called the homomorphism induced by the inductive limit. One verifies that the diagram

Transcription

1 Inductive limits of C -algebras 51 sequences {a n } suc tat a n, and a n 0. If A = A i for all i I, ten A i = C b (I,A) and i I A i = C 0 (I,A). i I 1.10 Inductive limits of C -algebras Definition Let {G n } be a sequence of groups and φ n : G n G n+1 be omomorpisms. We write φ n,n =id Gn and (if n m) φ n,m = φ m 1 φ n : G n G m. If k n m, ten φ k,m = φ n,m φ k,n. If F is a group and we ave omomorpisms ψ n : G n F suc tat te diagram G n φ n Gn+1 ψn+1 F commutes for eac n, ten ψ n = ψ m φ n,m for all m n. Te product n=1 G n wit pointwise operation is a group. Let F = {{g n } G n : g n+1 = φ n (g n ) for all sufficiently large n}. n=1 Ten F is a subgroup of n=1 G n. If ɛ n is te unit of G n, ten te set N of all {g n } n=1 G n suc tat g n = ɛ n for all sufficiently large n is a normal subgroup of F. Denote te quotient group F/N by G. We call G te inductive limit of te sequence {(G n,φ n )} and write G = lim n (G n,φ n ). Wen tere is no ambiguity, we may also write G = lim n G n. If g G n, define φ n (g) to be te sequence {g k} were g k = ɛ k for k<nand g n+i = φ n,n+i (g) for all i 0. ten φ n (g) F, and te map φ n, : G n G defined by te coset φ n (g)n is a omomorpism. Tis is called te omomorpism induced by te inductive limit. One verifies tat te diagram G n World Scientific Publising Co. Pte. Ltd. ttp:// φ n φn, Gn+1 φn+1, G

2 52 Te asics of C -algebras commutes for eac n and tat G is te union of te increasing sequence {φ n, (G n )}. From te definition, if φ n, (g) =ɛ, te unit of G, ten for some m n, φ n,m (g) =ɛ m. Teorem Let G = lim n (G n,φ n ), were G n are groups. (1) if g G n and f G m and φ n, (g) =φ m, (f), ten tere exists k n, m suc tat φ n,k (g) =φ m,k (f). (2) If G is a group and for eac n tere is a omomorpism ψ n : G n G suc tat te diagram G n φ n Gn+1 ψn+1 G commutes, ten tere is a unique omomorpism ψ : G G suc tat te diagram G n φ n, G ψ G commutes for eac n. Proof. Part (1) follows from te definition immediately. Suppose tat G and ψ n : G n G are as in (2). If g G n and f G m and φ n, (g) =φ m, (f), ten by (1), tere is k n, m suc tat φ n,k (g) = φ m,k (f). So ψ n (g) =ψ k φ n,k (g) =ψ k φ m,k (f) =ψ m (f). Terefore te map ψ : G G defined by ψ(φ n, (g)) = ψ n (g) is well defined. It is ten easily cecked tat ψ is a omomorpism and, by definition, ψ φ n, = ψ n. Uniqueness of ψ is also clear now. Example Let G n = Z and φ n : Z Z be defined by φ n (k) =2k for k Z. Let G = lim n (G n,φ n ). OnecancecktatG = Z[1/2] (see 2.7.4). Definition Let p be a seminorm on a -algebra A satisfying p(ab) p(a)p(b), p(a )=p(a) andp(a a)=p(a) 2. Set N = p 1 (0). It is a self-adjoint ideal of A. Ten A/N is -algebra wit norm a + N = p(a). Let be te anac space completion of A/N wit tis norm. It is easy to ceck tat is a C -algebra. Te map j : A defined by j(a) =a + N is called te canonical map from A to. Note tat j(a) isdensein. World Scientific Publising Co. Pte. Ltd. ttp://

