Non-separable AF-algebras

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1 Non-separable AF-algebras Takeshi Katsura Department of Mathematics, Hokkaido University, Kita 1, Nishi 8, Kita-Ku, Sapporo, 6-81, JAPAN Summary. We give two pathological phenomena for non-separable AF-algebras which do not occur for separable AF-algebras. One is that non-separable AF-algebras are not determined by their Bratteli diagrams, and the other is that there exists a non-separable AF-algebra which is prime but not primitive. 1 Introduction In this paper, an AF-algebra means a C -algebra which is an inductive limit of finite dimensional C -algebras on any directed set. Equivalently, Definition 1. A C -algebra A is called an AF-algebra if it has a directed family of finite dimensional C -subalgebras whose union is dense in A. When an AF-algebra A is separable, we can find an increasing sequence of finite dimensional C -subalgebras whose union is dense in A. Thus for separable C -algebras, the above definition coincides with the one in many literatures (for example, [E76]). For separable C -algebras, there exists one more equivalent definition of AF-algebras: Proposition 2 (Theorem 2.2 of [B72]). A separable C -algebra A is an AF-algebra if and only if it is a locally finite dimensional C -algebra, which means that for any finite subset F of A and any ε >, we can find a finite dimensional C -subalgebra B of A such that dist(x, B) < ε for all x F. To the best of the author s knowledge, it is still open that the above lemma is valid in general. For each positive integer n Z +, M n denotes the C -algebra of all n n matrices. Any finite dimensional C -algebra A is isomorphic to k i=1 M n i for some k Z + and t (n 1,..., n k ) Z k +. Let B = k j=1 M n be another j finite dimensional C -algebra. A -homomorphism ϕ: A B is determined up to unitary equivalence by the k k matrix N whose (j, i)-entry is the

2 2 Takeshi Katsura multiplicity of the composition of the restriction of ϕ to M ni natural surjection from B to M n j. A λ and the Definition 3. Let Λ be a directed set with an order. An inductive system of finite dimensional C -algebras (A λ, ϕ µ,λ ) over Λ consists of a finite dimensional C -algebra A λ for each λ Λ, and a -homomorphism ϕ µ,λ : A λ A µ for each λ, µ Λ with λ µ such that ϕ ν,µ ϕ µ,λ = ϕ ν,λ for λ µ ν. A Bratteli diagram of (A λ, ϕ µ,λ ) is the system (n λ, N µ,λ ) where n λ = t ((n λ ) 1,..., (n λ ) kλ ) Z k λ + satisfies A λ kλ = i=1 M (n λ ) i and N µ,λ is k µ k λ matrix which indicates the multiplicities of the restrictions of ϕ µ,λ as above. A Bratteli diagram (n λ, N µ,λ ) satisfies N µ,λ n λ n µ for λ µ, and N ν,µ N µ,λ = N ν,λ for λ µ ν. It is not difficult to see that when the directed set Λ is Z +, any system (n λ, N µ,λ ) satisfying these two conditions can be realized as a Bratteli diagram of some inductive system of finite dimensional C -algebras (see 1.8 of [B72]). This does not hold for general directed set: Example 4. Let Λ = a, b, c, d, e} with an order a b, c d, e. Let us define ( ) ( ) ( ) ( ) n a = (24), n b =, n 4 c =, n 6 d =, n 3 e =, 2 and N a,b = ( 3 3 ) ( ) ( ) , N b,d =, N 1 1 b,e =, N 1 1 a,d = ( 6 6 ), N a,c = ( 2 2 ) ( ) ( ) , N c,d =, N 2 c,e =, N 2 1 a,e = ( 6 6 ). These matrices satisfy N µ,λ n λ = n µ for λ, µ Λ with µ λ, and N a,b N b,d = N a,c N c,d = N a,d, N a,b N b,e = N a,c N c,e = N a,e. Thus the system (n λ, N µ,λ ) satisfies the two conditions above. However, one can see that this diagram never be a Bratteli diagram of inductive systems of finite dimensional C -algebras. In 1.8 of [B72], O. Bratteli showed that when the directed set Λ is Z +, a Bratteli diagram of an inductive system of finite dimensional C -algebras determines the inductive limit up to isomorphism. This is no longer true for general directed set Λ as the following easy example shows. Example 5. Let X be an infinite set, and Λ be the directed set consisting of all finite subsets of X with inclusion as an order. We consider the following two inductive systems of finite dimensional C -algebras. For each λ Λ, we define a C -algebra A λ = K(l 2 (λ)) = M λ whose matrix unit is given by e x,y } x,y λ. For λ, µ Λ with λ µ, we define a -homomorphism ϕ µ,λ : A λ A µ by ϕ µ,λ (e x,y ) = e x,y. It is clear to see that

