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)
Ali Taghavi
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