Surjectivity of unitization functor of C*-algebras

Question

Let $A$ be a unital $C^*$-algebra.

Question: Is $A$ isomorphic to the unitization of a $C^*$-algebra $A_0$?

Clarification: Here, I am using the functorial definition of unitization. The unitization of $A_0$ is the algebra with underlying vector space $A_0\oplus\mathbb C$ and norm $$||(a,\lambda)\|=\max\Big\{|\lambda|,\sup_{\|b\|=1}\{\|ab+\lambda b\|\}\Big\}.$$

Thoughts: Suppose $A$ is commutative. Then by Gelfand duality, $A$ is isomorphic to $C(X)$ for some compact Hausdorff space $X$. So in the commutative case it would suffice that any such $X$ is the one-point compactification of some space $X_0$; but I'm not even sure if this is true.

Answer 1

No. The algebra $A_0$ in question would necessarily be an ideal -- the multiplication you have on the unitization is $$(a,\mu)(b,\nu) = (ab + \mu b + \nu a, \mu \nu).$$ Thus if $\mu$ or $\nu$ is 0, then the right component will be 0, i.e., $A_0$ is an ideal. Are there C*-algebras without proper ideals? (try $M_n$)

As for the commutative case, see Which spaces can be seen as to be the one-point compactification of some other space.

Of course, the unitization you describe is a "minimal" unitization. However every unital C*-algebra $A$ is in fact the multiplier algebra of itself (which is in some sense a "maximal unitization" -- it corresponds to the Stone–Čech compactification in the commutative case).

Answer 2

No, but if $A_0$ is unital, then $A\cong A_0 \oplus_\infty \mathbb{C}$ as $C^*$-algebras. The map $$A_0\oplus_\infty \mathbb{C}\to A: (a, \lambda) \mapsto (a-\lambda 1_{A_0})+ \lambda 1_{A}$$ is an isomorphism of $C^*$-algebras.

Answer 3

The first time I came up with this question is when I learned K-theory. I realized that $B(H)$ cannot be such a C*-algebra.

If $A$ is the unitization of $B$, then we have a short exact sequence which is necessarily split: $$0\rightarrow B\rightarrow A\rightarrow \mathbb{C}\rightarrow0.$$ So by the split exactness of $K_0$, we have a split exact sequence: $$0\rightarrow K_0(B)\rightarrow K_0(A)\rightarrow K_0(\mathbb{C})\rightarrow 0.$$ Or equivalently, $K_0(A)\cong K_0(B)\oplus\mathbb{Z}$. The conclusion is: if $A$ is a unital C*-algebra whose $K_0$ group cannot admit $\mathbb{Z}$ as a direct summand, then $A$ cannot be a unitalization of another one.

For example, the Cuntz algebras (but they are also simple), $B(H)$, and more generally, $M(A\otimes\mathbb{K})$, where $M$ denotes the multiplier algebra, $\mathbb{K}$ is the algebra of compact operators on a infinite-dimensional separable Hilbert space.