Sometimes I see the requirement that approximate units $(u_\lambda)_\lambda$ in $C^*$-algebras must be eventually bounded, that means for some $C>0$ there is a $\lambda_0$ such that $\|u_\lambda\|<C$ for all $\lambda\ge\lambda_0$. Yet I can not come up with an example of an approximate unit that is not eventually bounded, so I'm asking for an example here. Bonus points when the approximate unit is a sequence and not just a net.
For this question I define an approximate unit of a $C^*$-algebra $A$ as a net $(u_\lambda)_\lambda$ in $A$ such that $u_\lambda a\to a$ and $a u_\lambda\to a$ for all $a\in A$.
I have looked at a few $C^*$-algebras I know:
- $C_0(\mathbb R)$: I'm quite positive that all approximate units are bounded here, but I have not worked out the details yet.
- $C_0(X)$ (and therefore all commutative $C^*$-algebras): I guess only something considerably larger than $X=\mathbb R$ can work, maybe $X=\ell^2(\mathbb N)$.
- group algebras $C^*(G)$: With the multiplication being a form of convolution my guess is that all approximate units converge to some "dirac" function with the weight being centered at the unit element of $G$. My gut feeling tells me again that these converge to $1$ in norm, but I could very well be wrong.
- compact operators on a Hilbert space $\mathcal K(H)$. The projections on finite subspaces are an approximate unit (bounded by 1).
- General case: Some $C^*$-subalgebra $A\subset\mathcal B(H)$.
The fact that $u_\lambda a \to a$ in norm implies that $u_\lambda a$ is eventually bounded and therefore for every $a$ the operator $M_{\lambda}$ given by $a \mapsto u_\lambda a$ satisfies that $M_\lambda(a)$ is bounded. But, by the Uniform Boundedness Principle, that holds iff $M_{\lambda}$ is bounded. Since we are in a $C^\ast$ algebra, the norm of $a \mapsto u_\lambda a$ coincide with the norm of $u_\lambda$.
There are indeed Banach algebras without bounded approximations.