# constructing a C*-algebra

#### Mauritzvdworm

Is it possible to construct a C*-algebra $$\displaystyle \mathcal{A}$$ such that there exists $$\displaystyle x\in\mathcal{A}$$ which cannot be decomposed into $$\displaystyle x=x_1x_2$$ with $$\displaystyle x_1\geq0$$ and $$\displaystyle x_2$$ a partial isometry?

It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some $$\displaystyle B(H)$$ and any operator $$\displaystyle T\in B(H)$$ can be decomposed using the polar decomposition, or am I missing something?

#### Opalg

MHF Hall of Honor
Is it possible to construct a C*-algebra $$\displaystyle \mathcal{A}$$ such that there exists $$\displaystyle x\in\mathcal{A}$$ which cannot be decomposed into $$\displaystyle x=x_1x_2$$ with $$\displaystyle x_1\geq0$$ and $$\displaystyle x_2$$ a partial isometry?

It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some $$\displaystyle B(H)$$ and any operator $$\displaystyle T\in B(H)$$ 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 $$\displaystyle \mathcal{A}$$, but the partial isometry does not necessarily belong to $$\displaystyle \mathcal{A}$$.

For example, if $$\displaystyle \mathcal{A}$$ is the commutative C*-algebra of continuous functions on [–1,1] and x is the function x(t) = t, then the positive part $$\displaystyle x_1$$ will be the function $$\displaystyle x_1(t) = |t|$$. But the partial isometry will be the function $$\displaystyle x_2$$, where $$\displaystyle x_2(t)$$ is –1 when x<0 and +1 when x>0. Obviously $$\displaystyle x_2$$ is not continuous and so cannot belong to $$\displaystyle \mathcal{A}$$.

• Mauritzvdworm

#### Mauritzvdworm

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.