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

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 ∈ S_i and any ε > 0, there exists an integer j ≥ i such that φ_ij(x) can be approximated by an element in L(A_j) to within ε.

• (iii)

  For any integer i, there exist a spanning set of A_i such that the images of all elements in that spanning set under φ_i∞ belong to L(A).

Proof. The implication (iii) ⇒ (i) is obvious.

To prove the implication (i) ⇒ (ii), it suffices to mention that every element (projection) in A can be arbitrarily closely approximated by elements (projections, resp.) in A_i.

Let us prove the implication (ii) ⇒ (iii). Clearly, without lots of generality we can assume that $$ for every i. For a fixed integer i, we put

It is evident that every element in A_i is a linear combination of elements in e ⊗ S_i ∪ D, where D is the set of all diagonal elements with coefficients in S_i. Thus, e  ⊗  S_i  ∪  D is the spanning set of A_i. Now, we claim that this spanning set satisfies the requirement of (iii).

Firstly, let $$ be an element in D. By (ii), φ_i∞(x_t) ∈ L(A) for every t. Hence φ_i∞(d) ∈ L(A). Lastly, let a ∈ e ⊗ S_i. By Identity (4), a can be assumed to be

Corollary 2. Let $$ be an AH algebra, where $$ and X_it are connected compact Hausdorff spaces. Then the following statements are equivalent.

• (i)

  A has the LP property.

• (ii)

  For any integer i, any $$ and any ε > 0, there exists an integer j ≥ i such that φ_ij(f) can be approximated by an element in L(A_j) to within ε.

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

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

• (1)

  The projections of B separate the traces on B.

• (2)

  For any self-adjoint element a in B_i and ε  > 0, there is a j ≥ i such that

  $$ ()

• (3)

  For any self-adjoint element a in B_i and any positive number ε, there is a j ≥ i such that

  $$ ()

• (4)

  B has real rank zero.

  • (i)

    The following implications hold in general:

    $$ ()

  • (ii)

    If B is simple, then the following equivalences hold:

    $$ ()

  • (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].

Lemma 5. Let X be a connected compact Hausdorff space and h = diag (f₁, f₂, …, f_n) a self-adjoint element in M_n(C(X)), where f₁, f₂, …, f_n are continuous maps from X to ℝ. For any positive number ε, there is a unitary u ∈ M_n(C(X)) such that

Proof. If f₁ ≤ f₂ ≤ ⋯≤f_n, then the unitary u is just the identity of M_n and λ_i = f_i. Therefore, to prove the lemma, we, roughly speaking, only need to rearrange the given family {f₁, f₂, …, f_n} to obtain an increasing ordered family. For n = 1, the lemma is obvious. Otherwise, using the idea of bubble sort, we can reduce to the case n = 2.

Let Z = (λ₁ − λ₂) ⁻¹(−ε/2, ε/2). Set E = {x ∈ X : f₁(x) ≤ f₂(x)}∩(X∖Z) and F = {x ∈ X : f₁(x) ≥ f₂(x)}∩(X∖Z).

It is clear that E and F are disjoint closed sets and X = E ∪ F ∪ Z. We have λ₁(x) = min {f₁(x), f₂(x)} and λ₂(x) = max {f₁(x), f₂(x)} for all x ∈ X. If E(F) is empty, then the unitary u can be chosen as $$ (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

For x ∈ X∖(E ∪ F) = Z, we have

Theorem 6. Given an AH algebra $$, where the ϕ_i are diagonal *-homomorphisms from A_i to A_i+1, where $$ and the X_it are connected compact Hausdorff spaces. If A has small eigenvalue variation in the sense of Bratteli and Elliott, then A has the LP property.

Proof. By Corollary 2, it suffices to show that ϕ_i∞(f) ∈ L(A) for every real-valued function f ∈ C(X_it). By the same argument in the proof of Theorem 1, we can assume that each A_t has only one component; that is, $$. Let ε > 0 be arbitrary. Since A has small eigenvalue variation in the sense of Bratteli and Elliott, there is an integer j ≥ i such that EV(ϕ_ij(f)) < ε. Let {μ₁, …, μ_n} be the eigenvalue pattern of ϕ_ij  (n = n_j/n_i). Then,

Put

Therefore,

Lemma 7. Let B be a C^*-algebra, and p and q projections in B. If p and q are Murray-von Neumann equivalent, then pBp is isomorphic to qBq.

In particular, if B = M_n(C(X)) (where X is a connected compact Hausdorff space) and q is a constant projection of rank m in B, then qBq is *-isomorphic to M_m(C(X)).