3 Inductive limits of C -algebras 53 Let { } be a sequence of C -algebras and n : +1 be a sequence of omomorpisms. Set A = {{a n } : a n+1 = n (a n ) for all sufficiently large n}. n=1 Ten A is a -subalgebra of n=1. It is important to note tat a k+1 a k for k N for some integer N > 0. So { a k } converges. We define p(a) = lim n a n. It is easy to ceck tat p is a seminorm on A satisfying p(ab) p(a)p(b), p(a )=p(a) andp(a a)=p(a) 2. We define A to be te completion of A /p 1 (0). As above, A is a C -algebra. We call it te inductive limit of te sequence (, n ) and write A = lim n (, n )(wen tere is no ambiguity, we may simply write A = lim n.) Remark As in te group case, if a, define n(a) ={a n } in A suc tat a 1 = a 2 = = a n 1 =0anda n+j = n,n+j (a) (j =0, 1,...,). If j : A A is te canonical map, ten te map n, : A defined by a j( n (a)) is a (C -algebra) omomorpism. It will be called te omomorpism from to A induced by te inductive limit. One cecks routinely tat te diagram n n, An+1 n+1, A commutes for eac n, and te union of te increasing sequence of C - subalgebras { n, ( )} is dense in A. Furtermore, if a. n, (a) = lim m n,n+m(a) (e 10.12) Teorem Let A = lim n (, n ) be an inductive limit of C - algebras. (1) If a,b A m,ε>0 and n, (a) = m, (b), ten tere is k 0 n, m suc tat n,k (a) m,k (b) <ε for all k k 0. World Scientific Publising Co. Pte. Ltd. ttp://

4 54 Te asics of C -algebras (2) If is a C -algebra and tere is a contractive positive linear map ψ n : suc tat te diagram n An+1 ψn+1 commutes for eac n, ten tere is a unique contractive positive linear map ψ : A suc tat te diagram n, A ψ commutes for eac n. Moreover, if ψ n is a omomorpism for eac n, ten ψ is a omomorpism. Proof. Part (1) follows from te equation (e 10.12) immediately. For part (2), let and ψ n be as in (2). Suppose tat a,b A m and n, (a) = m, (b). If ε>0isgiven, ten by (1), tere exists k 0 n, m suc tat n,k (a) m,k (b) <εfor all k k 0. Terefore, ψ n (a) ψ m (b) = ψ k ( n,k (a) m,k (b)) <ε for all ε. Tus ψ n (a) = ψ m (b). Tis sows tat te map ψ : n=1 n, ( ) defined by ψ( n, (a)) = ψ n (a) is well defined. Since we ave ψ n (a) = ψ n+i n,n+i (a) n,n+i (a), ψ( n, (a)) = ψ n (a) lim n,n+m(a) = n, (a). m Tus ψ 1 and it is easy to see tat it is a contractive positive linear map. Since n n, ( )isdenseina, ψ can be uniquely extended to a contractive positive linear map ψ : A suc tat ψ n, = ψ n for eac n. It is also clear tat, if ψ n is omomorpism for eac n, ψ is a omomorpism. Corollary Let A be a C -algebra and let { } be an increasing sequence of C -subalgebras of A wose union is dense in A. Set = lim n (,ı), were ı : A is te embedding. Ten A =. World Scientific Publising Co. Pte. Ltd. ttp://

5 Inductive limits of C -algebras 55 Proof. reader. Te proof is an easy application of (2) in and is left to te Example Let = M 2 n. Define n : +1 by defining n (a) = diag(a, a). A= lim n (, n ) is called te UHF-algebra of 2 - type. It is infinite dimensional and unital. A is a simple C -algebra by te following proposition, wic we leave to te reader as an easy exercise. Proposition ten A is simple. If A = lim n (, n ), were eac is simple, Definition An AF -algebra is an inductive limit of C -algebras of te form M k1 M k2 M kl. In oter words, an AF-algebra is an inductive limit of finite dimensional C -algebras. Te C -algebra A in is a simple AF -algebra. Definition Let A and be two C -algebras and φ i : A (i =1, 2) be two maps. Let ε>0andf A. We say φ 1 and φ 2 are approximately te same witin ε on F if φ 1 (a) φ 2 (a) <ε for all a F. Wen tis appens we write φ 1 ε φ 2 on F. Let u be a unitary. We denote adu : te automorpism a u au. We write if tere is a unitary u suc tat φ 1 ε φ 2 on F, adu φ 1 ε φ 2 on F, and say φ 1 and φ 2 are approximately unitarily equivalent witin ε on F. We say φ 1 and φ 2 are approximately unitarily equivalent if for any ε>0 φ 1 ε φ 2 on any finite subset F A and write φ 1 φ 2. We say φ 1 and φ 2 are unitarily equivalent if tere exists a unitary u suc tat adu φ 1 = φ 2. World Scientific Publising Co. Pte. Ltd. ttp://

