The range of operators on von Neumann algebras (Q2781304)

Transitivity is a concept that was introduced by \textit{G. D. Birkhoff} in 1926. He also gave the first example of a transitive linear operator on a topological vector space: The translation operator \(T_a: \mathcal{H} (\mathbb{C})\rightarrow \mathcal{H}(\mathbb{C})\), \((a \neq 0)\), on the Fréchet space of entire functions [C. R. Acad. Sci. Paris 189, 473-475 (1929; JFM 55.0192.07)]. In the context of separable and complete metric spaces X without isolated points, a continuous map \(f:X\rightarrow X\) is transitive if and only if it admits a dense orbit \(\{ x,f(x),f^2(x),\dots \}\) (i.e., f is hypercyclic). \textit{S. I. Ansari} [J. Funct. Anal. 148, No. 2, 384-390 (1997; Zbl 0898.47019)] and \textit{L. Bernal-Gonzales} [Proc. Am. Math. Soc. 127, No. 4, 1003-1010 (1999; Zbl 0911.47020)] independently showed that every (infinite dimensional) separable Banach space admits a hypercyclic operator. This result was extended for Fréchet spaces by \textit{J. Bonet} and the reviewer [J. Funct. Anal. 159, No. 2, 587-595 (1998; Zbl 0926.47011)]. The natural question that remained open was whether every non-separable Banach (or Fréchet) space admits a transitive operator. NEWLINENEWLINENEWLINEThe authors show in the paper under review that this is not the case by proving that, for every linear operator \(T:X\rightarrow X\) on a non-reflexive quotient of a von Neumann algebra X, the point spectrum of \(T^*\) is non-empty (which in particular implies that T is not transitive). Remarkable examples are \(X=\ell^\infty\), the space of bounded sequences, and \(X=L(\ell^2)\), the space of bounded operators on the Hilbert space \(\ell^2\) of 2-summable sequences.