In the discussion about the question Finite-dimensional subalgebras of $C^{\star}$-algebras the following separate question came up:
Suppose you have a
Let $C^{\star}$-algebra generated by self-adjoint elements H$ be a Hilbert space and $g_1 , a_1, \dots, g_n$a_n \in B(H)$ be self-adjoint operators. Consider the elements operators $x_1g_1+x_2g_2+\dots x_1a_1+x_2a_2+\dots + x_n g_n$ a_n$ , where the $x_i$'s are complex variables . If the spectra of these elements are the solutions of a polynomial equation in $x_1 , \dots, x_n$, is the algebra finite dimensional? More precisely, we want to and assume that there is a polynomial $p(z,x_1,\dots,x_n)$ p(z,x_1,\dots,x_n) \in \mathbb C[z,x_1,\dots,x_n]$ such that $z$ is in the spectrum of $x_1g_1+x_2g_2+\dots x_1a_1+x_2a_2+\dots + x_n g_n$ 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?
