Norms on unitizations of banach algebras revisited

Let A be an algebra without unit. If ∥ ∥ is a complete regular norm on A it is known that among the regular extensions of ∥ ∥ to the unitization of A there exists a minimal (operator extension) and maximal (ℓ₁-extension) which are known to be equivalent. We shall show that the best upper bound for the ratio of these two extensions is exactly 3. This improves the results represented by A. K. Gaur and Z. V. Kovářík and later by T. W. Palmer. The second author was partially supported by the grant No. 201/03/0041 of GAČR.