List questions from c-star-algebras

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For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$

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Closed unit ball of $B(H)$ with wot topology is compact

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Every normal operator on a separable Hilbert space has a square root that commutes with it

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Show that a nondegenerate -Banach algebra is a C-algebra

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Spectral measure of a unitary operator

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Irreducible C*-algebra $A$ implies that projection $p$ is rank one if $pAp=\Bbb C p$

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List Question

Positive Elements: Norm (Decomposition)

States: Approximate Unit

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$

Closed unit ball of $B(H)$ with wot topology is compact

States: Liouvilleans

On the weak closure of a sequence of projections

Every normal operator on a separable Hilbert space has a square root that commutes with it

Show that a nondegenerate -Banach algebra is a C-algebra

Von Neumann algebra generated by a subalgebra

Spectral measure of a unitary operator

What is so special about C*-Algebras?

Irreducible C*-algebra $A$ implies that projection $p$ is rank one if $pAp=\Bbb C p$

States: KMS-Condition

Compact operator space is the greatest ideal of $B(H)$

Hilbert- Schmidt class is an ideal

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