A Multiplier Approach to the Lance-Blecher Theorem

A new approach to the Lance-Blecher theorem is presented resting on theinterpretation of elements of Hilbert C*-module theory in terms of multipliertheory of operator C*-algebras: The Hilbert norm on a Hilbert C*-module allowsto recover the values of the inducing C*-valued inner product in a unique way,and two Hilbert C*-modules {M_1, <.,.>_1}, {M_2, <.,.>_2} are isometricallyisomorphic as Banach C*-modules if and only if there exists a bijectiveC*-linear map S: M_1 --> M_2 such that the identity <.,.>_1 \equiv_2 is valid. In particular, the values of a C*-valued inner producton a Hilbert C*-module are completely determined by the Hilbert norm inducedfrom it. In addition, we obtain that two C*-valued inner products on a BanachC*-module inducing equivalent norms to the given one give rise to isometricallyisomorphic Hilbert C*-modules if and only if the derived C*-algebras of''compact'' module operators are *-isomorphic. The involution and the C*-normof the C*-algebra of ''compact'' module operators on a Hilbert C*-module allowto recover its original C*-valued inner product up to the following equivalencerelation: <.,.>_1 \sim <.,.>_2 if and only if there exists an invertible,positive element $a$ of the center of the multiplier C*-algebra M(A) of A suchthat the identity <.,.>_1 \equiv a \cdot <.,.>_2 holds.