The authors study the algebraic \(K\)-theory of algebras of operators in Hilbert space \(H\) having trace class commutators. To do this they create two new objects: “perturbation vectors” \(\sigma_{T,S}\) and the joint torsion \(\tau(S_{1},S_{2},T_{2},T_{1})\) defined for two such pairs when \(S_{1}S_{2}=T_{2}T_{1}\). The perturbation vectors are obtained through the construction of a section \(\sigma\) of a canonical determinant bundle. The main results of this article are the proofs of the existence of this section and the basic properties of the perturbation vectors \(\sigma_{T,S}\) which give a trivialization of the bundle. Let \({\mathcal F}\) denote the Fredholm operators on \(H\) and \(Q\rightarrow {\mathcal F} \) be the Quillen det bundle.
Form \({\mathcal Q}+{\mathcal Q}^{*}\) as a bundle over \({\mathcal F}\times{\mathcal F}\). Let \(\wp= \operatorname {det} ({\mathcal Q}+{\mathcal Q}^{*})\) be the determinant bundle and \(M\equiv\{ (S,T)\mid (S,T)\in {\mathcal F}\times{\mathcal F}, S-T\in {\mathcal L}^{1}(H)\}\) with \(i_{M}:M\rightarrow {\mathcal F}\times {\mathcal F}\) the inclusion map. The authors prove the existence of a trivialization of the pullback \(i^{*}_{M}(\wp)\) by construction of a section \(\sigma\), called the perturbation section. The thereby constructed “perturbation vectors” \(\sigma_{S,T}\) generalize the perturbation determinant \(\operatorname {det} (S^{-1}T)\), when the operators \(S\),\(T\) are singular. Then, for pairs of elements \((A,D)\) and \((B,C)\) in \(M\) with \(AB=CD\) there is a Koszul complex \(K(A,B;C,D)\) with two torsion vectors built from its homology whose tensor product is a scalar multiple of \(\sigma_{A,D}\otimes \sigma_{B,C}.\) This gives a number \(\tau(A,B,C,D;H) \in C^{*}\) called the joint torsion. It has previously been shown that if \(A\) and \(B\) are commuting operators in \({\mathcal F}\) then the Steinberg symbol determinant \(\operatorname {det} _{*}\circ \partial\{ A+ L^{1}(H), B+L^{1}(H)\} =\tau(A,B,B,A;H).\) Corresponding results are now proved in the non-commuting case, and the section \(\sigma\) and the joint torsion are used to examine the \(n\times n\) subdeterminants of Toeplitz matrices with non-zero index as \(n\rightarrow \infty\).