2
$\begingroup$

Let $A$ be a finite dimensional $\mathrm{C}^*$-algebra.

We have the usual absolute value:

$$|a|:=\sqrt{a^*a},$$

and the cone of positive elements:

$$A^+:=\{a\in A:\exists \,b\in A, a=b^*b\}.$$

An operator $T\in B(A)$ is said to be positive if $T(A^+)\subset A^+$.

Does it hold that $|Tf|=T|f|$ for positive $T$?.

Context:

I don't have any reason to believe this but I have a $T$-invariant finite trace $h\in A^{'}$ ($h\circ T=h$) and if I have the above I have

$$\|Tf\|_{\mathcal{L}^1}=h(|Tf|)=h(T|f|)=h(|f|)=\|f\|_{\mathcal{L}^1},$$ which would be useful to me.

$\endgroup$
4
$\begingroup$

In general, we do not have $|Tf | = T |f|$.
To see this, consider the $2 \times 2$-matrices over $\mathbb{C}$ and let $T$ be taking the transpose of a matrix.
Then $T$ is a positive operator, but for $f = \left( \begin{matrix} 1 & 0 \\ 1 & 0\end{matrix} \right)$ we have \begin{align*} |Tf| =\frac{1}{\sqrt{2}}\left( \begin{matrix} 1 & 1 \\ 1 & 1\end{matrix} \right) \neq \left( \begin{matrix} \sqrt{2} & 0 \\ 0 & 0\end{matrix} \right) = T|f|. \end{align*}

$\endgroup$
  • $\begingroup$ Nice counter-example, thank you (you had them the wrong way around so I just edited that). $\endgroup$ – JP McCarthy Aug 14 '16 at 23:33
4
$\begingroup$

As Hetebrij pointed out, it's false. On a positive note, there is a nice (and useful!) characterisation of the linear maps that preserve the absolute value.

Theorem (Gardner 1979): For a linear map $\varphi\colon A \to B$ of C* algebras the following are equivalent

  1. $\varphi$ preserves absolute values
  2. $\varphi$ is positive and $\varphi(1)\varphi(ab)=\varphi(a)\varphi(b)$ for all $a,b\in A$.
  3. $\varphi$ is $2$-positive and disjoint (i.e. if $x,y \geq 0$ and $xy=0$ then $\varphi(x)\varphi(y)=0$).

http://www.ams.org/journals/proc/1979-076-02/S0002-9939-1979-0537087-0/S0002-9939-1979-0537087-0.pdf

With Gardner's Theorem we can prove the following characterisation of the positive unital maps that preserve absolute value.

Proposition. (WW) For a positive unital map $\varphi\colon A \to B$ between von Neumann algebras the following are equivalent:

  1. $\varphi$ preserves absolute values
  2. $\varphi$ is a $*$-homomorphism
  3. $\varphi$ preserves projections and is $2$-positive
  4. $\varphi$ preserves ranges and is $2$-positive

( Follows from Proposition 47 of http://scitation.aip.org/content/aip/journal/jmp/57/9/10.1063/1.4961526 )

$\endgroup$
  • $\begingroup$ Excellent - this may be very useful thank you. $\endgroup$ – JP McCarthy Oct 11 '16 at 6:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.