Proof. By assumption, there exists a partial isometry v such that p = v^*v and q = vv^*. Let us consider the following maps:

In the case B = M_n(C(X)) and q is a constant projection of rank m in B, we have qBq = M_m(C(X)). Therefore, pBp is *-isomorphic to M_m(C(X)).

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

Proof. We can assume that $$, for all i. It is easy to see that p₁ is Murray-von Neumann equivalent to $$, where m₁ is the rank of p₁. For i > 1, define q_i = ϕ_i−1(q_i−1). Then q_i = ϕ_1i(q₁) is constant, since q₁ is constant and ϕ_1i is diagonal. Let us denote by m_i the rank of q_i, then m_i+1∣m_i. By Lemma 7, there are *-isomorphisms Θ_i from $$ to $$ such that

On the other hand, it is straightforward to check that Θ_i+1∘ϕ_i = ψ_i∘Θ_i and hence $$. By Theorem 6, A has the LP property.

Example 9. Let $$ be a Goodearl algebra [6] and ω_t,1 the weighted identity ratio for ϕ_t,1. Suppose that X is not totally disconnected and has finitely many connected components, then the following statements are equivalent.

• (i)

  A has real rank zero.

• (ii)

  lim _t→∞ω_t,1 = 0.

• (iii)

  A has small eigenvalue variation in the sense of Bratteli and Elliott.

• (iv)

  A has the LP property.

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

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 $$ and $$, where A_n is a homogeneous algebra, ϕ_n is an injective diagonal homomorphism from A_n to A_n+1, and i_n is the embedding from M_n to M_n+1 which associates each a ∈ M_n to diag (a, 0) ∈ M_n+1 for each positive integer n. Let us consider the inductive limit $$. For each integer n ≥ 1, denote by i_n,∞ and ϕ_n,∞ the homomorphisms from M_n and A_n to K and A in the inductive limit of K and A, respectively. Then

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

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

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

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

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

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

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

Definition 15. Let A⊃P be an inclusion of unital C^*-algebras with a conditional expectation E from A to P.

• (1)

  A quasi-basis for E is a finite set $$ such that for every a ∈ A,

  $$ ()

• (2)

  When $$ is a quasi-basis for E, we define index E by

  $$ ()
  When there is no quasi-basis, we write Index E = ∞. 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 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 $$, which shows that Index E is a positive element [17, Lemma 2.1.6].

• (3)

  By the above statements, if A is a simple C^*-algebra, then Index E is a positive scalar.

• (4)

  If Index E < ∞, then E is faithful; that is, E(x^*x) = 0 implies x = 0 for x ∈ A.

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

• (1)

  Index E is finite if and only if $$ has the identity (equivalently $$) and there exists a constant c > 0 such that E(x^*x) ≥ cx^*x for x ∈ A; that is, $$ 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 $$ has a form

  $$ ()
  for some x_i and y_i in A.

• (3)

  Let $$ be the unreduced C^*-basic construction defined in Definition 2.2.5 of [17], which has the certain universality (cf.(5) below). If index E is finite, then there exists an isomorphism from $$ to $$ [17, Proposition 2.2.9]. Therefore, we can identify $$ with $$. So we call $$ the C^*-basic construction and denote it by C^*〈A, e_P〉. Moreover, we identify λ(A) with A in $$, and we define it as

  $$ ()

• (4)

  The C^*-basic construction C^*〈A, e_p〉 is isomorphic to qM_n(P)q for some n ∈ ℕ and projection q ∈ M_n(P) [17, Lemma 3.3.4]. If index E is finite, then index E is a central invertible element of A and there is the dual conditional expectation $$ from C^*〈A, e_P〉 to A such that

  $$ ()
  by [17, Proposition 2.3.2]. Moreover, $$ has a finite index and faithfulness. If A is simple unital C^*-algebra, index E ∈ A by Remark 16(4). Hence $$ by [17, Proposition 2.3.4].

• (5)

  Suppose that index E is finite and A acts on a Hilbert space ℋ faithfully and e is a projection on ℋ such that eae = E(a)e for a ∈ A. If a map P∋x ↦ xe ∈ B(ℋ) is injective, then there exists an isomorphism π from the norm closure of a linear span of AeA to C^*〈A, e_P〉 such that π(e) = e_P and π(a) = a for a ∈ A [17, Proposition 2.2.11].

Definition 18. Let α be an action of a finite group G on a unital C^*-algebra A. α is said to have the Rokhlin property if there exists a partition of unity {e_g} _g∈G ⊂ A_∞ consisting of projections satisfying

Let A⊃P be an inclusion of unital C^*-algebras. For a conditional expectation E from A to P, we denote by E^∞ the natural conditional expectation from A^∞ to P^∞ induced by E. If E has a finite index with a quasi-basis $$, then E^∞ also has a finite index with a quasi-basis $$ and Index (E^∞) = Index E.

