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Projective spectrum and cyclic cohomology. (English) Zbl 1297.46051

Summary: For a tuple \(A = (A_1, A_2, \dots, A_n)\) of elements in a unital algebra \(\mathcal B\) over \(\mathbb C\), its 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 \(A(z) = z_1A_1 + z_2A_2 + \cdots + z_nA_n\) is not invertible in \(\mathcal B\). The \(\mathcal B\)-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 Chern-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 formula.

MSC:

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46H05 General theory of topological algebras
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
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