Projective Spectrum and Cyclic Cohomolgy

Abstract

For a tuple $A=(A_1,A_2,...,A_n)$ of elements in a unital algebra $\mathcal{B}$ over $\mathbb{C}$, its {\em projective spectrum} $P(A)$ or $p(A)$ is the collection of $z\in {\mathbb{C}}^n$, or respectively $z\in {\mathbb{P}}^{n-1}$ such that the multi-parameter pencil $A(z)=z_1{A}_1+z_2{A}_2+\cdots +z_n{A}_n$ is not invertible in $\mathcal{B}$. $\mathcal{B}$-valued $1$-form $A^{-1}(z)dA(z)$ contains much topological information about $P^c(A):={\mathbb{C}}^n\setminus P(A)$. In commutative cases, invariant multi-linear functionals are effective tools to extract that information. This paper shows that in non-commutative cases, the cyclic cohomology of $\mathcal{B}$ does a similar job. In fact, a Chen-Weil type map $\kappa$ from the cyclic cohomology of $\mathcal{B}$ to the de Rham cohomology $H_d^{\ast }(P^c(A),\mathbb{C})$ is established. As an example, we prove a closed high-order form of the classical Jacobi's formula.

References in corpus (2)

• Proper Analytic Free Maps
• Interpolation in the noncommutative Schur-Agler class