Lemma 4. Let I_φ= (z_j)∈ℓ^∞(M):z_j_φ^#→0 , and let C_φ= t∈ℓ^∞(M): t I_φ⊆ I_φ,\ I_φt⊆ I_φ . Then C_φ is a unital C^* -algebra and I_φ is a closed two-sided ideal in it. Constant sequences from M and all sequences in ℓ^∞(A) belong to C_φ .

The set I_φ is self-adjoint and norm closed. If s,t∈ I_φ , then s_jt_j_φ≤ s_jt_j_φ, (s_jt_j)^*_φ≤ t_js_j^*_φ, so st∈ I_φ . The two multiplier conditions now show directly that C_φ is a unital, norm-closed * -algebra, contains I_φ , and makes it an ideal. Multiplication by a fixed bounded operator is continuous on bounded sets for strong* convergence, so constant sequences are multipliers. If a∈A , then left multiplication has state-norm bound a , and the centralizer property gives ta_φ^2 =φ(a^*t^*ta) =φ(t^*t aa^*) ≤a^2φ(t^*t). The last inequality follows from positivity, since φ(t^*t b)=φ(b^ 1/2 t^*t b^ 1/2 )≥0 for b≥0 in A . The same estimates apply to adjoints, uniformly on bounded A -sequences. Thus these sequences are multipliers.