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Let A be $C^*$-algebra with unit $e$ and $a\in A$ normal. We define $$alg(a,a^*)=\overline{ \{ \sum\limits_{k,l=0}^n\lambda_{k,l}a^k\overline{a}^l; \lambda_{k,l}\in\mathbb{C}, n\in\mathbb{N}\} \\}$$ to be the smallest $C^*$-subalgebra of A which contains $a$. $alg(a,a^*)$ is a commutative $C^*$-algebra with unit $a^0=e$.

Here are my questions related to this definition:

1.Is the non-unital version of $alg(a,a^*)$ the $C^*$-algebra $\overline{ \{ \sum\limits_{k,l=1}^n\lambda_{k,l}a^k\overline{a}^l; \lambda_{k,l}\in\mathbb{C}, n\in\mathbb{N}\} \\}$, i.e. all the polynoms without constant term, and is the unitarization of $\overline{ \{ \sum\limits_{k,l=1}^n\lambda_{k,l}a^k\overline{a}^l; \lambda_{k,l}\in\mathbb{C}, n\in\mathbb{N}\} \\}$ again $alg(a,a^*)$?


Now, let $V$ be a nonunital $C^*$-algebra, $v\in V$. In lecture we defined the spectrum of v : $\sigma_{V}(v):=\sigma_{V_1}(v)=\{\lambda\in\mathbb{C}; v-\lambda e_{V_1}\; \text{is not invertible in}\; V_1\}$. Here is $V_1$ the unitarization of V and $e_{V_1}$ it's unit. The following facts about the spectrum of elements are known (I will mention only this which could be important for my questions):

1)If $v\in V$, then $v$ is not invertible in $V_1$ and therefore $0\in \sigma_{V_1}(v)$.
2)If $V$ is unital, it is $\sigma_{V_1}(v)=\sigma_V(v)\cup \{0\}$.
3)If $V$ is unital, $\sigma_V(v)$ is a compact subset of $\mathbb{C}$.

  1. My question related to this definitions and properties are: Are $ \sigma_V(v)$ and $\sigma_{V_1}(v)$ compact subsets of $\mathbb{C}$ in both cases, if V is unital and if V is non-unital? I'm not sure in the non-unital case:

If $V$ is non-unital, by definition it is $\sigma_{V}(v)=\sigma_{V_1}(v)$ and the spectrum of an element in a unital $C^*$-algebra is a compact subset of $\mathbb{C}$. Or is $\sigma_V(v)$ only a locally compact, noncompact subset of the complex numbers in this case ? But then there is no relation of the one-point-compactification of $\sigma_V(v)$ and the unitalization of $V$ I would say, is it correct?
If $V$ is unital, here I would say that both are compact subsets, because of the property 3).

I appreciate your help. Regards

banach-c
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1 Answers1

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  1. Not exactly. You only need to omit the term with $k=l=0$. As for the unitization yes, if you algebra $B$ is embedded in a unital algebra $A$, then the unitization of $B$ is isomorphic to the subalgebra of $A$ generated by $B$ and $e$.

  2. $\sigma_V(v)$ is compact regardless; it is just defined as a the spectrum of $v$ in $V_1$, and this is a compact subset of $\mathbb C$.

Martin Argerami
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