A unital $C^*$ algebra $A$ is said a Tietze algebra if it satisfies the following:
For every ideal $I$ of $A$ and every unital morphism $\phi: C[0,1] \to A/I$ there is a unital morphism $\tilde{\phi}:C[0,1] \to A$ with $\phi=q\circ \tilde{\phi}$ where $q:A\to A/I$ is the quotion map.
Obviousely every commutative algebra is a Tietze algebra. But what is an example of non Tietze algebra? Furthermore: Is the familly of Tietze algebras closed under the minimal or Maximal $C^*$ tensor product?
Every unital C*-algebra has this property.
Let $f \in C[0,1]$ be the function $f(t) = t$. If $\phi: C[0,1] \to \mathcal{A}$ is a unital $*$-homomorphism then $\phi(f)$ is a positive element $x$ of $\mathcal{A}$ whose norm is at most $1$. Conversely, given any positive element $x$ of $\mathcal{A}$ whose norm is at most $1$, there is a unique unital $*$-homomorphism from $C[0,1]$ to $\mathcal{A}$ taking $f$ to $x$. (This is a good exercise.)
In those terms, we are asking if every positive element of $\mathcal{A}/\mathcal{I}$ whose norm is at most $1$ can be lifted to a positive element of $\mathcal{A}$ whose norm is at most $1$. This is true, and it's also a good exercise.
