In what follows, $X$ is a topological space and $\mathscr{C}^{b}(X)$ is the space of bounded, continuous functions on $X$ (with values in $\mathbb{C}$). I give you an adaptation of the definition of a mean as stated in Reiter, H. - Classical Harmonic Analysis and Locally Compact groups:
A mean $\mathscr{M}$ on $\mathscr{C}^{b}(X)$ is a positive, linear functional $\phi\mapsto\mathscr{M}(\phi)$ such that $\mathscr{M}(1_{X})=1$.
I wanted to define a mean on a closed, unital subalgebra, i.e. assume that $\mathcal{A}\subseteq\mathscr{C}^{b}(X)$ is a closed subspace of $\mathscr{C}^{b}(X)$ which is closed under multiplication and contains $1_{X}$, then I say that:
A mean $\mathscr{M}$ on $\mathcal{A}$ is a positive, linear functional such that $\mathscr{M}(1_{X})=1$.
No magic up to now. However I have the following problem: in Reiter's definition it follows quite easily (by using the same standard tricks as for integration), that $|\mathscr{M}(\phi)|\leq\lVert\phi\rVert_{\infty}$ for all $\phi\in\mathscr{C}^{b}(X)$ so that $\mathscr{M}$ is of norm less than 1. Unfortunately I do not necessarily have the functions $|\phi|$, $\Re(\phi)$ (real part) and $\Im(\phi)$ (imaginary part) available in $\mathcal{A}$. It works for real-valued functions because $-\lVert\phi\rVert_{\infty}\leq \phi\leq\lVert\phi\rVert_{\infty}$ but I do not know how to extend this to $\mathbb{C}$-valued functions. Do I need that $\mathcal{A}$ is a $C^{\ast}$-subalgebra (i.e. closed under conjugacy)?
Follow-up question:
Is the assumption of positivity sufficient to guarantee that $\mathscr{M}(f)\in\mathbb{R}$ whenever $f$ is real-valued? I can not guarantee $\max\{f,0\}$, $\min\{-f,0\}\in\mathcal{A}$ anymore.
