connection between unital $C^*$ algebra and finite von neumann algebra

Question

Let $A$ be a unital $C^*$-algebra with a tracial state $\tau$, $L^2(A,\tau)$ is the Hilbert space induced by the GNS constructtion.Suppose $\lambda$ is the left action of $A$ on $L^2(A,\tau)$ ,does there exist a projection $p$ and a finite Von Neumann algebra $\lambda(A)^{"}$ such that $A=p\lambda(A)^{"}$?

Answer 1

Since $A$ contains adjoints, you would have $A=p\lambda(A)''p$, so $A$ would be a von Neumann algebra. Thus the answer is no for any C$^*$-algebra that is not a W$^*$-algebra.