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Abstract

The classical Bohr inequality states that for complex numbers a, b and real numbers p, q > 1 such that 1/p + 1/q = 1, we have |a + b| 2 p|a| 2 + q|b| 2 with equality if and only if b = (p − 1)a. Various generalizations of the Bohr inequality occur for scalars, vectors, matrices and operators. In this paper, this inequality is generalized from Hilbert space operators to the context of C * -algebras and some extensions and related inequalities are obtained. For each inequality, the necessary and sufficient condition for the equality is also determined. The idea of transforming problems in operator theory to problems in matrix theory, which are easy to handle, plays a key role.