Cyclic cohomology of projective limits of topological algebras

Transcription

1 Cyclic cohomology of projective limits of topological algebras Zinaida Lykova Newcastle University 9 August 2006

2 The talk will cover the following points: We present some relations between Hochschild, cyclic and periodic cyclic (co)homologies of mixed complexes in the category of locally convex spaces and continuous linear operators. Applications of these results and some wellknown results from the homology of topological algebras to cyclic-type continuous homology and cohomology of some classes of Fréchet and Banach algebras. Let (A α, T α,β ) (Λ, ) be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications, and let A = lim α A α. We establish relations between the cyclic-type continuous cohomology and homology of A and A α, α Λ.

3 Mixed Complexes C. Kassel, 1987; J. Cuntz and D. Quillen, We shall use the mixed complex approach to continuous cyclic type theories in the category LCS of locally convex spaces and continuous linear operators. A mixed complex (M, b, B) in the category LCS is a family M = {M n } n 0 of locally convex spaces M n equipped with continuous linear operators b n : M n M n 1 and B n : M n M n+1, which satisfy the identities b 2 = bb + Bb = B 2 = 0. We assume that in degree zero the differential b is identically equal to zero.

4 We arrange the mixed complex (M, b, B) as a double complex b b b b b b M 4 B M3 B M2 B M1 B M0 B 0 b b b b b M 3 B M2 B M1 B M0 B 0 b b b b M 2 B M1 B M0 B 0 b b b M 1 B M0 B 0 b b M 0 B 0 b 0

5 Homology of Mixed Complexes There are three homology theories that can be naturally associated with a mixed complex. The Hochschild homology H b (M) of (M, b, B) is the homology of the chain complex (M, b), that is, H b n (M) = H n(m, b) = Ker {b n : M n M n 1 }/Im {b n+1 : M n+1 M n }.

6 To define the cyclic homology of (M, b, B), let us denote by B c M the total complex of the above double complex, that is, (B c M) n b+b (B c M) n 1... b+b (B c M) 0 0, where the spaces (B c M) 0 = M 0,..., and (B c M) 2k 1 = M 1 M 3... M 2k 1 (B c M) 2k = M 0 M 2... M 2k. are equipped with the product topology, and continuous linear operators b + B are defined by and (b + B)(y 0,..., y 2k ) = (by 2 + By 0,..., by 2k + By 2k 2 ) (b + B)(y 1,..., y 2k+1 ) = (by 1,..., by 2k+1 + By 2k 1 ).

7 The cyclic homology of (M, b, B) is by definition H c (M, b, B) = H (B c M, b + B). The periodic cyclic homology of (M, b, B) is defined in terms of the complex b+b b+b (B p M) ev (B p M) odd (B p M) ev b+b (B p M) odd, where even/odd chains are elements of the product spaces (B p M) ev = and (B p M) odd = M 2n n 0 n 0 M 2n+1, respectively. The spaces (B p M) ev/odd are locally convex spaces with respect to the product topology. The continuous differential

8 b + B is defined as an obvious extension of the above. The periodic cyclic homology of (M, b, B) is H p ν (M, b, B) = H ν(b p M, b + B), where ν Z/2Z. The three different homologies of a mixed complex are related in the following interesting way. Lemma 1 with Let (M, b, B) be a mixed complex Then also Hn b (M) = 0 for all n 0. Hn c (M, b, B) = Hp n (M, b, B) = 0 for all n 0.

9 Proposition 1 Let (M, b, B) be a mixed complex of locally convex spaces. Then, for any even integer N, say N = 2K, and the following assertions, we have (i) N (ii) N (i) N+1 and (i) N (iii) N : (i) N for all n N, Hn b (M) = {0}; (ii) N for all k K, up to isomorphism of linear spaces, H2k c (M, b, B) = Hc N (M, b, B) and H2k+1 c (M, b, B) = Hc N 1 (M, b, B); (iii) N up to isomorphism of linear spaces, H p 0 (M, b, B) = Hc N (M, b, B) and H p 1 (M, b, B) = Hc N 1 (M, b, B). Remark 1 If (M, b, B) is a mixed complex of Fréchet spaces, the above isomophisms are topological.

10 There are also three types of cyclic cohomology theory associated with the mixed complex obtained when one replaces a chain complex of locally convex spaces by its dual complex of strong dual spaces. For example, the cyclic cohomology associated with the mixed complex (M, b, B) is defined to be the cohomology of the dual complex ((B c M), b + B ) of strong dual spaces and dual operators; it is denoted by H c (M, b, B ).

