Abstract.Equivalence of various norms on the unitization of a nonunital Banach algebra is established, with bounds (1 and 6exp(l)) uniform over the class of such algebras. A tighter bound, 3, is obtained in C*-algebras for elements with Hermitian nonunital parts.The algebra norm || • || on a nonunital Banach algebra A can be extended to an algebra norm on the unitization A+ in many ways. Proposition 4.3 in [3] states that among these extensions, the /i-norm ||Ae + a||i = |A|-(-||a|| is maximal and the operator norm \\Xe + a\\op = sup{||Ax + ax\\ : \\x\\ < 1} is minimal, provided that it does extend || • ||, i.e., that || • || is a regular (= operator) norm.In the latter case, A+ is complete under both || • ||, and || • ||op, so by the "two-norm lemma" [2, II.2.5] these two norms are equivalent; the pure existence nature of the lemma does not yield an explicit bound M in || • ||i < M\\ • ||op and such a bound seems to depend on the algebra A .The present theorem establishes uniform equivalence of the two unitization norms over the class of nonunital Banach algebras with regular norms.Theorem. For every nonunital Banach algebra A with unitization A+ and with regular norm, and for every X £ C and a £ A, we have \\Xe + a\\op < ||/te + a||i < (6exp 1)||A^ + a||op.If A is a C*-algebra, a £ A is hermitian, and X is complex then \\Xe + a\\x <3||Ae + a||op and the constant 3 is best (minimal) possible. Proof. In a general algebra A with a regular norm, we have an extension of the classical inequality for the numerical radius v(a) [1, Theorem 4.1]:v(a) < \\a\\ < (expl)v(a).
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