Definition 19. A conditional expectation E of a unital C^*-algebra A with a finite index is said to have the Rokhlin property if there exists a projection e ∈ A_∞ satisfying

Proposition 20 (see [19].)Let α be an action of a finite group G on a unital C^*-algebra A and E the canonical conditional expectation from A to the fixed point algebra P = A^α defined by

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

Proposition 22. Let P ⊂ A be an inclusion of unital C^*algebras and E conditional expectation from A to P with a finite index. Suppose that A is simple. Consider the basic construction

If E : A → P has the Rokhlin property with a Rokhlin projection e ∈ A_∞, then the double dual conditional expectation $$ has the Rokhlin property.

Proof. Note that from Remark 17(4) and [19, Corollary 3.8], C^*-algebras C^*〈A, e_P〉 and C^*〈B, e_A〉 are simple.

Since e_Pee_P = E^∞(e)e_P = (Index E) ⁻¹e_p, (Index E)ee_pe ≤ e, and

Let $$ be a quasi-basis for $$ and e_A be the Jones projection of $$. Set $$. Then g is a projection and $$. Indeed, since

Consider the following:

To prove that the double dual conditional expectation $$ has the Rokhlin property, we will show that g is the Rokhlin projection of $$. Since $$ for any z ∈ C^*〈A, w_P〉, by Remark 17(5), there exists an isomorphism π : C^*〈C^*〈A, e_P〉, e_A〉→C^*〈C^*〈A, e_P〉, e〉 such that π(e_A) = e and π(z) = z for z ∈ C^*〈A, e_P〉. Then

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

Proof. Let x ∈ P and ε > 0. Since A has the LP property, x can be approximated by a line sum of projection $$ such that $$.

Since β : A^∞ → P^∞ is an injective *-homomorphism by Proposition 21, we have

Theorem 24. Let α be an action of a finite group G on a simple unital C^*-algebra A and E be canonical conditional expectation from A to the fixed point algebra P = A^α defined by

Lemma 25. Under the same conditions in Theorem 24 consider the following two basic constructions:

• (1)

  π(a) = a for all a ∈ A,

• (2)

  π(e_p) = q, where $$,

• (3)

  A⋊_αG = C^*〈A, q〉,

• (4)

  $$ for all b ∈ B,

• (5)

  $$.

• (6)

  $$ and $$.

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

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

Since for any a ∈ A

By the similar steps we will show conditions (4) and (5). Since for any x, y, a, b ∈ A

The condition (6) comes from the direct computation.

Lemma 26. Under the same conditions in Lemma 25  C^*〈A ⋊_α G, e_F〉 is isomorphic to M_|G|(A).

Proof. Note that $$ is a quasi-basis for F. By [17, Lemma 3.3.4], there is an isomorphism from C^*〈A ⋊_α G, e_F〉 to rM_|G|(A)r, where $$. Hence C^*〈A ⋊_α G, e_F〉 is isomorphic to M_|G|(A).

Proof of Theorem 24. Let {e_g} _g∈G be the Rokhlin projection of E. From Proposition 20,  E : A → A^G is of index finite and has a projection e ∈ A^′ ∩ A^∞ such that E^∞(e) = (1/|G|)1. Note that index E = |G| and e = e₁. Consider the basic construction

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

Since C^*〈A ⋊_α G, e_F〉 is isomorphic to M_|G|(A) by Lemma 26 and A has the LP property, C^*〈A ⋊_α G, e_F〉 has the LP property. Hence, C^*〈B, e_A〉 has the LP property, because that C^*〈A ⋊_α G, e_F〉 is isomorphic to C^*〈B, e_A〉 from Lemma 25. From Proposition 22,  $$ has the Rokhlin property, hence we conclude that C^*〈A, e_P〉 has the LP property by Theorem 23. Since C^*〈A, e_P〉 is isomorphic to A ⋊_α G by Lemma 25, we conclude that A ⋊_α G has the LP property.

Remark 27.

• (1)

  When an action α of a finite group G does not have the Rokhlin property, we have an example of simple unital C^*-algebra with the LP property such that the fixed point algebra A^G does not have the LP property by Example 14. Note that the action α does not have the Rokhlin property.

• (2)

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

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

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

Since α is a pointwise outer (i.e., for any g ∈ G∖{0}  α_g is outer) by [23, Lemma 1.5], A ⋊_α G is simple.

Therefore, by [1, Corollary 4], A ⋊_α G has the LP property.

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