An $E_0$-semigroup of $B(H)$ is a one parameter strongly continuous semigroup of unital $*$-endomorphisms of $B(H)$. The classification of E$_0$-semigroups up to cocycle conjugacy remains an intriguing problem. In this talk we will discuss it in a slightly different guise: the search for a rich class where classification is possible.

Robert T. Powers showed that every $E_0$-semigroup that possesses a strongly continuous intertwining semigroup of isometries arises (up to cocycle conjugacy) from a boundary weight map over $K$ separable Hilbert space. The class has attracted our attention is that of $q$-pure boundary weight maps, i.e.\ those which have totally ordered sets of subordinates.

As we have shown previously, the $q$-pure class is classifiable in the appropriate sense when $K$ is finite-dimensional: as it turns out that the case of range rank one boundary weight maps tell the whole story.

The situation is still elusive in this class when $K$ is infinite dimensional. We discuss some new examples which indicate some new phenomena. They are not $q$-pure, but seem to indicate that the range rank one case may not tell the whole story when $\dim K =\infty$.

This talk is based on joint work in progress with C. Jankowski and R.T.~Powers, and the paper: C. Jankowski, D. Markiewicz and R.T. Powers, ``Classification of q-pure q-weight maps over finite dimensional Hilbert spaces", J. Funct. Anal. 277 (2019), no. 6, pp. 1763–1867.