Remark 2

Let M be a left A-module. The B-linear mappings

$\beta_{\mathscr{H}(M)}:\mathscr{H}(M)\rightarrow \mathscr{H}(\mathscr{T}(\mathscr{H}(M)))$ and $\mathscr{H}(\alpha_M) :\mathscr{H}(\mathscr{T}(\mathscr{H}(M)))\rightarrow \mathscr{H}(M)$ satisfy the relation $\mathscr{H}(\alpha_M)\circ \beta_{\mathscr{H}(M)}= 1_{\mathscr{H}(M)}$. They are not bijective in general.

Let V be a left B-module. The A-linear mappings

$\mathscr{T}(\beta_V) :\mathscr{T}(V)\rightarrow \mathscr{T}(\mathscr{H}(\mathscr{T}(V)))$ and $\alpha_{\mathscr{T}(V)}:\mathscr{T}(\mathscr{H}(\mathscr{T}(V)))\rightarrow \mathscr{T}(V)$ satisfy the relation $\alpha_{\mathscr{T}(V)}\circ \mathscr{T}(\beta_V) = 1_{\mathscr{T}(V)}$. They are not bijective in general.