Toeplitz and Hankel operators associated with subdiagonal algebras

Abstract:

Let $\mathcal M$ be a $\sigma$-finite von Neumann algebra and let $\mathcal A\subset \mathcal M$ be a maximal subdiagonal algebra with respect to some faithful normal expectation $\mathcal E$ on $\mathcal M.$ Let $\phi$ be a normal faithful $\mathcal E$-invariant state on $\mathcal M$, let $L^2(\mathcal M,\phi )$ be the non-commutative Lebesgue space in the sense of U. Haagerup, and consider the Hardy space $H^2(\mathcal A,\phi )\subset L^2(\mathcal M,\phi )$ associated with the pair $(\mathcal A,\phi ).$ For each $x\in \mathcal M$, the Toeplitz operator $T_x\in B(H^2(\mathcal A,\phi ))$ and the Hankel operator $H_x\in B(H^2(\mathcal A,\phi ),H^2(\mathcal A,\phi )^\perp )$ are defined as in the classical case of the unit circle. We show that the mapping $x\mapsto T_x$ is completely isometric on $\mathcal M$ and therefore $\sigma (x)\subset \sigma (T_x)$ for all $x\in \mathcal M.$ We also show that $\|H_x\|=dist(x,\mathcal A)$ for every $x\in \mathcal M.$