Finite dimensionality of certain $C^{\star}$-algebras

Question

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?

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 "http://www.worldscinet.com/jta/01/0103/S1793525309000126.html". The paper didn't address this particular question, but said something in general.

Ron