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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 $$a\Delta(m_\alpha)\to a,\qquad \|a\Delta(m_\alpha)\|\leq K\|a\|,\qquad (a\in A)$$ Could we conclude that $K-\|m\|\geq C$, for some non-negative real number $C$ ?

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 $C$ ?

Suppose that $A$ is a finite dimensional unital Banach algebra, $x\in A$ and $\{x_\alpha\}$ is a net (not necessarily sequence) in $A$ such that $\|x_\alpha-x\|\to 0$. Also we have a real number $K>0$ such that $$\|ax_\alpha-x_\alpha a\|\leq K\|a\|,\qquad (a\in A)$$ Could we conclude that $K\geq\|x\|$ ?

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 $$a\Delta(m_\alpha)\to a,\qquad \|a\Delta(m_\alpha)\|\leq K\|a\|,\qquad (a\in A)$$ Could we conclude that $K-\|m\|\geq C$, for some non-negative real number $C$ ?