We prove an operator-theoretic reconstruction of the Beilinson-Bloch regulator for compact Riemann surfaces, using loop operators and the Connes-Karoubi character for Fredholm modules. The proof includes a new computation of the Connes-Karoubi character for Steinberg symbols of the circle, which relies on the Helton-Howe determinant theory, but not on the Carey-Pincus theory of joint torsion.
A key computation in our work is the evaluation of the Connes--Karoubi character @math on Steinberg symbols @math . This is known to equal the joint torsion of Toeplitz operators [Prop. 1] Carey-Pincus:1999 , thanks to a combination of recent results of Kaad @cite_8 and Migler @cite_26 that enable the joint torsion, in this case, to be matched with the Connes--Karoubi character. (For details, see the remark at the end of .) However, since our work does not concern joint torsion, we choose to execute a direct, and considerably shorter, computation of @math , which employs the determinant theory of Helton and Howe @cite_4 , but is otherwise fairly elementary. The validity of the reconstruction formula for the full @math -group is, to the best of our knowledge, a new result.
@cite_26 @cite_4 @cite_8
1687736749 2075162834 1546311336
A determinant in algebraic @math -theory is associated to any two almost commuting Fredholm operators. On the other hand, one can calculate a homologically defined invariant known as joint torsion. We answer in the affirmative a conjecture of Richard Carey and Joel Pincus, namely that these two invariants agree. In particular, this implies that joint torsion is norm continuous, depends only on the images of the operators modulo trace class, and satisfies the expected Steinberg relations. Moreover, we show that the determinant invariant of two commuting operators can be computed simply as a determinant on a finite dimensional vector space. In the context of 2-summable Fredholm modules, we prove that the Connes-Karoubi multiplicative character coincides with Brown's determinant invariant on algebraic K -theory. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation.