Journal of the
Korean Mathematical Society JKMS

We discuss a certain geometric property $X_{\theta, \gamma}$ of dual algebras generated by a polynomially bounded operator and property $({\bf A}_{\aleph_0,\aleph_0})$; these are central to the study of $\aleph_0 \times \aleph_0$-systems of simultaneous equations of weak*-continuous linear functionals on a dual algebra. In particular, we prove that if $T \in {\bf A}^M$ satisfies a certain sequential property, then $T \in \bf A^M_{\aleph_0}({\mathcal H}) $ if and only if the algebra ${\mathcal A}_T$ has property $X_{0, 1/M}$, which is an improvement of Li-Pearcy theorem in [8].

Keywords: polynomially bounded operator, dual operator algebra

MSC numbers: Primary 47D27; Secondary 47A53