Linear Transformation from C*-Algebra is Bounded if Bounded on Positive Elements

Theorem

Let $\struct {A, \ast, \norm {\, \cdot \,}_A}$ be a $\text C^\ast$-algebra.

Let $\struct {B, \norm {\, \cdot \,}_B}$ be a normed vector space over $\C$.

Let $T : A \to B$ be a linear transformation such that there exists $M > 0$ with:

$\norm {T x}_B \le M \norm x_A$ for each $x \in A$ positive.

Then $T$ is bounded.

$\norm {T x}_B \le M' \norm x_A$ for each $x \in A$ Hermitian.

Let $x \in A$ be Hermitian.

$x = x^+ - x^-$

with:

$\norm {x^+}_A \le \norm x_A$ and $\norm {x^-}_A \le \norm x_A$

Then we have:

$\ds \norm {T x}_B$

$=$

$\ds \norm {T x^+ - T x^-}_B$

$\ds $

$\le$

$\ds \norm {T x^+}_B + \norm {T x^-}_B$

$\ds $

$\le$

$\ds M \norm {x^+}_A + M \norm {x^-}_A$

$\ds $

$\le$

$\ds M \norm x_A + M \norm x_A$

$\ds $

$=$

$\ds 2 M \norm x_A$

for all $x \in A$.

Hence $M' = 2 M$ satisfies:

$\norm {T x}_B \le M' \norm x_A$ for each $x \in A$ Hermitian.

$\blacksquare$