Given a $C^*$ algebra $A$, and $f:A\to \mathbb{C}$ a continuous linear functional in $A^*$, we may consider $f$ to be a normal element of the dual of the enveloping Von Neumann algebra $\tilde{A}$ of $A$, i.e. $f\in \tilde{A}^*$.
In this case, by the Polar decomposition theorem (See Takesaki Theorem III.4.2), it follows that $$f= v\cdot w$$ for some partial isometry $v\in \tilde{A}^*$ and some positive $w\in A^*$.
My question is, when can it be guaranteed that $v$ is actually in $A$ (i.e. in the copy of $A$ sitting inside $\tilde{A}$)?
In other words, when is it possible to write a linear functional on $A$ as the translate of a positive functional? Are there any conditions apriori that will guarantee this is true?
Thanks!
The problem you have in a C$^*$-algebra, is that you may have few or even no partial isometries. This is already the problem with the polar decomposition.
As an extreme case, consider $A=C_0(\mathbb R)$, where the only partial isometry is $0$. If $A$ is unital, then all unitaries are partial isometries, but there might not be others: this will be the case in any projectionless algebra, like $C[0,1]$, of $C_r^*(\mathbb F_2)$.
That's precisely what Takesaki achieves by working on the enveloping von Neumann algebra: a von Neumann algebra is flooded by projections and partial isometries. In particular, the polar decomposition is always possible in a von Neumann algebra.