Finite dimensionality of certain $C^{\star}$-algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:21:38Z http://mathoverflow.net/feeds/question/36184 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/36184/finite-dimensionality-of-certain-c-star-algebras Finite dimensionality of certain $C^{\star}$-algebras Andreas Thom 2010-08-20T13:03:26Z 2011-01-29T12:15:09Z <p>In the discussion about the question <a href="http://mathoverflow.net/questions/35207/finite-dimensional-subalgebras-of-c-star-algebras" rel="nofollow">Finite-dimensional subalgebras of $C^{\star}$-algebras</a> the following separate question came up:</p> <p>Let <code>$H$</code> be a Hilbert space and <code>$a_1, \dots, a_n \in B(H)$</code> 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 $p(z,x_1,\dots,x_n)=0$. </p> <blockquote> <p><strong>Question:</strong> Is the subalgebra of $B(H)$ which is generated by the operators $a_1 , \dots, a_n$ finite dimensional? </p> </blockquote> http://mathoverflow.net/questions/36184/finite-dimensionality-of-certain-c-star-algebras/36671#36671 Answer by Ron for Finite dimensionality of certain $C^{\star}$-algebras Ron 2010-08-25T13:51:24Z 2010-08-25T13:51:24Z <p>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.</p> <p>Ron</p>