6 56 Te asics of C -algebras For two elements a and b in we write a ε b if a b <ε.we write a ε b if tere is a unitary u A suc tat u au ε b and we write a b if tere is a unitary u A suc tat u au = b. Definition Let L : A be a positive linear map. Given ε>0 and a subset F A, L is said to be F-ε-multiplicative if for all a, b F. L(ab) L(a)L(b) <ε Definition Let 1, 2,A 1 and A 2 be C -algebras. Consider te (not necessary commutative) diagram: 1 φ 2 L1 A 1 ψ L2 A2 Let F 1 and ε>0. We say te above diagram approximately commutes on F witin ε if ψ L 1 ε L 2 φ on F. Note tat tis includes special case tat A 1 = A 2 and ψ =id A1. Let A = lim n (, n )and = lim n ( n,φ n ) be two inductive limits of C -algebras. Suppose tat we ave te following (not necessary commutative) diagram: World Scientific Publising Co. Pte. Ltd. ttp:// L1 L2 L3 A 1 φ 1 A2 φ 2 A3 φ 3 A were L i is a contractive positive linear map for eac n. Suppose tat tere are finite subsets F 1 in te unit ball of 1, F 2 in te unit ball of 2,..., wit n (F n ) F n+1 suc tat te closure of n n, (F n ) contains te unit ball of. Suppose also tat tere is a decreasing sequence of positive numbers {r n } wit n=1 r n < suc tat te nt-square of te above diagram approximately commutes on F n witin r n, i.e., φ n L n rn L n+1 n on F n. Ten we say te diagram is one-sided approximately intertwining.

7 Inductive limits of C -algebras 57 If eac square is actually commutative, ten we say te diagram is intertwining. Te following teorem will be used in Capter 4 and Capter 6. Teorem Let A = lim n (, n ) and = lim n ( n,φ n ) be two inductive limits of C -algebras. Suppose tat tere are contractive positive linear maps L n : n suc tat te diagram L1 L2 L3 A 1 φ 1 A2 φ 2 A3 φ 3 A is one-sided approximately intertwining. Ten tere is a contractive positive linear map L : A suc tat te diagram n Ln n, φ n, L A approximately commutes on F n witin k=n r k, and L n, (b) = lim k φ k, L k n,k (b) for all b n. If furtermore, L n is F n -ε n -multiplicative for some ε n > 0 wit n=1 ε n <, ten L : A is a omomorpism. Proof. For eac b j consider te sequence {φ k, L k j,k (b)}. We claim tat te above sequence is Caucy. We assume tat b 0and set b = b/ b. Given ε>0, tere is a n=1 n, (F n ) suc tat j, (b ) a <ε/4( b +1). y , coose a F i (i j) ands i suc tat i, (a )=a and i,s j,i (b ) i,s (a ) <ε/4( b +1). World Scientific Publising Co. Pte. Ltd. ttp://

