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In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up:

Let $H$ be a Hilbert space and $a_1, \dots, a_n \in B(H)$ be self-adjoint operators. Consider the operators $x_1a_1+x_2a_2+\dots + x_n a_n$ , where the $x_i$'s are complex variables and assume that there is a polynomial $p(z,x_1,\dots,x_n) \in \mathbb C[z,x_1,\dots,x_n]$ such that $z$ is in the spectrum of $x_1a_1+x_2a_2+\dots + x_n a_n$ if and only if $p(z,x_1,\dots,x_n)=0$.

Question: Is the subalgebra of $B(H)$ which is generated by the operators $a_1 , \dots, a_n$ finite dimensional?

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1 Answer 1

This is an interesting question. The set is call multiparameter spectrum of the tuple g, it also called projective spectrum in my paper "". The paper didn't address this particular question, but said something in general.


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Thanks for the reference. – Andreas Thom Aug 25 '10 at 14:29

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