# Fuglede-Putnam-Rosenblum theorem

Let $A$ be a $C^{\ast}$-algebra with unit $e$.

The Fuglede-Putnam-Rosenblum theorem makes the assertion that for a normal element $a\in A$ the kernel of the commutator mapping $[a,-]\colon A\to A$ is a $\ast$-closed set.

The general formulation of the result is as follows:

Theorem. Let $A$ be a $C^{\ast}$-algebra with unit $e$. Let two normal elements $a,b\in A$ be given and $c\in A$ with $ac=cb$. Then it follows that $a^{\ast}c=cb^{\ast}$.

Lemma. For any $x\in A$ we have that $\exp(x-x^{\ast})$ is a element of $A$.

Proof. We have for $x\in A$ that $\exp(x-x^{\ast})^{\ast}\exp(x-x^{\ast})=\exp(x^{\ast}-x+x-x^{\ast})=\exp(0)=e$. And similarly $\exp(x-x^{\ast})\exp(x-x^{\ast})^{\ast}=e$. ∎

With this we can now give a proof the Theorem.

Proof. The condition $ac=cb$ implies by induction that $a^{k}c=cb^{k}$ holds for each $k\in\mathbb{N}$. Expanding in power series on both sides yields $\exp(a)c=c\exp(b)$. This is equivalent to $c=\exp(-a)c\exp(b)$. Set $U_{1}:=\exp(a^{\ast}-a),U_{2}:=\exp(b-b^{\ast})$. From the Lemma we obtain that $\|U_{1}\|_{A}=\|U_{2}\|_{A}=1$. Since $a$ commutes with $a^{\ast}$ und $b$ with $b^{\ast}$ we obtain that

 $\exp(a^{\ast})c\exp(-b^{\ast})=\exp(a^{\ast})\exp(-a)c\exp(b)\exp(b^{\ast})$

which equals $\exp(a^{\ast}-a)c\exp(b-b^{\ast})=U_{1}cU_{2}$.

Hence

 $\|\exp(a^{\ast})c\exp(-b^{\ast})\|\leq\|c\|.$

Define $f\colon\mathbb{C}\to A$ by $f(\lambda):=\exp(\lambda a^{\ast})c\exp(-\lambda b^{\ast})$. If we substitute $a\mapsto\lambda a,b\mapsto\lambda b$ in the last estimate we obtain

 $\|f(\lambda)\|\leq\|c\|,\ \lambda\in\mathbb{C}.$

But $f$ is clearly an entire function and therefore Liouville’s theorem implies that $f(\lambda)=f(0)=c$ for each $\lambda$.

This yields the equality

 $c\exp(\lambda b^{\ast})=\exp(\lambda a^{\ast})c.$

Comparing the terms of first order for $\lambda$ small finishes the proof. ∎

Title Fuglede-Putnam-Rosenblum theorem FugledePutnamRosenblumTheorem 2013-05-08 21:47:27 2013-05-08 21:47:27 karstenb (16623) karstenb (16623) 1 karstenb (16623) Theorem msc 47L30