11 Proposition 2 Let (M, b, B) be a mixed complex of locally convex spaces and let (M, b, B ) be the dual complex of strong dual spaces. Then, for any even integer N, say N = 2K, and the following assertions, we have (i) N (ii) N (i) N+1 and (i) N (iii) N : (i) N for all n N, H n b (M, b ) = {0}; (ii) N for all k K, up to isomorphism of linear spaces, Hc 2k (M, b, B ) = Hc N (M, b, B ) and Hc 2k+1 (M, b, B ) = Hc N 1 (M, b, B ); (iii) N up to isomorphism of linear spaces, H 0 p (M, b, B ) = H N c (M, b, B ) and H 1 p (M, b, B ) = H N 1 c (M, b, B ).

12 Hochschild homology for nonunital locally convex algebras By a locally convex algebra we shall mean an algebra A which is a locally convex topological vector space in such a way that the ring multiplication in A is jointly continuous. A is not necessarily unital. The continuous bar and naive Hochschild homology of A are defined respectively as and H bar (A) = H (C(A), b ) = Ker b n /Im b n+1 H naive (A) = H (C(A), b) = Ker b n /Im b n+1, where C n (A) = Aˆ (n+1), ˆ is the completed projective tensor product, and the differentials b, b are given by b n : C n(a) C n 1 (A),

13 b n (a 0... a n ) = and n 1 i=0 ( 1) i (a 0... a i a i+1... a n ) b n : C n (A) C n 1 (A), b(a 0... a n ) = b (a 0... a n )+ ( 1) n (a n a 0... a n 1 ). Note that H naive n (A) = H n (A, A), n 0, the continuous homology of A with coefficients in A, and H bar n (A) = H n+1 (A, C), n 0, where C is the trivial A-bimodule.

14 Cyclic type homology for nonunital topological algebras A + denotes the unitization of a complete locally convex algebra A. Consider the mixed complex in LCS, where ( ΩA +, b, B) Ω n A + = Aˆ (n+1) Aˆ n, ˆ is the completed projective tensor product, and with b = ( b 1 λ 0 b ) ; B = ( 0 0 N 0 λ(a 1... a n ) = ( 1) n 1 (a n a 1... a n 1 ) and N = id + λ λ n 1. ) (See Loday s book.)

15 The mixed complex ( ΩA +, b, B) is the following b b b b B Aˆ 3 Aˆ 2 B Aˆ 2 A B A B b b b b B Aˆ 2 A B A B 0 b b b B B 0 Aˆ 4 Aˆ 3 Aˆ 3 Aˆ 2 Aˆ 2 A b A b 0 A b B 0 The continuous Hochschild homology of A, the continuous cyclic homology of A and the continuous periodic cyclic homology of A are defined by HH (A) = H b ( ΩA +, b, B), and HC (A) = H c ( ΩA +, b, B) HP (A) = H p ( ΩA +, b, B).

16 There is a cyclic type cohomology theory of locally convex algebras obtained when one replaces chain complexes of locally convex spaces by their dual complexes of strong dual spaces. For example, continuous bar cohomology Hbar n (A) of A is the cohomology of the dual complex of (C(A), b ).

17 We note also the following short exact sequence of complexes of locally convex spaces 0 (C(A), b) ( ΩA +, b) (C(A)[1], b [1]) 0, which leads to a long exact homology sequence connecting the three homologies H bar n (A) Hn naive (A) HH n (A) This shows that H bar n 1 (A). H naive n (A) = HH n (A) for all n 0 if and only if Hn bar (A) = 0 for all n 0. Example 1 Banach algebras A such that Hn bar (A) 0 for some n 0. (i) The maximal ideal of the disc algebra. A 0 (D) = {w : w(0) = 0}

18 (ii) l 2 with coordinatewise multiplication. (iii) the algebra HS(H) of Hilbert-Schmidt operators on a Hilbert space H. Example 2 Fréchet algebras A such that Hn bar (A) = 0 for all n 0. (i) Unital Fréchet algebras. For example, the algebra O(C) of holomorphic functions on C; the algebra C (M) of smooth functions. (ii) By extending results of B. E. Johnson one can prove that a Fréchet algebra A with a left or right bounded approximate identity has this property. For example, C -algebras; the Banach algebra A = K(E) of compact operators on a Banach space E with the bounded compact approximation property (P. Dixon).

