Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

Abstract:

A classical result of W. Bade states that if $\mathcal {M}$ is any $\sigma$-complete Boolean algebra of projections in an arbitrary Banach space $X$ then, for every $x_0\in X,$ there exists an element $x’$ (called a Bade functional for $x_0$ with respect to $\mathcal {M})$ in the dual space $X’$, with the following two properties: (i) $M\mapsto \langle Mx_0,x’\rangle$ is non-negative on $\mathcal {M}$ and, (ii) $Mx_0=0$ whenever $M\in \mathcal {M}$ satisfies $\langle Mx_0,x’\rangle =0.$ It is shown that a Fréchet space $X$ has this property if and only if it does not contain an isomorphic copy of the sequence space $\omega = \mathbb {C}^{\mathbb {N}}.$