Representation Of The Taf Algebra

Let A be TAF algebra and ρ : A �' AlgN be a nest representation where N = Latρ(A) is a nest. In this paper,we obtain the following results:1. The following are equivalent:(i) kerρ is a completely meet irreducible ideal.(ii) There exist non-zero vectors g, h ∈ H so that 0₊ = [g], I_-^⊥ = [h] and R = g (?) h .2. If ρ(A) contains non-zero compact operators,then (i) N is similar to a completely atomic nest;(ii) ρ(A) is *-dense in AlgN.3. Let ρ(A) be closed, and let e ∈ U_i∈NA_i be an element of A so that ρ(e) is a compact operator.Then ρ(e) can be expressed as the sum of finitely many rank one operators inρ(U_i∈NA_i).4. Let ρ(A) be closed.Then the compact operators in ρ(A) form a closed ideal of AlgN.The above results generalize the corresponding results written by Elias Katsoulis and Justin R. Peters in [20] from strongly maximal TAF algebra to TAF algebra.