19 Let ω n (x) = (1 + x ) 1 1 n, ω 0 (x) = 1 + x and let A n = { f L 1 (R) : p n (f) = n = 0, 1,.... Then A n under the norm p n. Let A = + f(x) ω n(x)dx < is a Banach algebra A n n=1 topologised by the collection of norms p n, n = 0, 1,.... Then A is a Fréchet algebra with bounded approximate identity (I. G. Craw) and H bar n (A) = 0 for all n 0. (iii) If A is flat as a right Fréchet A-module and A 2 = A, then H bar n (A) = 0 for all n 0. Here A 2 is the closed linear span of {ab : a, b A}. For example, l 1, N(H). },

20 Proposition 3 Let A be a complete locally convex algebra. Then, for any even integer N, say N = 2K, and the following assertions, we have (i) N (ii) N (iii) N (ii) N+1 and (ii) N (iv) N : (i) N Hn naive (A) = {0} for all n N and Hn bar (A) = {0} for all n N 1; (ii) N HH n (A) = {0} for all n N; (iii) N and for all k K, HC 2k+2 (A) = HC N (A) HC 2k+1 (A) = HC N 1 (A); (iv) N HP 0 (A) = HC N (A) and HP 1 (A) = HC N 1 (A). For Fréchet algebras, the above isomorphisms are topological.

21 Applications to Fréchet and operator algebras Recall the definition of H n (A, X). Let A be a Fréchet algebra and let X be a Fréchet A-bimodule. We denote by C n (A, X), n = 0, 1,..., the Fréchet space X ˆ Aˆ n. We also set C 0 (A, X) = X. From the chains we form the standard homology complex d 0 0 C 0 (A, X)... C n (A, X) C n+1 (A, X)..., (C (A, X)) where the differential d n is given by the formula d n (x a 1 a 2... a n+1 ) = (x a 1 a 2... a n+1 )+ ni=1 ( 1) i (x a 1... a i a i+1... a n+1 ) + ( 1) n+1 (a n+1 x a 1... a n ). d n The n-th homology of C (A, X), denoted by H n (A, X), is called the n-th continuous homology group of the Fréchet algebra A with coefficients in X.

22 We shall need later the following technical result for Fréchet spaces, which extends known results of B.E. Johnson for the Banach case. Lemma 2 (i) A short sequence 0 Y i Z j W 0 in F r is exact if and only if the dual short sequence of the strong dual Fréchet spaces is exact. (ii) A complex (X, d) in Fr is exact if and only if its dual complex (X, d ) is exact. Thus for all n if and only if for all n. H n (X, d) = 0 H n (X, d ) = 0

23 Therefore, for a Fréchet algebra A with a left or right bounded approximate identity, one can prove that Hbar n (A) = {0} for all n 0, and so, by Lemma 2, Hn bar (A) = {0} for all n 0. Proposition 4 Let (M, b) be a complex of Fréchet spaces and let N N. Then the following statements are equivalent: (i) H n (M, b) = {0} for all n N and H N 1 (M, b) is Hausdorff; (ii) H n (M, b ) = {0} for all n N. The proofs depend on the open mapping principle and the Hahn-Banach theorem.

24 A Fréchet algebra A is amenable if, for all Fréchet A-bimodules X, H n (C (A, X)) = {0} for all n 1 and H 0 (C (A, X)) is Hausdorff. A Fréchet algebra A is called N-amenable if H n (C (A, X) ) = {0} for all n N for all Fréchet A-bimodules X. The weak bidimension of a Fréchet algebra A is db w A = inf{n : H n+1 (C (A, X) ) = {0} for all Fréchet A bimodules X}. Recall that the algebra A is said to be biflat if it is flat in the category of Fréchet A- bimodules. It is known that any amenable Fréchet algebra is biflat.

