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LSpice
LSpice
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Let $(A,\,^\ast,\Vert\cdot\Vert)$$(A,\,^\ast,\lVert\cdot\rVert)$ be a Banach $\ast$-algebra with the property that $\Vert a \Vert^2=\rho(a^\ast a)$$\lVert a \rVert^2=\rho(a^\ast a)$ holds for all $a\in A$, where $\rho(x)$ denotes the spectral radius of an element $x\in A$.

Is $(A,\,^\ast,\Vert\cdot\Vert)$$(A,\,^\ast,\lVert\cdot\rVert)$ then a $C^\ast$-algebra?

Let $(A,\,^\ast,\Vert\cdot\Vert)$ be a Banach $\ast$-algebra with the property that $\Vert a \Vert^2=\rho(a^\ast a)$ holds for all $a\in A$, where $\rho(x)$ denotes the spectral radius of an element $x\in A$.

Is $(A,\,^\ast,\Vert\cdot\Vert)$ then a $C^\ast$-algebra?

Let $(A,\,^\ast,\lVert\cdot\rVert)$ be a Banach $\ast$-algebra with the property that $\lVert a \rVert^2=\rho(a^\ast a)$ holds for all $a\in A$, where $\rho(x)$ denotes the spectral radius of an element $x\in A$.

Is $(A,\,^\ast,\lVert\cdot\rVert)$ then a $C^\ast$-algebra?

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B K
B K
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Can C*-algebras be characterized among Banach *-algebras by the spectral radius?

Let $(A,\,^\ast,\Vert\cdot\Vert)$ be a Banach $\ast$-algebra with the property that $\Vert a \Vert^2=\rho(a^\ast a)$ holds for all $a\in A$, where $\rho(x)$ denotes the spectral radius of an element $x\in A$.

Is $(A,\,^\ast,\Vert\cdot\Vert)$ then a $C^\ast$-algebra?