I would also invite people to check out Chu spaces (http://chu.stanford.edu/ http://chu.stanford.edu/). They are essentially matrices with closure under norm (m

I would also invite people to check out Chu spaces (http://chu.stanford.edu/ http://chu.stanford.edu/). They are essentially matrices with closure under norm (meaning you always have an inverse). This puts them on the same side as probability, linear logic, C*-algebra, and constructive mathematics.