Every C^*-algebra has this property. Indeed if A is a untial Banach algebra which is C^*-algebra with respect to involutions * and #, then if x = x^* and f be a state on A (i.e., by [K&R1, Theorem 4.3.2] is a bounded linear functional satisfying ||f|| = f(1) = 1) then, f(x) = [`(f(x<sup>*</sup>))] = [`f(x)], so that f(i(x-x^#)) = i(f(x)-[`f(x)]) = 0. Therefore, by [K&R1, Proposition 4.3.3] sp(i(x-x^#)) = {0}. Hence i(x-x^#) = 0, by [K&R1, Proposition 4.1.1.(i)]. So x^# = x = x^*. For an arbitrary element x with the real and imaginary parts x₁ and x₂, we have x^* = x₁-ix₂ = x^#. ( If A doesn't have a unit, it is enough to consider its unitization).

Ref.
[K&R1] R.V. Kadilon, J.R. Ringrase, fundamentals of the theory of operator algebras (I), Acad. Press, 1983.