25 Corollary 1 Let A be a Fréchet algebra. Suppose that db w A = m and m = 2L is an even integer. Then, (i) for all l L, HC 2l+2 (A) = HC m (A) and HC 2l+3 (A) = HC m+1 (A); (ii) HP 0 (A) = HC m (A), HP 1 (A) = HC m+1 (A); (iii) for all l L, HC 2l+2 (A) = HC m (A) and HC 2l+3 (A) = HC m+1 (A); (iv) HP 0 (A) = HC m (A), HP 1 (A) = HC m+1 (A). (v) suppose also that A is a Banach algebra; then HE 0 (A) = HP 0 (A) = HC m (A) and HE 1 (A) = HP 1 (A) = HC m+1 (A). If m is odd, one can get similar formulae. Here HE k (A), k = 1, 2, is the entire cyclic cohomology of A, - see Connes book. Statement (v) follows from the result of M. Khalkhali that, for a Banach algebra A of a finite weak bidimension db w A, we have HE k (A) = HP k (A) for k = 0, 1.

26 Example 3 Fréchet algebras of finite homological bidimension. (i) Let O(U) be the Fréchet algebra of holomorphic functions on a polydomain U = U 1 U 2... U m C m. Then db O(U) = m (J.L. Taylor). (ii) Let M be any infinitely smooth manifold of topological dimension m, and let C (M) be the Fréchet algebra of all infinitely smooth functions on M. Then db C (M) = m (O.S.Ogneva). For relations between the continuous cyclic cohomology of C (M) and de Rham homology of M, see Connes book. (iii) Let S(R m ) be the Fréchet algebra of rapidly decreasing infinitely smooth functions on R m. Then db S(R m ) = m (O.S. Ogneva and A.Ya. Helemskii).

27 Hence, for these Fréchet algebras: O(U), C (M) and S(R m ), the conditions of Corollary 1 are satisfied. For example, by Corollary 1 and a result of A. Wassermann, for an even m, HP 0 (S(R m )) = HC m (S(R m )) = the one-dimensional linear space generated by the fundamental m-trace φ(f 0, f 1,..., f m ) = R m f 0 df 1... df m, and HP 1 (S(R m )) = HC m+1 (S(R m )) = {0}.

28 Corollary 2 Let A be a biflat Banach algebra. Then, up to topological isomorphism, (i) for all l 0, HC 2l (A) = A/[A, A] and HC 2l+1 (A) = {0}; (ii) HP 0 (A) = A/[A, A] and HP 1 (A) = {0}; (iii) for all l 0, HC 2l (A) = A tr HC 2l+1 (A) = {0}; and (iv) HE 0 (A) = HP 0 (A) = A tr HE 1 (A) = HP 1 (A) = {0}. and Here A tr = {f A : f(ab) = f(ba) for all a, b A} and [A, A] = the closure in A of the linear span of elements of the form {ab ba : a, b A}. Statements (i) and (iii) are shown by A.Ya. Helemskii.

29 Example 4 Biflat Banach algebras. (i) Amenable Banach algebras. Each nuclear C -algebra is amenable (U. Haagerup). Some examples of amenable C -algebras are: (a) GCR C -algebras; in particular, commutative C -algebras and the C - algebra of compact operators K(H) on a Hilbert space H; (b) Uniformly hyperfinite algebras (UHF-algebras). Let G be a locally compact group with a leftinvariant Haar measure ds. The group algebra L 1 (G) of Haar integrable functions on an amenable locally compact group G with convolution product is amenable too (B.E. Johnson). Therefore, for all l 0, HE 0 (L 1 (G)) = HP 0 (L 1 (G)) = HC 2l (L 1 (G)) = L 1 (G) tr = {f L 1 (G) : f(ab) = f(ba) for all a, b L 1 (G)} and

30 HE 1 (L 1 (G)) = HP 1 (L 1 (G)) = HC 2l+1 (L 1 (G)) = {0}. (ii) Biprojective Banach algebras. The definition and examples of biprojective Banach algebras are given in a book of A.Ya. Helemskii. For the Banach algebra l 1 of summable complex sequences (ξ n ) with coordinatewise operations, by Corollary 2, HP 0 (l 1 ) = HC 2k (l 1 ) = l 1, for all k 0, HP 1 (l 1 ) = HC 2k+1 (l 1 ) = {0}, HE 0 (l 1 ) = HP 0 (l 1 ) = HC 2k (l 1 ) = l, where l is the Banach space of bounded sequences, and HE 1 (l 1 ) = HP 1 (l 1 ) = HC 2k+1 (l 1 ) = {0}, for all k 0.

