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Suppose that A = C([0,1]). For each unitary u Î A, the mapping t\longmapsto ut from [0,1] to the unitary group of
G of A with ut(x) = u((1-t)x) connects u to u(0)1. If u(0) = exp(iq) for some real number q, {exp(itq)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 Î Ah.
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.