Suppose that $A$ is a finite dimensional unital commutative Banach algebra and $A\hat{\otimes} A$ is its Projective tensor product by itself. Define $\Delta:A\hat{\otimes} A\to A$ with 
$$\Delta(\sum_{i=1}^\infty a_i\otimes b_i)=\sum_{i=1}^\infty a_i b_i.$$
There is $m\in A\hat{\otimes} A$ with 
$$a\Delta(m)=a,\qquad (a\in A)$$
Also there is a net $\{m_\alpha\}\subset A\hat{\otimes} A$ such that $\|m_\alpha-m\|\to 0$ and there is $K>0$ such that 
$$a\Delta(m_\alpha)\to a,\qquad  \|a\Delta(m_\alpha)\|\leq K\|a\|,\qquad \|a\cdot m_\alpha-m_\alpha\cdot a\|\leq K\|a\|,\qquad (a\in A)$$
Could we conclude that $K-\|m\|\geq C$, for some non-negative real number $C$ ?