Invertible bimodules and projectivity

Question

Let $A$ be a noncommutative algebra over a field, say $\mathbb{C}$ or $\mathbb{R}$, and let $L$ be a bimodule over $A$. If $L$ is invertible, that is, if the dual right $A$-module $L^*$ satisfies $$ L^* \otimes_A L \simeq A $$ then will $L$ necessarily be projective a left module over $A$?

Answer 1

Yes. If $\varphi :L^*\otimes _A L\rightarrow A$ is an isomorphism, there exists an element $t =\sum\limits_{i=1}^{n} x_i^*\otimes x_i$ of $L^*\otimes _A L$ such that $\varphi (t )=1$. For any $x\in L$, we have then $x\varphi (t)=x$, that is, $\sum_i \langle x,x_i^*\rangle x_i=x$. Now consider the homomorphisms $u:A^n\rightarrow L$ and $v:L\rightarrow A^n$ defined by $u(e_i)=x_i$ and $v(x)=(\langle x,x_1^*\rangle,\ldots ,\langle x,x_n^*\rangle)$; we have $u\circ v=\operatorname{Id}_L $, thus $L$ is a direct summand of $A^n$, hence projective.