Non self adjoint subalgebra of a C*-algebra.

Question

Can you give an example of a closed (unital) non self adjoint subalgebra of a unital C*algebra?

Answer 1

Take your favorite C*-algebra $A$ and an element $a\in A$ that is not self-adjoint. Then the closed subalgebra generated by $a$ will usually not contain $a^*$, and hence will not be self-adjoint.

For an explicit example, let $D\subset\mathbb{C}$ be the closed unit disk, let $A=C(D)$, and let $a\in A$ be the inclusion function $D\to\mathbb{C}$ (i.e., $a(z)=z$). The closed subalgebra generated by $a$ is then the closure of the set of functions $D\to\mathbb{C}$ which are polynomials in $z$. If $a^*$ were in the closed subalgebra generated by $a$, then the function $a^*(z)=\overline{z}$ would be a uniform limit of polynomials in $z$. This is impossible by some basic complex analysis (for instance, the fact that a uniform limit of holomorphic functions is holomorphic).

Answer 2

Take $A=M_2(\mathbb C)$, and consider the subalgebra $$ B=\left\{\begin{bmatrix}\alpha&\beta\\ 0&\gamma\end{bmatrix}:\ \alpha,\beta,\gamma\in\mathbb C\right\}. $$