Triviality of a trace on the space of commuting trace-class self-adjoint operators

TL;DR

This paper proves that any real-valued map on tuples of commuting trace-class self-adjoint operators that mimics the trace must be trivial, linking it to Milnor's K-group and continuous homomorphisms.

Contribution

It establishes the triviality of such trace-like maps by connecting them to Milnor's K-group, a novel insight in operator theory and algebraic K-theory.

Findings

Any such trace map must be trivial.

Connection established with Milnor's K-group.

Shows the non-existence of non-trivial trace-like maps.

Abstract

In the present article, we investigate a possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with continuous group homomorphisms from the Milnor's $K$-group of the real numbers into the additive group of real numbers. Using this connection, it is shown that any such trace map must be trivial.