31 Corollary 3 Let A be a C -algebra without non-zero bounded traces. Then HH n (A) = HH n (A) = {0} for all n 0, HC n (A) = HC n (A) = {0} for all n 0 and HP k (A) = HP k (A) = {0} for k = 0, 1. The corollary is based on a result by E. Christensen and A.M. Sinclair. Example 5 C -algebras without non-zero bounded traces. (i) The C -algebra K(H) of compact operators on an infinite-dimensional Hilbert space H (J.H. Anderson). We can also show that C(Ω, K(H)) tr = {0}, where Ω is a compact space. (ii) Properly infinite von Neumann algebras U. This class includes the C -algebra B(H) of all bounded operators on an infinite-dimensional Hilbert space H. Therefore, for the above algebras, the continuous homologies and cohomologies HH, HC, HP are trivial.

32 Projective limits of topological algebras A projective system of topological algebras is, by definition, a projective system (A α, T α,β ) (Λ, ) such that A α, α Λ, are topological algebras and T α,β, with α β in Λ, continuous morphisms of algebras. The projective limit A = lim α A α = {(a α ) α Λ A α : T α,β a β = a α for all α β} is a closed subalgebra of the cartesian product topological algebra α Λ A α. In particular, A is complete if this is the case for each A α, α Λ; A is Hausdorff if this is the case for each A α, α Λ. The projective limit A = lim α A α of locally convex algebras with jointly continuous multiplications is a topological algebra of the same type (see A.Mallios book).

33 Let (E α, T α,β ) (Λ, ) be a projective system of topological vector spaces. Define T β as the restrictions of the projections P r β : α Λ E α E β, β Λ, on lim α E α. Note that T α = T α,β T β if α β. Recall that a projective system (E α, T α,β ) (Λ, ) of topological vector spaces, as well as the corresponding projective limit, is said to be reduced if the maps T β : lim α E α E β have dense range for every β Λ.

34 Fréchet algebras as projective limits of Banach algebras Let A be a Fréchet locally m-convex algebra. We may assume that the sequence of submultiplicative semi-norms Γ = (p i ) i N on A is increasing; that is, for every a A, p 1 (a) p 2 (a)... p n (a) p n+1 (a).... By the Arens-Michael decomposition, up to topological isomorphism of algebras, A = lim i A i, the projective limit of a sequence of Banach algebras A i. Here A i is the completion of the normed algebra (A/p 1 i (0), p i ). We shall denote the closed two-sided ideal p 1 i (0) = Ker p i = {a A p i (a) = {0}} of A by N i. For any n, the quotient map ρ n : A A/N n

35 is a continuous surjective algebra morphism. Consider also the mapping ρ n : A A n obtained by composing ρ n with the natural embedding i n : A/N n A n. Recall that, for any n, m, with n m, the mapping f nm : A/N m A/N n : f nm (a + N m ) a + N n is a continuous surjective algebra morphism and one has the relation ρ n = f nm ρ m for any n, m, with n m. Furthermore, by considering the continuous extensions f nm of the maps f nm i n, one can see that the family {(A n, f nm ) N } is a reduced countable projective system of Banach algebras.

36 Theorem 1 Let (A α, T α,β ) (Λ, ) be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications, and let A be the projective limit algebra A = lim α A α. Then, up to isomorphism of linear spaces, for all n 0, H n naive (A) = lim α H n naive (A α), H n bar (A) = lim α H n bar (A α), HC n (A) = lim α HC n (A α ), and for k = 0, 1, HH n (A) = lim α HH n (A α ) HP k (A) = lim α HP k (A α ).

37 We deduce the theorem from a sequence of lemmas. Lemma 3 Let (A α, T α,β ) (Λ, ) be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications and let A = lim A α. Then α (i) for all n 1, (Aˆ n α, T ˆ n α,β ) (Λ, ) is a reduced projective system of complete Hausdorff locally convex algebras; (ii) for all n 0, up to topological isomorphism, C n (A) def = Aˆ (n+1) = lim α Aˆ (n+1) α, where the isomorphism is the linear continuous extension of (a (1) α 1 ) α1 (a (2) α 2 ) α2... (a (n+1) ) αn+1 (a (1) α a (2) α α n+1... a (n+1) α ) α ;

38 (iii) for all n 1, the dual system ((Aˆ n α ), (T ˆ n α,β ) ) (Λ, ) is a reduced inductive system and, up to isomorphism of linear spaces, C n (A) def = (Aˆ (n+1) ) = lim α (Aˆ (n+1) α ). Lemma 4 Let (A α, T α,β ) (Λ, ) be a reduced projective system of complete Hausdorff locally convex spaces. Then, up to topological isomorphism, and N l=1 l 1 lim α Aˆ (k l ) α = lim α N l=1 lim α Aˆ (k l ) α = lim α ( l 1 Aˆ (k l ) α, (T ˆ (k ) l ) α,β ) l l 1 (Λ, ) Aˆ (k l ) α, Aˆ (k l ) α, is a reduced projective system of complete Hausdorff locally convex spaces.

