+1 for this interesting answer. honestly I am not familiar with "universal algebra generated by elements and relations" but I am curious since 1 decade. For example what would we lose or gain if we add extra relation $[a,b]=-log(q)$ in this or in any other universal algebra? because this condition consist with any commutative algebra in a deformation of $A_q$ with q close to 1

@No-one You are well come! Thank you for sharing the Yosida paper

@JochenGlueck Yes I see. thank you!

Let consider $P=\infty, X=\mathbb{Z}, \mu=$counting measure and $\theta$ the shift operator and $f$ be the characteristic function of the set prime integers. In this case what is the limit you are talking about? does the density of prime number is needed? a prime integer is an integer whose absolute value is a prime number

Unfortunately I do not access to paper by Yosida "Mean ergodic theorem in Banach spcae'. But do you think that this paper can help for your question. If you have a PDF of this paper please send me via email

@ThomasRot Your interesting MO answer below (My +1 for that) contains materials about real analytic objects the paper by Michor and Vizman mathoverflow.net/a/466437/36688 So I think you already familiar with analytic structures

@DmitriiKorshunov Wonderful reference! I realy appreciate your comment which upgrade the value of your beautiful answer to my post

@ThomasRot Ok Thanks

@Z.M I was talking about exercise 15 page 21 Differential topology Hirsch, 1976

Can one modify this idea to answer your original question?

With a little modification of algebra you are considering the answer is negative. Instead of polynomial algebra let's consider the algebra of lorentz polynomials with $*$ operation as follows: $(az^n)^*=\bar{a}z^{-n}$ then this is the group algebra of all function on $\mathbb{Z}$ with compact support so according to its pre $C^*$ algebra structure every positive functional must be Heremitian.

Thank you very much for your very perfect and interesting answer.

@ThomasRot in chapter 2 he frequently apply smooth approximation of $C^r$ maps for positive r but he apply to $C^0$ with caution and very rare