Remarks on Tauberian Theorem for Quasicentral Banach Algebras

Let A be a semisimple quasicentral Banach algebra with bounded approximate identity and suppose the center Z(A) of A is completely regular. We denote by suppB(x) the hull-kernel closure of {P∈Prim B: x〓P} whenever B is an algebra with structure space Prim B and x∈B. suppB(x) is called the support of x. Also denote by B00 the set of all x∈B such that suppB(x) is quasicompact, namely it satisfies the Borel-Lebesgue axiom without necessarily being Hausdorff. Let Φ be Dixmier's representation of Z(M(A)), the central double centralizer algebra of A (see [4] for definition) and Z00(A) the set of all z∈Z(A) such that supp (ΦLz) is quasicompact. Here Lz(x)=zx, x∈A and supp (ΦLz) is the hull-kernel closure of {P∈Prim A: ΦLz(P)≠0}. Actually Z00(A)=A00∩Z(A)⊂(Z(A))00 (cf. [5, Lemma 3.4 and 3.7]). However Z00(A)=(Z(A))00 was left open in [5]. The following result asserts that this is true.