Trace Class Operators, Regulators, and Assembly Maps in K-Theory

Let $G$ be a group and let $KH$ be homotopy algebraic $K$-theory. We prove that if $G$ satisfies the rational $KH$ isomorphism conjecture for the group algebra $L^1[G]$ with coefficients in the algebra of trace-class operators in Hilbert space, then it also satisfies the $K$-theoretic Novikov conjecture for the group algebra over the integers, and the rational injectivity part of the Farrell-Jones conjecture with coefficients in any number field.

2010 Mathematics Subject Classification: Primary 19D55, Secondary 19F27, 19K99.

Keywords and Phrases: Borel regulator, Homotopy algebraic K-theory, Multiplicative K-theory, trace-class operators.

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