39 Conclusion of the proof of Theorem 1. Let us prove the result for HC. By definition, HC n (A) = H n ((B c ΩA + ), b + B ), where (B c ΩA + ) 0 = A,..., (B c ΩA + ) 2k 1 = (Aˆ 2 A) (Aˆ 4 Aˆ 3 )... (Aˆ (2k) Aˆ (2k 1) ) and (B c ΩA + ) 2k = A (Aˆ 3 Aˆ 2 )... (Aˆ (2k+1) Aˆ (2k) ). By Lemma 4, one can see that lim α ( and (Aˆ 2 α (B c ΩA + ) 2k 1 = A α)... (Aˆ (2k) α ) Aˆ (2k 1) α ) (B c ΩA + ) 2k = lim α ( A α... (Aˆ (2k+1) α ) Aˆ (2k) α ).

40 In view of Lemma 3, the cochain complex ((B c ΩA + ), b + B ) is an inductive limit of the reduced inductive system of cochain complexes ((B c Ω(A α ) + ), b + B ). Using the fact that cohomology commutes with inductive limits we have HC n (A) = lim α HC n (A α ). The same method works for the cohomology groups Hnaive, H bar, HH and HP.

41 The topological derived functors lim j (n) One can find the definition of the topological derived functors lim (n), n 0, of lim in a j j paper of V.P. Palamodov or in a book by J. Eschmeier and M. Putinar. Recall that one possible calculation of the derived functor lim (1) is as follows. Firstly, ev- j ery projective system of Fréchet spaces (E j, T j,k ) (N, ) possesses a free resolution of the form 0 (E j ) j N u (Lj ) j N v (Lj 1 ) j N 0, (1) where L 0 = {0} and, for all j 1, L j = j k=1 E k is the free projective system with generators E 1,..., E k,..., and the morphisms u = (u j ) and v = (v j ) of projective systems are defined by the formula u j : E j L j : x (T 1,j x,..., T j,j x)

42 and v j : L j L j 1 : (x 1,..., x j ) (x 1 T 1,2 x 2,..., x j 1 T j 1,j x j ) (x j E j ). Since the projective limit is a left-exact functor, the short exact sequence (1) induces a sequence of Fréchet spaces and continuous operators Proj(u) 0 lim E j j j N Proj(v) E j j N which is left exact. Then one can deduce that lim j (1) E j = CokerProj(v) and E j lim j (n) E j = {0} for n 2, up to an isomorphism of linear spaces. Therefore, on ignoring the topology in lim j (1) E j, we obtain the algebraic lim j (1) E j ; see, for example, Weibel s book for a similar construction in the algebraic case.

43 Recall that a projective system (E j, T j,k ) (N, ) with the property that all structural maps T j,k : E k E j, j k, have dense range is called a topological Mittag-Leffler system. Note that every reduced projective system (E j, T j,k ) (N, ) is a topological Mittag-Leffler system. Therefore, for a reduced projective system (E j, T j,k ) (N, ), one has lim (1) E j j = {0} (see Eschmeier and Putinar s book).

44 Theorem 2 Let (A j, T j,k ) (J, ) be a countable reduced projective system of Fréchet locally convex algebras, and let A be the projective limit of (A j, T j,k ) (J, ), A = lim A i i. Then, for all n 0, the following short sequences of linear spaces and linear operators 0 lim i (1) H naive n+1 (A i) H naive n (A) lim i H naive n (A i ) 0, 0 lim i (1) H bar n+1 (A i) H bar n (A) lim i H bar n (A i) 0, 0 lim i (1) HH n+1 (A i ) HH n (A) lim i HH n (A i ) 0, 0 lim i (1) HC n+1 (A i ) HC n (A) lim i HC n (A i ) 0, 0 lim (1) HP 1 (A i ) HP 0 (A) lim HP 0 (A i ) 0, i i and 0 lim (1) HP 0 (A i ) HP 1 (A) lim HP 1 (A i ) 0 i i are exact.

