When Does a $C^*$-algebra have a Subalgebra Invariant under Involution?

Question

I know very little about $C^*$-algebras but am intrigued by this question. In the trivial case of the complex field itself, the reals are such a subalgebra. Are there any other cases?

Answer 1

Consider for example $M_n(\mathbb{C})$ and let $A \in M_n(\mathbb{C})$ be self-adjoint. Then the set

$$ \{ p(A) \, | \, p \in \mathbb{R}[X] \} $$

which consists of the matrices that are real polynomials in $A$ is a real algebra invariant under conjugation. For example if $A = \operatorname{diag}(1,1,2)$ then this algebra is isomorphic to $\mathbb{R} \oplus \mathbb{R}$.

Answer 2

When $A$ is abelian, the selfadjoint elements form a real algebra. So, for an arbitrary C$^*$-algebra $A$, take $x\in A$ selfadjoint, and then $C^*(x)$ is an abelian C$^*$-algebra and its selfadjoint elements (which amount to $f(x)$ for $f$ continuous, as in Levap's answer) gives a real subalgebra invariant under the involution.