A multiplier approach to the Lance-Blecher theorem

TL;DR

This paper introduces a novel approach to the Lance-Blecher theorem by interpreting Hilbert C*-modules through multiplier theory, revealing how inner products and module isomorphisms are characterized by C*-algebraic structures.

Contribution

It presents a new perspective on the Lance-Blecher theorem using multiplier theory, characterizing inner products and isomorphisms of Hilbert C*-modules via multiplier C*-algebras.

Findings

Inner products are uniquely determined by the Hilbert norm.

Isometric isomorphisms correspond to C*-linear maps preserving inner products.

Inner products are equivalent if related by an invertible central element in the multiplier algebra.

Abstract

A new approach to the Lance-Blecher theorem is presented resting on the interpretation of elements of Hilbert C*-module theory in terms of multiplier theory of operator C*-algebras: The Hilbert norm on a Hilbert C*-module allows to 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 isometrically isomorphic as Banach C*-modules if and only if there exists a bijective C*-linear map S: M_1 --> M_2 such that the identity <.,.>_1 \equiv <S(.),S(.)>_2 is valid. In particular, the values of a C*-valued inner product on a Hilbert C*-module are completely determined by the Hilbert norm induced from it. In addition, we obtain that two C*-valued inner products on a Banach C*-module inducing equivalent norms to the given one give rise to isometrically isomorphic Hilbert C*-modules if and only if the derived C*-algebras of ''compact'' module operators are *-isomorphic. The involution and the C*-norm of the C*-algebra of ''compact'' module operators on a Hilbert C*-module allow to recover its original C*-valued inner product up to the following equivalence relation: <.,.>_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 such that the identity <.,.>_1 \equiv a \cdot <.,.>_2 holds.