3 Non-separable AF-algebras 3 this defines an inductive system of finite dimensional C -algebras, and the inductive limit is K(l 2 (X)). For each λ Λ with n = λ, we set A λ = M n whose matrix unit is given by e k,l } 1 k,l n. For λ, µ Λ with λ µ, we define a -homomorphism ϕ µ,λ : A λ A µ by ϕ µ,λ (e k,l) = e k,l. It is clear to see that this defines an inductive system of finite dimensional C -algebras, and the inductive limit is K(l 2 (Z + )). The above two inductive systems give isomorphic Bratteli diagrams, but the AF-algebras K(l 2 (X)) and K(l 2 (Z + )) determined by the two inductive systems are isomorphic only when X is countable. In a similar way, we can find two inductive systems of finite dimensional C -algebras whose Bratteli diagrams are isomorphic, but the inductive limits are x X M 2 and k=1 M 2 which are not isomorphic when X is uncountable. By Example 5, we can see that G. A. Elliott s celebrated theorem of classifying (separable) AF-algebras using K -groups (Theorem 6.4 of [E76]) does not follow for non-separable AF-algebras, because K -groups are determined by Bratteli diagrams. Example 5 is not so interesting because the inductive system (A λ, ϕ µ,λ ) has many redundancies and does not come from directed families of finite dimensional C -subalgebras. More interestingly, we can get the following whose proof can be found in the next section: Theorem 6. There exist two non-isomorphic AF-algebras A and B such that they have directed families of finite dimensional C -subalgebras which define isomorphic Bratteli diagrams. The author could not find such an example in which every finite dimensional C -subalgebras are isomorphic to full matrix algebras M n (cf. Problem 8.1 of [D67]). As another pathological fact on non-separable AF-algebras, we prove the next theorem in Section 3. Theorem 7. There exists a non-separable AF-algebra which is prime but not primitive. It had been a long standing problem whether there exists a C -algebra which is prime but not primitive, until N. Weaver found such a C -algebra in [W3]. Note that such a C -algebra cannot be separable. Acknowledgments. The author is grateful to the organizers of the Abel Symposium 24 for giving him opportunities to talk in the conference and to contribute in this volume. He is also grateful to George A. Elliott and Akitaka Kishimoto for useful comments. This work was partially supported by Research Fellowship for Young Scientists of the Japan Society for the Promotion of Science.

4 4 Takeshi Katsura 2 Proof of Theorem 6 In this section, we will prove Theorem 6. Let X be an infinite set, and Z be the set of all subsets z of X with z = 2. For each z Z, we define a C -algebra M z by M z = M 2. Elements of the direct product z Z M z will be considered as norm bounded functions f on Z such that f(z) M z for z Z. For each z Z, we consider M z z Z M z as a direct summand. We denote by z Z M z the direct sum of M z s which is an ideal of z Z M z. Definition 8. For each z Z, we fix a matrix unit e z i,j }2 i,j=1 of M z = M 2. For each x X, we define a projection p x z Z M z by p x (z) = e z 1,1 if x z, if x / z. We denote by A the C -subalgebra of z Z M z generated by z Z M z and p x } x X. Definition 9. For each z = x 1, x 2 } Z, we fix a matrix unit e z x i,x j } 2 i,j=1 of M z = M 2. For each x X, we define a projection q x z Z M z by q x (z) = e z x,x if x z, if x / z. We denote by B the C -subalgebra of z Z M z generated by z Z M z and q x } x X. The following easy lemma illustrates an difference of A and B. Lemma 1. For x, y X with x y, we have p x p y q x q y =. = e x,y} 1,1, and Proof. Straightforward. Definition 11. Let λ be a finite subset of X. We denote by A λ the C -subalgebra of A spanned by z λ M z and p x } x λ, and by B λ the C -subalgebra of B spanned by z λ M z and q x } x λ, Lemma 12. There exist isomorphisms A λ = Bλ = z λ M 2 x λ for each finite set λ X such that two inclusions A λ A µ and B λ B µ have the same multiplicity. C