8 58 Te asics of C -algebras y assumption, k+l φ k,k+l L k s,k ( i,s (a )) L k+l i,k+l (a ) < r t. Terefore k+l φ k,k+l L k s,k ( j,s (b )) L k+l s,k+l ( j,s (b )) <ε/2( b +1)+ r t. Tis implies tat {φ n, L n j,n (b )} is Caucy in. So {φ n, L n j,n (b)} is Caucy. Tis proves te claim. Define ψ j : j A by ψ j (b) = lim k φ k, L k j,k (b). It is clear tat ψ j is positive and contractive. It is also clear tat it is linear. From te definition, we obtain te following commutative diagram (for eac n) t=k t=k n n n+1 ψn+1 A Tus by , tere is a unique contractive positive linear map L : A suc tat n n, L A commutes for eac n. Now if n=1 ε n < (ε n > 0) and L n are F n -ε n -multiplicative, ten ψ j (ab) ψ j (a)ψ j (b) lim φ n, L n j,n (ab) φ n, L n j,n (a)φ n, L n j,n (b) n = lim φ n, (L n ( j,n (ab)) L n ( j,n (a))l n ( j,n (b))) <ε n n if j,n (a), j,n (b) F n. So ψ j (ab) =ψ j (a)ψ j (b) if j,n (a), j,n (b) F n. Tis of course implies tat ψ j is multiplicative. A positive multiplicative linear map is a -omomorpism. Terefore eac ψ j is a omomorpism. y , L is a omomorpism. Definition In , for eac n 1, let G n be a finite subset of te unit ball suc tat φ n (G n ) G n+1 and n φ n, (G n )isdense World Scientific Publising Co. Pte. Ltd. ttp://

9 Inductive limits of C -algebras 59 in te unit ball of A, and let s n > 0 wit n=1 s n <. If in addition, we ave contractive positive linear maps: H n : n+1 suc tat n n n+1 Ln Hn Ln+1 φ n An+1 is approximately commutative on F n and on G n witin r n and witin s n, respectively, i.e., H n L n rn n on F n and L n+1 H n sn φ n on G n, ten we say te diagram L1 H1 L2 H2 L3 A 1 φ 1 A2 φ 2 A3 φ 3 A is two-sided approximately intertwining. If eac triangle above is actually commutative, ten we say te diagram is intertwining. Teorem Let A = lim n (, n ) and = lim n ( n,φ n ) be two inductive limits of C -algebras. Suppose tat tere are contractive positive linear maps L n : n and H n : n+1 suc tat te diagram L1 H1 L2 H2 L3 A 1 φ 1 A2 φ 2 A3 φ 3 A is two-sided approximately intertwining. Ten tere are contractive positive linear maps L : A and H : A suc tat L H =id A and H L =id and suc tat te diagram n n, Ln Hn, L H φ n, A approximately commutes on F n witin r n and approximately commutes on G n witin s n. World Scientific Publising Co. Pte. Ltd. ttp://

10 60 Te asics of C -algebras If, furtermore, L n are F n -ε n -multiplicative, H n is G n -δ n -multiplicative wit n=1 ε n < and n=1 δ n <, ten L and H are isomorpisms. Proof. It follows from tat tere are L : A and H : A suc tat L n, (b) = lim φ k, L k n,k (b) andh φ n, (a) = lim k+1, H k φ n,k (a) k k for all a and b n. Terefore Also k+1, H k φ n,k (a) H φ n, (a) 0ask. φ k+1, L k+1 (H k φ n,k (a)) L k+1, (H k φ n,k (a)) 0 as k. Since L k+1 (H k φ n,k (a)) φ n,k+1 (a) 0 as k (tis can be verified in a similarly manner as in te argument used in te proof of using te fact tat n φ n, (G n ) is dense in te unit ball of A and L k+1 H k sn φ k on G n ), we conclude tat L H(φ n, (a)) φ k+1, (φ n,k+1 (a)) 0 as k. Tis implies tat L H =id A. Similarly, H L =id Exercises Prove tat in a unital algebra, every proper ideal is contained in a proper maximal ideal Let T be a bounded operator on a Hilbert space H. If sp(t )is disconnected, ten T as a non-trivial invariant subspace, i.e., tere is a proper closed subspace H 0 H suc tat T (H 0 ) H Let T be in Sow tat lim n T n 1/n =0. Is zero an eigenvalue of T? Let H be a Hilbert space wit ortonormal basis {e n } n=1. Define a bounded operator S by S(e n )=e n+1. Tis operator S is called te sift operator. Sow tat S as no square roots. World Scientific Publising Co. Pte. Ltd. ttp://