Two elements x, y (xy ≠ yx) of a Banach algebra A such that e^{x}.e^{y} ≠ e^{x+y}.

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Consider A = B(l^{2}) and the unilateral shift operator T on l^{2}, defined by T (x_{1}, x_{2}, ...) = (0, x_{1}, x_{2}, ...) and its adjoint T^{∗}(x_{1}, x_{2}, ...) = (x_{2}, x_{3}, ...). Assuming ξ_{k} = (δ_{kn})_{n∈N} , k ∈ N; < e^{T} e^{T^{∗}}ξ_{1}, ξ_{1} >=< e^{T} ξ_{1}, ξ_{1} >=< ξ_{1}, ξ_{1} >= 1, since T^{∗}ξ_{1} = 0 and T ξ_{1} = ξ_{2}. Also (T + T^{∗})(ξ_{1}) = ξ_{2}, (T + T^{∗})^{2}(ξ_{1}) = ξ_{1} + ξ_{3}, … and so

< e^{T +T^{∗}}ξ_{1}, ξ_{1} >=< ξ_{1}, ξ_{1} > + < ξ_{2}, ξ_{1} > + < \frac{1}{2!}(ξ_{1} + ξ_{3}), ξ_{1} > + … > 1. Hence

e^{T} .e^{T^{∗}} ≠ e^{T +T^{∗}}.