A C^{∗}-algebra A where every unitary element is of the form exp(ih) for a self-adjoint h ∈ A.

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Suppose that A = C([0, 1]). For each unitary u ∈ A, the mapping t ⟼ u_{t} from [0, 1] to the unitary group of G of A with u_{t}(x) = u((1 − t)x) connects u to u(0)1. If u(0) = exp(iθ) for some real number θ, {exp(itθ)1; 0 ≤ t ≤ 1} in G connects 1 to u(0)1. Therefore u is connected to 1. Now by [K&R3, Exercise 4.6.7], u = exp(ih) for some h ∈ A_{h}.

Comment. By [K&R1, Theorem 5.2.1], A isn’t W^{∗}-algebra. Ref. [K&R1] R.V. Kadilon, J.R. Ringrase, Fundamentals of the theory of operator algebras (I), Acad. Press, 1983. [K&R3] R.V. Kadilon, J.R. Ringrase, Fundamentals of the theory of operator algebras (III), Acad. Press, 1991.