Is there an example of two non-isomorphic complex $ * $-algebras, $ A $ and $ B $, described by generators and relations, such that the following properties hold?
- $ A $ and $ B $ possess universal enveloping $ C^{*} $-algebras $ {C^{*}}(A) $ and $ {C^{*}}(B) $ respectively.
- $ {C^{*}}(A) $ is $ * $-isomorphic to $ {C^{*}}(B) $.
We can think of enveloping C*-algebras as "completing" the $*$-algebra with respect to some norm, so we could perhaps try to take some $B$ with $A\not\subseteq B\not\subseteq C^*(A)$, but I can't think of an easy example in terms of (finitely many) generators and equations.
Alternatively, there are some equalities/implications which are true in C*-algebras but which aren't true in $*$-algebras, so the construction of the enveloping C*-algebra will have some quotients involved.
With that in mind, take $A$ to be the $*$-algebra generated by a unit $1$ and and element $a$ satisfying $$a=a^*,\qquad a^2=0.$$ Then $A$ is two-dimensional.
But in $C^*(A)$, we have $a=0$, so $C^*(A)$ is one-dimensional and thus $$C^*(A)=\mathbb{C}=C^*(\mathbb{C}),$$ and $\mathbb{C}$ and $A$ are not isomorphic $*$-algebras.