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Let $T$ $\in \mathfrak{B}(\mathbb{H})$ be normal. Let $A$ be the closed subalgebra generated by $T$, $T^{*}$ and $I$. Suppose $T$ can be approximated in norm by finite linear combinations of projections in $A$. What is the necessary and sufficient condition on $\sigma(T)$ for this to happen?

I am not sure how to approach this question. Do we use the spectral theorem for normal operators? Does it follow that since $T$ can be norm-approximated by linear combinations of projections, its spectrum must be finite? I would be grateful for some hints on how to start.

Arundhathi
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1 Answers1

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Have you seen that $A\cong C(\sigma(T))$? This is equivalent to asking for conditions on $\sigma(T)$ such that the function $z\mapsto z$ can be uniformly approximated by linear combinations of continuous characteristic functions on $\sigma(T)$, which is asking for a lot of clopen subsets of $\sigma(T)$ to exist. For example, this would hold if $\sigma(T)$ were the Cantor set.

(I noticed the problem asks for hints so I turned this comment into an answer. Related: Is a von Neumann algebra just a C*-algebra which is generated by its projections?)

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Jonas Meyer
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