A Banach algebra with a unique C^{∗}-involution.

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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) = \overline{f(x^{∗})} = \overline{f(x)}, so that f(i(x − x^{#})) = i(f(x)−\overline{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 x_{1} and x_{2}, we have x^{∗} = x_{1} − ix_{2} = 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.