Derivations on Toeplitz Algebras

Abstract

AbstractLet H2(Ω) be the Hardy space on a strictly pseudoconvex domain Ω ⊂ ℂn, and let A ⊂ L∞(∂Ω) denote the subalgebra of all L∞-functions ƒ with compact Hankel operator Hƒ. Given any closed subalgebra B ⊂ A containing C(Ω), we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra 𝒯(B) ⊂ B(H2(Ω). In particular, we show that every derivation on 𝒯(A) is inner. These results are new even for n = 1, where it follows that every derivation on T(H∞ +C) is inner, while there are non-inner derivations on T(H∞ + C(∂ℝn)) over the unit ball Bn in dimension n > 1.