  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}.
