Toeplitz Algebras over Fock and Bergman Spaces

In this paper, we investigate Toeplitz subalgebras generated by certain class of Toeplitz operators on the p -Fock space and the p -Bergman space with 1 < p < infinity . Let BUC( C (n) ) and BUC( B (n) ) denote the collections of bounded uniformly continuous functions on C n and B n (the unit ball in C-n), respectively. On the p -Fock space, we show that the Toeplitz algebra which has a translation -invariant closed subalgebra of BUC( C-n ) as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols. This answers an open question recently raised by Fulsche [3]. On the p -Bergman space, we study Toeplitz algebras with symbols in some translationinvariant closed subalgebras of BUC( B (n) ). In particular, we obtain that the Toeplitz algebra generated by all Toeplitz operators with symbols in BUC( B (n) ) is equal to the closed linear space generated by Toeplitz operators with such symbols. This generalizes the corresponding result for the case of p = 2 obtained by Xia [11], which was proven by a di ff erent approach.