Cyclic-type cohomology of strict inductive limits of Fréchet algebras

We present methods for the computation of the Hochschild and cyclic-type continuous homology and cohomology of some locally convex strict inductive limits A=limm→Am of Fréchet algebras AmAm. In the pure algebraic case it is known that, for the cyclic homology of A  , HCn(A)=limm→HCn(Am) for all n⩾0n⩾0 [Cyclic Homology, Springer, Berlin, 1992, E.2.1.1]. We show that, for a locally convex strict inductive system of Fréchet algebras (Am)m=1∞ such that0→Am→Am+1→Am+1/Am→00→Am→Am+1→Am+1/Am→0is topologically pure for each m and for continuous Hochschild and cyclic homology, similar formulas hold. For such strict inductive systems of Fréchet algebras we also establish relations between the continuous cohomology of A   and AmAm, m∈Nm∈N. For example, for the continuous cyclic cohomology HCn(A)HCn(A) and HCn(Am)HCn(Am), m∈Nm∈N, we show the exactness of the following short sequence, for all n⩾0n⩾0,0→limm←(1)HCn-1(Am)→HCn(A)→limi←HCn(Am)→0,where limm←(1) is the first derived functor of the projective limit. We give explicit descriptions of continuous periodic and cyclic homology and cohomology of a LF  -algebra A=limm→Am which is a locally convex strict inductive limit of amenable Banach algebras AmAm, where for each m  , AmAm is a closed ideal of Am+1Am+1.