According  to the interesting answer of Rado, trace  is  an  algebraic way to represent the dimension of fibres of  a  vector  bundle on a compact Hausdorff space $X$. Every  vector  bundle on $X$ corresponds to an idempotent  matrix  valued function $A(x),\;x\in X$ . The  dimension of  fibre of  vector bundle is equal to $tr(A(x))$. I learned this  from "Very basic  noncommutative  geometry"  By  Masoud  Khalkhali (Can be find in arxiv)