Remarks on quasicentral approximate identities (Q799905)

Let A be a Banach algebra with bounded approximate identity \(\{e_{\lambda}\}\) and B a Banach algebra containing A as a subalgebra. Set \(\sup p(A)=\sigma (B^{**},B^*)-\lim_{\lambda}e_{\lambda}.\) Then it is shown that (I) \(\sup p(A)\) commutes with each element of B if and only if A has a bonded approximate identity \(\{u_{\alpha}\}\) which is contained in the convex hull of \(\{e_{\lambda}\}\) such that \(\lim_{\alpha}\| u_{\alpha}b-bu_{\alpha}\| =0\) for all \(b\in B\) and (II) If A is a two-sided ideal of B and if \(A^*=A^*A+AA^*\) then \(\sup p(A)\) commutes with each element of B. This results give a new interpretation of the theorem of \textit{R. S. Doran} and \textit{J. Wichman} concerning quasicentral approximate identities, [Approximate identities and factorization in Banach modules, Lecture Notes Mat. 768 (1979; Zbl 0418.46039)].