Let $A$ be a (possibly non-unital) algebra over $\mathbb C$. We say that $A$ is self-induced if the product map $m:A \otimes_A A \rightarrow A$ is an isomorphism. Here $A \otimes_A A$ is the balanced tensor product, the quotient of $A\otimes A$ by the linear span of elements of the form $ab\otimes c - a\otimes bc$.
This notion seems to have been introduced by Gronbaek in Morita equivalence for self-induced Banach algebras and was further studied by Meyer in Smooth and rough modules over self-induced algebras. If $A$ is unital, or more generally, has local, one-sided, units, then $A$ is self-induced. I am interested in non-trivial examples of $A$ which are not self-induced.
What is an example of $A$ which is not self-induced, but such that the product map is surjective, and such that the product is non-degenerate (so for each non-zero $a\in A$ there are $b,c\in A$ with $ba\not=0, ac\not=0$).
I have tagged this functional analysis, as the notion seems to have arisen in the context of topological algebras (and so people working in this area might know examples). But the question does not ask about the topological case. I would be interested to know if this idea is studied in non-topological contexts under a different name?