Kadison–Schwarz inequality

Statement

Let $A$ and $B$ be unital $C^*$ -algebras and let $\Phi:A\to B$ be a unital $2$ -positive map. The Kadison–Schwarz inequality states that every $a\in A$ satisfies

\Phi(a)^*\Phi(a)\leq\Phi(a^*a) \quad\text{and}\quad \Phi(a)\Phi(a)^*\leq\Phi(aa^*).

In particular the inequalities hold for every unital completely positive map. The order is the one determined by positive elements of $B$ . These estimates are operator-algebraic analogues of the scalar Cauchy–Schwarz inequality.

Matrix proof

The matrix

\begin{pmatrix}a^*a&a^*\\a&1_A\end{pmatrix} =\begin{pmatrix}a^*\\1_A\end{pmatrix} \begin{pmatrix}a&1_A\end{pmatrix}

is positive in $M_2(A)$ . Applying $\Phi^{(2)}$ and taking the Schur complement of its lower-right unit gives $\Phi(a^*a)-\Phi(a)^*\Phi(a)\geq0$ . Applying the same argument to $a^*$ gives the second inequality.

Equality and multiplicativity

For a unital completely positive $\Phi$ , the elements $a$ for which equality holds in both displayed inequalities form its multiplicative domain. On this $C^*$ -subalgebra,

\Phi(ab)=\Phi(a)\Phi(b),\qquad \Phi(ba)=\Phi(b)\Phi(a)

for all $b\in A$ . Equality therefore identifies the portion of the domain on which a generally nonmultiplicative map behaves like a $*$ -homomorphism.

Conventions and sharpness

Kadison's original inequality requires only unital positivity when $a$ is self-adjoint: $\Phi(a)^2\leq\Phi(a^2)$ . The two inequalities for arbitrary $a$ follow from unital $2$ -positivity; this stronger form is often called the Kadison–Schwarz or Choi–Schwarz inequality. Omitting unitality changes the estimate and generally introduces a factor involving $\Phi(1_A)$ .

References

Richard V. Kadison, “A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras,” Annals of Mathematics 56 (1952), 494–503. DOI record. Relevant: the Schwarz inequality for unital positive maps on self-adjoint elements.
Man-Duen Choi, “A Schwarz Inequality for Positive Linear Maps on $C^*$ -Algebras,” Illinois Journal of Mathematics 18 (1974), 565–574. DOI record. Relevant: the arbitrary-element form and its relation to higher positivity.
Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapter 3 on completely positive maps and the Schwarz inequality.