List questions from c-star-algebras

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Must there always be a normal element with non-discrete spectrum in an infinite-dimensional C*-algebra?

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If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

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Element $a$ of a C-algebra such that $f(a^a) = 0$ but $f(aa^*) \neq 0$ for some positive linear functional $f$

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List Question

Existence of faithful state in $C^\ast$-algebras

Existence of a completely supported probability measure

C*-algebras as Banach lattices?

A question about pure state

A problem on $C^\ast$-algebras and $W^\ast$-algebras

Must there always be a normal element with non-discrete spectrum in an infinite-dimensional C*-algebra?

If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

Element $a$ of a C-algebra such that $f(a^a) = 0$ but $f(aa^*) \neq 0$ for some positive linear functional $f$

How does this imply surjectivity? An application of the Tietze extension theorem.

Approximate identity in a $C^*$-algebra

Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

Reflexive Banach algebras?

Spatial tensor product of algebras and normed spaces

Simplicity of Cuntz algebra

Involutive and C* Banach Algebras.

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