Quasi-free subalgebras of the Toeplitz algebra

The Toeplitz algebra (i.e., the universal $C^*$-algebra $\mathcal T$ generated by an isometry) has several interesting dense locally convex subalgebras, for example, the algebraic Toeplitz algebra and the smooth Toeplitz algebra. Such subalgebras, as well as the Toeplitz algebra itself, play an important role in bivariant $K$-theory and in cyclic homology. The motivation for our talk comes from the fact (observed independently by R. Meyer and O. Yu. Aristov) that the algebraic Toeplitz algebra is quasi-free in the sense of Cuntz and Quillen. On the other hand, the $C^*$-Toeplitz algebra itself is not quasi-free by a general result of O. Yu. Aristov. Now a natural question is whether or not the smooth Toeplitz algebra is quasi-free. To answer this question, we introduce a family $\{ \mathcal T_{P,Q} \}$ of dense locally convex subalgebras of $\mathcal T$ indexed by Köthe power sets $P,Q$ satisfying some natural conditions. Our main result gives a sufficient condition for $\mathcal T_{P,Q}$ to be quasi-free in terms of $P$ and $Q$. As a corollary, we show that the smooth Toeplitz algebra and the holomorphic Toeplitz algebra are quasi-free.
This is a part of a joint project with O. Yu. Aristov.