5 Non-separable AF-algebras 5 Proof. For x λ, let us denote p x A λ by p x = p x y λ\x} ex,y} 1,1. Then we have an orthogonal decomposition A λ = z λ M z + x λ Cp x. This proves A λ = z λ M 2 x λ C. Similarly we have B λ = z λ M 2 C. Now it is routine to check the last statement. x λ Proposition 13. Two C -algebras A and B are AF-algebras, and the directed families A λ } and B λ } of finite dimensional C -subalgebras give isomorphic Bratteli diagrams. Proof. Follows from the facts A λ = A, λ X λ X B λ = B and Lemma 12. Remark 14. From Proposition 13, we can show that K (A) and K (B) are isomorphic as scaled ordered groups. In fact, they are isomorphic to the subgroup G of z Z Z generated by z Z Z and g x} x X, where g x z Z Z is defined by 1 if x z, g x (z) = if x / z. The order of G is the natural one, and its scale is g G g(z) 2 for all z Z}. From this fact and Elliott s theorem (Theorem 6.4 of [E76]), we can show the next lemma, although we give a direct proof here. Proposition 15. When X is countable, A and B are isomorphic. Proof. Let us list X = x 1, x 2,...}. We define a -homomorphism ϕ: A B as follows. For z = x k, x l }, we define ϕ(e z i,j ) = ez x ni,x nj where n 1 = k, n 2 = l when k < l and n 1 = l, n 2 = k when k > l. For x k X, we set k 1 ( ϕ(p xk ) = q xk + e x i,x k } x i,x i e xi,x ) k} x k,x k. Now it is routine to check that ϕ is an isomorphism from A to B. i=1 Proposition 15 is no longer true for uncountable X. To see this, we need the following lemma.

6 6 Takeshi Katsura Lemma 16. There exists a surjection π A : A x X C defined by π A(M z ) = for z Z and π A (p x ) = δ x for x X. Its kernal is z Z M z which coincides with the ideal generated by the all commutators xy yx of A. The same is true for B. Proof. Let π A be the quotient map from A to A/ z Z M z. Then A/ z Z M z is generated by π A (p x )} x X which is an orthogonal family of non-zero projections. This proves the first statement. Since x X C is commutative, the ideal z Z M z contains all commutators. Conversely, the ideal generated by the commutators of A contains z Z M z because M 2 is generated by its commutators. This shows that z Z M z is the ideal generated by the all commutators of A. The proof goes similarly for B. Proposition 17. When X is uncountable, A and B are not isomorphic. Proof. To the contrary, suppose that there exists an isomorphism ϕ: A B. By Lemma 16, z Z M z is the ideal generated by the all commutators in both A and B. Hence ϕ preserves this ideal z Z M z. Thus we get the following commutative diagram with exact rows; z Z M z A x X C ϕ ϕ z Z M z B π A π B x X C. Since the family of projections q x } x X in B is mutually orthogonal, the surjection π B : B x X C has a splitting map σ B : x X C B defined by σ B (δ x ) = q x. Hence by the diagram above, the surjection π A : A x X C also has a splitting map σ A : x X C A. Let us set p x = σ A (δ x ) for x X. Choose a countable infinite subset Y of X. For each y Y, the set F y = x X x y, (py p y)(x, y}) 1/2 } is finite, because p y p y ker π A = z Z M z. Since X is uncountable, we can find x X with x / Y y Y F y. Since F x = x X x x, (p x p x )(x, x }) 1/2 } is finite, we can find y Y \F x. We set z = x, y }. From y / F x, we have (p x p x )(z) < 1/2, and from x / F y, we have (p y p y )(z) < 1/2. However, p x (z) = p y (z) = e z 1,1 and p x (z) is orthogonal to p y (z). This is a contradiction. Thus A and B are not isomorphic. Combining Proposition 13 and Proposition 17, we get Theorem 6.

