On the set of invertible elements in Banach Jordan algebras (Q1911986)

Let \(A\) be a real or complex Banach Jordan algebra with a unit element \textbf{1}, let \(\Omega\) denote the open set of all invertible elements of \(A\), and \(\Omega_1\) the connected component of \(\Omega\) containing \textbf{1}. The author proves the equality \[ \Omega_1= \{ U(\exp x_1) \dots U(\exp x_n) (\mathbf{1}): n\in \mathbb{N},\;x_1,\dots, x_n\in A\}. \] We recall that, for \(x\) in \(A\), the operator \(U(x): A\to A\) is related to the binary Jordan product \(\circ\) of \(A\) by \(U(x)y= 2x \circ (x\circ y)- (x\circ x)\circ y\) for all \(y\) in \(A\). The equality above generalizes a well-known result for associative Banach algebras and implies that \(\Omega_1\) is analytically arcwise connected. This answers a question raised by \textit{A. Aupetit} [Math. Proc. Camb. Philos. Soc. 117, 479-489 (1995; Zbl 0837.46040)].