation has the Rokhlin property, we will show that is the Rokhlin projection of . Since for any , by Remark 17, there exists an isomorphism such that and for . Then