7 Non-separable AF-algebras 7 3 A prime AF-algebra which is not primitive In this section, we construct an AF-algebra which is prime but not primitive. Although we follow the idea of Weaver in [W3], our construction of the C - algebra and proof of the main theorem is much easier than the ones there. A similar construction can be found in [K4], but the proof there uses general facts of topological graph algebras. Let X be an uncountable set, and Λ be the directed set of all finite subsets of X. For n N, we set Λ n = λ X λ = n }. We get Λ = n= Λ n. Definition 18. For n Z + and λ Λ n, we define For Λ, we define l( ) = }. l(λ) = t: 1,..., n} λ t is a bijection}. Note that l(λ) = n! for λ Λ n and n N. Definition 19. For n N and λ Λ n, we define M λ = Mn! whose matrix unit is given by e (λ) s,t } s,t l(λ). Definition 2. Take λ Λ n and µ Λ m with λ µ =. For t l(λ) and s l(µ), we define ts l(λ µ) by t(i) for i = 1,..., n (ts)(i) = s(i n) for i = n + 1,..., n + m. Note that when µ =, we have t = t. Definition 21. For λ, µ Λ with λ µ, we define a -homomorphism ι µ,λ : M λ M µ by ( (λ)) ι µ,λ e s,t = e (µ) su,tu u l(µ\λ) for s, t l(λ). Note that ι λ,λ is the identity map of M λ, and that ι λ3,λ 2 ι λ2,λ 1 ι λ3,λ 1 for λ 1 λ 2 λ 3. For λ 1, λ 2 Λ n and µ Λ m with λ 1 λ 2 and λ 1 λ 2 µ, the images ι µ,λ1 (M λ1 ) and ι µ,λ2 (M λ2 ) are mutually orthogonal. Definition 22. For λ Λ, we define a -homomorphism ι λ : M λ µ Λ M µ by ι µ,λ (x) if λ µ, ι λ (x)(µ) = otherwise, for x M λ. We set N λ = ι λ (M λ ) µ Λ M µ and f (λ) s,t s, t l(λ). = ι λ (e (λ) s,t ) N λ for

