Tensor products of Banach algebras. II

Abstract

In [l], [2] there are descriptions of an alleged bijection 31Z3 <->9TCiX3TC2, where 9K%-is the set of regular maximal ideals of the Banach algebra ^4;, * = 1, 2, 3, and where A3 = A1 ®7 ^ is the greatest cross-norm tensor product of Ai and Ai.The given constructions for the bijection are valid if both Ai and A2 are commutative.Otherwise, the constructions are in error (e.g., [l, p. 301, line 12]; [2, p. 753, line 4 and p. 755, Case 2]).The author knows of no counterexamples to the conclusions inThe purpose of this note is to offer (correct) proofs of modified results in more general contexts and to apologize for the regrettable oversights.New research of K. Laursen, whose work (in prepublished form) the author had the privilege to examine, is based on the existence of the bijection mentioned above.Exposure to the new research led to the author's discovery of his earlier errors.