Cauchy–Schwarz inequality for Hilbert C*-modules

Statement

Let $E$ be a right Hilbert $C^*$ -module over a $C^*$ -algebra $A$ , with inner product linear in the second variable. For all $x,y\in E$ ,

\langle x,y\rangle_A^*\langle x,y\rangle_A \leq \lVert\langle x,x\rangle_A\rVert\,\langle y,y\rangle_A

in the order on positive elements of $A$ . Taking norms gives

\lVert\langle x,y\rangle_A\rVert \leq \lVert x\rVert\,\lVert y\rVert, \qquad \lVert x\rVert=\lVert\langle x,x\rangle_A\rVert^{1/2}.

This is the Hilbert-module Cauchy--Schwarz inequality. It makes the algebra-valued inner product jointly continuous and justifies the triangle inequality for the induced norm.

Proof idea

After adjoining a unit if necessary, positivity of $\langle y-xa,y-xa\rangle_A$ is applied with

a=(\langle x,x\rangle_A+\varepsilon1)^{-1}\langle x,y\rangle_A.

Expanding the positive element and letting $\varepsilon\downarrow0$ yields the displayed order inequality. The scalar proof by direct division cannot simply be copied, because $\langle y,y\rangle_A$ need not be invertible or commute with $\langle x,y\rangle_A$ Lance, Proposition 1.1.

Consequences

For fixed $x$ , the map $y\mapsto\langle x,y\rangle_A$ is bounded with norm at most $\lVert x\rVert$ . Hence the inner product extends uniquely across completion, and

\lVert\langle x,y\rangle_A-\langle x',y'\rangle_A\rVert \leq \lVert x-x'\rVert\lVert y\rVert +\lVert x'\rVert\lVert y-y'\rVert.

The theorem also proves the triangle inequality for the module norm by expanding $\langle x+y,x+y\rangle_A$ .

Conventions and scope

If the inner product is linear in the first variable, the order-valued formula is written with the variables and factors reversed. Unlike the scalar Cauchy--Schwarz inequality, equality does not in general admit a simple characterization by linear dependence, because coefficients lie in a possibly noncommutative algebra.

References

E. C. Lance, Hilbert $C^*$ -Modules: A Toolkit for Operator Algebraists, Cambridge University Press, 1995. DOI record. Relevant: Proposition 1.1 on the module Cauchy--Schwarz inequality and induced norm.