8 8 Takeshi Katsura For λ Λ n, We have N λ = Mn! and f (λ) s,t } s,t l(λ) is a matrix unit of N λ. Lemma 23. For λ, µ Λ with λ µ, s, t l(λ) and s, t l(µ), we have f (λ) s,t f (µ) s,t = f (µ) su,t when s = tu with some u l(µ \ λ), and f (λ) s,t f (µ) s,t = otherwise. Proof. Straightforward. Lemma 24. For λ, µ Λ, we have N λ N µ N µ if λ µ, N λ N µ N λ if λ µ, and N λ N µ = otherwise. Proof. If λ µ, we have N λ N µ N µ by Lemma 23. Similarly we have N λ N µ N λ if λ µ. Otherwise, we can easily see N λ N µ = from the definition. Corollary 25. For each n, the family N λ } λ Λn orthogonal. of C -algebras is mutually Corollary 26. Take λ, λ Λ with λ λ. Let p λ be the unit of N λ. Then N λ a ap λ N λ is an injective -homomorphism. Definition 27. We define A = λ Λ N λ µ Λ M µ. Proposition 28. The set A is an AF-algebra. Proof. For each µ Λ, A µ = λ µ N λ is a finite dimensional C -algebra by Lemma 24. For λ, µ Λ with λ µ, we have A λ A µ. Hence A = µ Λ A µ is an AF-algebra. Lemma 29. Every non-zero ideal I of A contains N λ for some λ Λ. Proof. As in the proof of Proposition 28, we set A µ = λ µ N λ for µ Λ. Since A = µ Λ A µ, we have I = µ Λ (I A µ) for an ideal I of A. Hence if I is nonzero, we have I A µ for some µ Λ. Thus we can find a non-zero element a I in the form a = λ µ a λ for a λ N λ. Since a, we can find λ Λ with λ µ such that a λ and a λ = for all λ λ. Take x X with x / µ. Set λ = λ x }. Let p λ be the unit of N λ. For λ µ, a λ p λ only when λ λ. Hence we have ap λ = a λ p λ. By Corollary 26, a λ p λ is a non-zero element of N λ. Hence we can find a non-zero element in I N λ. Since N λ is simple, we have N λ I. We are done. Lemma 3. If an ideal I of A satisfies N λ I for some λ Λ, then N λ I for all λ λ. Proof. Clear from Lemma 24 and the simplicity of N λ. Proposition 31. The C -algebra is prime but not primitive.

9 Non-separable AF-algebras 9 Proof. Take two non-zero ideals I 1, I 2 of A. By Lemma 29, we can find λ 1, λ 2 Λ such that N λ1 I 1 and N λ2 I 2. Set λ = λ 1 λ 2 Λ. By Lemma 3, we have N λ I 1 I 2. Thus I 1 I 2. This shows that A is prime. To prove that A is not primitive, it suffices to see that for any state ϕ of A we can find a non-zero ideal I such that ϕ(i) = (see [W3]). Take a state ϕ of A. By Corollary 25, the family N λ } λ Λn of C -algebras is mutually orthogonal for each n N. Hence the set Ω n = λ Λ n the restriction of ϕ to N λ is non-zero} is countable for each n N. Since X is uncountable, we can find x X such that x / λ for all λ n N Ω n. Let I = λ x N λ. Then I is an ideal of A by Lemma 24. Since λ x implies ϕ(n λ ) =, we have ϕ(i) =. Therefore A is not primitive. This finishes the proof of Theorem 7. Remark 32. Let (A λ, ϕ µ,λ ) be an inductive system of finite dimensional C - algebras over a directed set Λ, and A be its inductive limit. It is not hard to see that the AF-algebra A is prime if and only if the Bratteli diagram of the inductive system satisfies the analogous condition of (iii) in Corollary 3.9 of [B72]. Hence, the Bratteli diagram of an inductive system of finite dimensional C -algebras determines the primeness of the inductive limit, although it does not determine the inductive limit itself. However the primitivity of the inductive limit is not determined by the Bratteli diagram. In fact, in a similar way to the construction of Example 5, we can find an inductive system of finite dimensional C -algebras whose Bratteli diagram is isomorphic to the one coming from the directed family A λ } constructed in the proof of Proposition 28, but the inductive limit is separable. This AF-algebra is primitive because it is separable and prime (see, for example, Proposition of [P79]). References [B72] Bratteli, O. Inductive limits of finite dimensional C -algebras. Trans. Amer. Math. Soc. 171 (1972), [D67] Dixmier, J. On some C -algebras considered by Glimm. J. Funct. Anal. 1 [E76] (1967) Elliott, G. A. On the classification of inductive limits of sequences of semisimple finite-dimensional algebras. J. Algebra 38 (1976), no. 1, [K4] Katsura, T. A class of C -algebras generalizing both graph algebras and homeomorphism C -algebras III, ideal structures. Preprint 24, math.oa/4819. [P79] Pedersen, G. K. C -algebras and their automorphism groups. London Mathematical Society Monographs, 14. Academic Press, Inc., London-New York, [W3] Weaver, N. A prime C -algebra that is not primitive. J. Funct. Anal. 23 (23), no. 2,