Thread: constructing a C*-algebra

Is it possible to construct a C*-algebra such that there exists which cannot be decomposed into with and a partial isometry?

It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some and any operator can be decomposed using the polar decomposition, or am I missing something?

Originally Posted by Mauritzvdworm

Is it possible to construct a C*-algebra such that there exists which cannot be decomposed into with and a partial isometry?

It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some and any operator can be decomposed using the polar decomposition, or am I missing something?

The decomposition certainly works in B(H), giving a positive operator and a partial isometry in B(H). The positive operator will be in , but the partial isometry does not necessarily belong to .

For example, if is the commutative C*-algebra of continuous functions on [–1,1] and x is the function x(t) = t, then the positive part will be the function . But the partial isometry will be the function , where is –1 when x<0 and +1 when x>0. Obviously is not continuous and so cannot belong to .

That is what I thought, interestingly in the case of von Neumann algebras the partial isometry is part of the von Neumann algebra. Thank you.