## Recovering “$n$” from $M_n(\mathbb{C})$

Is there an example of an infinit dimensional $$C^*$$ algebra $$A$$ which admits the following structure:

The $$C^*$$ algebra $$A$$ admits a faithful trace $$tr$$ such that the multiplication $$m: A\otimes A \to A$$ would be a bounded operator between pre Hilbert spaces (with respect to the inner product $$tr(ab^*)$$ on $$A$$ and its natural extension to the algebraic tensor product $$A\otimes A$$) such that $$mm^*=\lambda Id$$ for some $$\lambda \in \mathbb{C}$$, where $$m^*$$ is the adjoint of $$m$$ after completion of the above pre Hilbert space? Is there an example of this situation for which $$\lambda$$ is not necessarilly an integer number?

Motivation from finit dimensional case: For $$A=M_n(\mathbb{C})$$ with the standard trace we have $$\lambda=n$$ so this process recover $$n$$ from $$M_n(\mathbb{C})$$ that is $$mm^*=nId$$. Namely the matric multiplication is a rescaled “Partial Isometry”.