Toeplitz algebras over Fock and Bergman spaces

In this paper, we study Toeplitz algebras generated by certain class of Toeplitz operators on the $p$-Fock space and the $p$-Bergman space with $1<p<\infty$. Let BUC($\mathbb C^n$) and BUC($\mathbb B_n$) denote the collections of bounded uniformly continuous functions on $\mathbb C^n$ and $\mathbb B_n$ (the unit ball in $\mathbb C^n$), respectively. On the $p$-Fock space, we show that the Toeplitz algebra which has a translation invariant closed subalgebra of BUC($\mathbb C^n$) as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols. This answers a question recently posed by Fulsche \cite{Robert}. On the $p$-Bergman space, we study Toeplitz algebras with symbols in some translation invariant closed subalgebras of BUC($\mathbb B_n)$. In particular, we obtain that the Toeplitz algebra generated by all Toeplitz operators with symbols in BUC($\mathbb B_n$) 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 \cite{Xia2015}.