45 Proof. By Lemma 3 and Lemma 4, the chain complex (B p ΩA +, b + B) is the projective limit of a reduced projective system of chain complexes (B p Ω(A i ) +, b + B). The system is reduced, and therefore is a topological Mittag-Leffler system. By J. Eschmeier and M. Putinar and V. P. Palamodov, for the countable topological Mittag-Leffler system of chain complexes (B p Ω(A i ) +, b+ B), we have lim i (1) (B p Ω(A i ) + ) ev = {0} and lim i (1) (B p Ω(A i ) + ) odd = {0}. It is well known that in the algebraic case, for any tower of chain complexes {C i } and for C = lim C i i such that lim (1) C i i = {0}, there is an exact sequence for each n 0 lim i (1) H n+1 (C i ) H n (C) lim i H n (C i ) 0.

46 Therefore, for the tower of chain complexes (B p ΩA +, b + B) = lim i (B p Ω(A i ) +, b + B), the short sequence of linear spaces and linear operators 0 lim i (1) HP n+1 (A i ) HP n (A) lim i HP n (A i ) 0 is exact for each n. The same method works for the homology groups H naive, H bar, HH and HC.

47 Theorem 3 Let A be a Fréchet locally m- convex algebra and A = lim A i i. Suppose that each Banach algebra A i is biflat. Then, (i) for all l 0, HC 2l (A) = A/[A, A] and HC 2l+1 (A) = {0}; (ii) HP 0 (A) = A/[A, A] and HP 1 (A) = {0}; (iii) for all l 0, HC 2l (A) = A tr and HC 2l+1 (A) = {0}. (iv) HP 0 (A) = A tr and HP 1 (A) = {0}.

48 Example 6 Let C(R) be the Fréchet locally m-convex algebra of continuous complex-valued functions on R. Here the Arens-Michael decomposition is C(R) = lim C[ k, k]. Each k C -algebra C[ k, k] is amenable and therefore biflat. By Theorem 3, since C(R) is commutative, for all l 0, HP 0 (C(R)) = HC 2l (C(R)) = C(R), HP 1 (C(R)) = HC 2l+1 (C(R)) = {0}, HP 0 (C(R)) = HC 2l (C(R)) = C(R) = lim M([ k, k]), where M([ k, k]) is the Ba- k nach space of C-valued measures on [ k, k], and HP 1 (C(R)) = HC 2l+1 (C(R)) = {0}.

49 Corollary 4 Let A be a Fréchet locally C - algebra and A = lim i A i. Suppose that each C -algebra A i has no non-zero bounded trace. Then, Hnaive n (A) = Hnaive n (A) = {0}, for all n 0, HC n (A) = HC n (A) = {0}, for all n 0, and, HP k (A) = HP k (A) = {0}, k = 0, 1. Example 7 Let H = lim i H i be a strict inductive limit of Hilbert spaces, that is, H = m=1 H m where (H m, <, > m ) m=1 an increasing sequence of Hilbert spaces H 1 H 2... H m H m+1... such that <, > i =<, > j on H i for all i < j. We endow H with the LF -topology. Suppose that H 1 and H m+1 /H m, m = 1, 2,..., are infinite-dimensional spaces. Consider the Fréchet locally C -algebra L(H) of continuous linear operators T on H that leave each H i invariant and satisfy T j P ij = P ij T j for all

50 i < j where T j = T H j : T j (η) = T (η) for η H j and P ij is the projection from H j onto H i. By the Arens-Michael decomposition, up to topological isomorphism of algebras, L(H) = lim m L(H m ) the projective limit of a sequence of C -algebras L(H m ) where, for each m, L(H m ) is isomorphic to the C -direct sum B(H 1 ) B(H 2 H 1 )... B(H m H m 1 ). Here B(K) is the C -algebra of all bounded linear operators on the Hilbert space K and has no bounded trace when K is infinite-dimensional. Therefore, by Corollary 4, for all n 0, HH n (L(H)) = HH n (L(H)) = {0}, HC n (L(H)) = HC n (L(H)) = {0} and, HP k (L(H)) = HP k (L(H)) = {0}, k = 0, 1, Recall that locally C -algebras of the type L(H) are very important; they play a similar role in the theory of locally C -algebras as that of the C -algebra of all bounded linear operators on a Hilbert space in the theory of C -algebras (see A. Inoue).