Fuglede-Putnam-Rosenblum theorem

Let A be a C-algebra with unit e.

The Fuglede-Putnam-Rosenblum theorem makes the assertion that for a normal element aA the kernel of the commutator mapping [a,-]:AA is a -closed setPlanetmathPlanetmath.

The general formulation of the result is as follows:

Theorem. Let A be a C-algebra with unit e. Let two normal elements a,bA be given and cA with ac=cb. Then it follows that ac=cb.

Lemma. For any xA we have that exp(x-x) is a element of A.

Proof. We have for xA that exp(x-x)exp(x-x)=exp(x-x+x-x)=exp(0)=e. And similarly exp(x-x)exp(x-x)=e. ∎

With this we can now give a proof the Theorem.

Proof. The condition ac=cb implies by inductionMathworldPlanetmath that akc=cbk holds for each k. Expanding in power seriesMathworldPlanetmath on both sides yields exp(a)c=cexp(b). This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to c=exp(-a)cexp(b). Set U1:=exp(a-a),U2:=exp(b-b). From the Lemma we obtain that U1A=U2A=1. Since a commutes with a und b with b we obtain that


which equals exp(a-a)cexp(b-b)=U1cU2.



Define f:A by f(λ):=exp(λa)cexp(-λb). If we substitute aλa,bλb in the last estimate we obtain


But f is clearly an entire functionMathworldPlanetmath and therefore Liouville’s theorem implies that f(λ)=f(0)=c for each λ.

This yields the equality


Comparing the terms of first order for λ small finishes the proof. ∎

Title Fuglede-Putnam-Rosenblum theorem
Canonical name FugledePutnamRosenblumTheorem
Date of creation 2013-05-08 21:47:27
Last modified on 2013-05-08 21:47:27
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 1
Author karstenb (16623)
Entry type Theorem
Classification msc 47L30