I have the following question and since I am not an expert on C\*-algebras, I thought I ask it here:


I know that in general the sum and product of normal elements need not be normal. It is even true that every element in a C\*-algebra is the sum of two normal elements. What I do not know is if that is true for multiplication, too, i.e. if every element in a C\*-algebra is a product of normal elements.


I think I know that every invertible element can be written as a product of a unitary and a positive element but this decomposition does not work for non-invertible elements.


My standard example is the right shift operator on the l^2(N)-space. This is a non-normal operator, which is not invertible and I have no idea how to write it as a product of two normal operators.


I would be very grateful if anyone could help.


Greetings, Tom.



is every element in a C* algebra a product of normal elements?
Since the question was asked wrt C$\*$-algebras, I guess there is room for a general remark. Suppose that $xy = 1$ where $x$ is a product of normal elements, say
$v\_1v\_2\cdots v\_n$. Then $v\_1$ is normal and has a right inverse;
therefore it is (two-sided) invertible. Thus $v\_2\dots v\_n y $ is
invertible, so $v\_2$ has a right inverse, and thus is invertible. By
induction, we obtain that all the $v\_i$ are two-sided invertible, so that
$x$ is too.


The upshot of this argument is that a product of normal elements that is
one-sided invertible, is actually invertible. In particular, the right
shift is not a product of normal elements---although of course, the index argument above is decisive in this case.


If every element of a C$\*$-algebra were a product of normals,
then it would be directly finite (all one-sided inverses are two-sided).
Whether this is sufficient is unclear (as is the question as to whether the product of normals property extends to matrix rings over the original). But if the C\*-algebra has one in
the stable range, then every element is an invertible times a positive, so
is a product of (three) normals. 


And for von Neumann algebras, if of finite type, then every element is a product of two normals, as the isometry in the polar decomposition argument (below) can be modified to be a unitary in that case. If not of finite type, the property fails. 


