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We'll analyse some cases where we have equality. Consider $A = M_{n}(\mathbb{C})$ with the operator norm induced by the $L^2$ norm on $\mathbb{C}^n$. The norm of a matrix $a$ is $\max ( \sigma_l)$ the largest of the singular values of $a$. Assume that $a$ is invertible. Then the norm of $a^{-1}$ is $\max( \sigma_l^{-1})$. Therefore $$||a|| = \max ...


Assume that $A$ is unital (otherwise we can take $A$ to be zero-dimensional). We need to prove that every non-zero element $x$ is invertible and then apply Gelfand-Mazur theorem. The set of non-invertible elements is closed. If it contains any non-zero element $x$ then its boundary also contains some non-zero element, call it $y$. Since $y$ is a boundary ...


Hint:$|f(x)g(x)|=|f(x)|\,|g(x)|\leq\|f\|_{\infty} |g(x)|,\forall x\in X\,\,\,$


The property does not follow from the others, but is useful and satisfied by enough examples to make it often worth assuming. Aside from C*-algebras, $L^1$ algebras on locally compact groups satisfy this definition. One of the useful consequences of the property is that it implies that $*$-representations of Banach $*$-algebras on Hilbert space are ...

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