MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ denotes the commutant of $A$). I can't find any reference to why this is true anywhere earlier in the chapter. Am I missing something, or is this referring to some other result?

For context, 4.1.10 states:

Let $A$ be a von Neumann algebra on a Hilbert space $H$ and $v$ an element of $A$ with polar decomposition $v = u|v|$. Then $u \in A$.

share|cite|improve this question
Does Murphy show in a previous chapter (perhaps an exercise) that in a unital C*-algebra, every element is a linear combination of 4 unitary elements? (If $x=x^*$ and $\|x\|\leq 1$, then $x\pm i\sqrt{1-x^2}$ are unitary and $x$ is their average$.) $A'$ is a unital C*-algebra. – Jonas Meyer Oct 16 '12 at 22:06
Thanks Jonas. Murphy showed decomposition into hermitian elements and into positive elements and that has been used heavily in the last chapter. However, I don't recall coming across that decomposition into unitaries (and I have done all the HW problems). Thanks for providing it, I figured I was missing something simple. – Jason Ekstrand Oct 17 '12 at 15:33

See Remark 2.2.2 on p.45 of Murphy's book. There he shows every element in a unital C*-algebra may be written as a linear combination of four unitary elements.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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