Derivations on Toeplitz Algebras

Let H-2(Omega) be the Hardy space on a strictly pseudoconvex domain Omega subset of C-n, and let A subset of L-infinity(partial derivative Omega) denote the subalgebra of all L-infinity-functions f with compact Hankel operator H-f. Given any closed subalgebra B subset of A containing C(partial derivative Omega), we describe the first Hochschild cohomology group of the corresponding Toeplitz algebra J(B) subset of B(H-2(Omega)). In particular, we show that every derivation on J(A) is inner. These results are new even for n = 1, where it follows that every derivation on J(H-infinity + C) is inner, while there are non-inner derivations on J(H-infinity + C(partial derivative B-n)) over the unit ball B-n in dimension n > 1.