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”.