Double centralizer properties

Question

Let $A$ be a finite dimensional algebra and $M$ a right $A$-module. Then $M$ is a left $B=End_A(M)$-module. $M$ is said to have the double centralizer property in case the canonical map given by right multiplication $F: A \rightarrow End_{B^{op}}(M)^{op}$ is an isomorphism (of algebras). That $F$ is injective means that $M$ is faithful as an $A$-module, meaning $Ma \neq 0$ for all nonzero $a \in A$. The most famous situation is when $M=fA$ for some idempotent $f$, such that $fA$ is minimal faithful projective-injective. Examples include Schur algebras and blocks of category $\mathcal{O}$. Questions: Let $M=fA$ for some idempotent $f$ (and add some properties if needed, such as M being injective or...)

• Is there an example where the two algebras $A$ and $End_{B^{op}}(M)^{op}$ are isomorphic as algebras but there is no double centralizer property? (answer is no,see comments in the answer)
• Is there an example where the two algebras $A$ and $End_{B^{op}}(M)^{op}$ have the same dimension as a vector-space but there is no double centralizer property? (Answer is yes: see the answer by Dag Oskar Madsen.)

Answer 1

An example for the second question: Let $A$ be the $4$-dimensional algebra $$A=\begin{pmatrix} \Bbbk & 0\\ \Bbbk & \Bbbk \end{pmatrix} \times \Bbbk$$ and $$f=(\begin{pmatrix} 0 & 0\\ 0 & 1 \end{pmatrix},0)$$ Then $B={\rm{End}}_A(fA) \cong \Bbbk$ and $${\rm{End}}_{B^{\rm{op}}}(fA)^{\rm{op}}={\rm{End}}_{\Bbbk}(\Bbbk^2) \cong M_2(\Bbbk),$$ which is also $4$-dimensional but not isomorphic to $A$.