Cauchy–Schwarz inequality for a positive functional

Let $A$ be a $C^*$ -algebra and let $\varphi:A\to\mathbb C$ be a positive linear functional. The Cauchy–Schwarz inequality for $\varphi$ is

\left|\varphi(b^*a)\right|^2 \leq \varphi(a^*a)\,\varphi(b^*b) \qquad(a,b\in A).

It is the ordinary Cauchy–Schwarz inequality for the positive semidefinite sesquilinear form

\langle a,b\rangle_\varphi=\varphi(b^*a).

No faithfulness, unitality, or normalization of $\varphi$ is required. When either factor on the right vanishes, the mixed term necessarily vanishes as well.

Derivation and equality

Positivity gives

0\leq \varphi\bigl((a+\lambda b)^*(a+\lambda b)\bigr) \qquad(\lambda\in\mathbb C).

If $\varphi(b^*b)>0$ , choosing $\lambda=-\varphi(b^*a)/\varphi(b^*b)$ yields the displayed inequality. If $\varphi(b^*b)=0$ , applying positivity first to $a+\lambda b$ shows directly that $\varphi(b^*a)=0$ .

After quotienting by the zero-length vectors, equality holds precisely when the classes of $a$ and $b$ are linearly dependent. This is the usual equality criterion in the resulting pre-Hilbert space; it need not mean that $a$ and $b$ are linearly dependent inside $A$ .

Structure and consequences

The null space

N_\varphi=\{a\in A:\varphi(a^*a)=0\}

is a left ideal. Indeed, Cauchy–Schwarz and $a^*c^*ca\leq\lVert c\rVert^2a^*a$ show that $ca\in N_\varphi$ whenever $a\in N_\varphi$ . Consequently the form descends to $A/N_\varphi$ ; its Hilbert-space completion is the space used in the GNS construction.

For a state on $M_n(\mathbb C)$ of the form $\varphi(x)=\langle x\xi,\xi\rangle$ , the inequality becomes the usual Cauchy–Schwarz inequality for the vectors $a\xi$ and $b\xi$ . Positivity is decisive: an arbitrary bounded linear functional need not make $\varphi(b^*a)$ into a positive semidefinite form and need not satisfy this inequality.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. Elsevier DOI record. Relevant: §3.3 on positive linear functionals and their Cauchy–Schwarz inequality.