Closure and homomorphic properties in \(B^*\)-algebras via functions which operate (Q788963)

The object of this paper is to provide an elementary and unified treatment and extension of results from \textit{J. A. Van Casteren} [Proc. Am. Math. Soc. 72, 54-56 (1978; Zbl 0383.46008)] and \textit{L. T. Gardner} [Proc. Am. Math. Soc. 73, 341-345 (1979; Zbl 0371.46021), Math. Scand. 44, 196-200 (1979; Zbl 0393.46045)] dealing with linear subspaces of \(B^*\)-algebras which are \(| \cdot |\)-closed and linear maps between \(B^*\)-algebras which commute with the absolute value \(| \cdot |\) or with some other non-affine map, respectively. The arguments involve the well-known technique of equating coefficients from the power series of \((1+z)^{1/2}\) [cf. \textit{F. Forelli}, Can. J. Math. 16, 721-728 (1964; Zbl 0132.094) and \textit{R. Schneider}, Can. J. Math. 27, 133-137 (1975; Zbl 0323.46026)] and a smoothing device used previously to great effect by \textit{K. deLeeuw} and \textit{Y. Katznelson} [J. Anal. Math. 11, 207-219 (1963; Zbl 0123.309)]. Three typical results are the following: Let \({\mathcal S}\) be a closed subspace of the \(B^*\)-algebra \({\mathcal A}\). If some non-polynomial continuous function f and some self-adjoint \(x\in {\mathcal S}\) satisfy f(x)\(\in {\mathcal S}\), then every positive power of x lies in \({\mathcal S}\). From this follows Casteren's result that \({\mathcal S}\) is a \({}^*\)-subalgebra if it is closed under \(| \cdot |\). If \({\mathcal H}\) is a Hilbert space and \(\Phi\) : \({\mathcal A}\to {\mathcal B}({\mathcal H})\) satisfies \(| \Phi(x)| =\Phi(| x|)\) for all x, then there is a unique \({}^*\)-algebra homomorphism \(\Psi\) of \({\mathcal A}\) into the weak operator closure of the algebra generated by \(\Phi\) (\({\mathcal A})\) such that \(\Phi(x)=\Psi(x)\Phi(1)=\Phi(1)\Psi(x)\) for all x (cf. Gardner).