Factorization of Matrices of Quaternions

We review known factorization results in quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, the QR factorization. We prove there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature. Rather than work directly with matrices of quaternions, we work with complex matrices with a specific symmetry based on the dual operation. We discuss related results regarding complex matrices that are self-dual or symmetric, but perhaps not Hermitian.Comment: Corrected proofs of Theorem 2.4(2) and Theorem 3.

Principal angles and approximation for quaternionic projections

We extend Jordan's notion of principal angles to work for two subspaces of quaternionic space, and so have a method to analyze two orthogonal projections in M_n(A) for A the real, complex or quaternionic field (or skew field). From this we derive an algorithm to turn almost commuting projections into commuting projections that minimizes the sum of the displacements of the two projections. We quickly prove what we need using the universal real C*-algebra generated by two projections.Comment: 11 pages, 4 figures, 4 auxiliary Matlab file

Quantitative K-Theory Related to Spin Chern Numbers

We examine the various indices defined on pairs of almost commuting unitary matrices that can detect pairs that are far from commuting pairs. We do this in two symmetry classes, that of general unitary matrices and that of self-dual matrices, with an emphasis on quantitative results. We determine which values of the norm of the commutator guarantee that the indices are defined, where they are equal, and what quantitative results on the distance to a pair with a different index are possible. We validate a method of computing spin Chern numbers that was developed with Hastings and only conjectured to be correct. Specifically, the Pfaffian-Bott index can be computed by the "log method" for commutator norms up to a specific constant

The Point in Weak Semiprojectivity and AANR Compacta

We initiate the study of pointed approximative absolute neighborhood retracts. Our motivation is to generate examples of C*-algebras that behave in unexpected ways with respect to weak semiprojectivity. We consider both weak semiprojectivity (WSP) and weak semiprojectivity with respect to the class of unital C*-algebras (WSP1). For a non-unital C*-algebra, these are different properties. One example shows a C*-algebra can fail to be WSP while its unitization is WSP. Another example shows WSP1 is not closed under direct sums.Comment: Corrected the statement of Theorem 4.16(b

Estimating Norms of Commutators

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y] assuming that x and y are contractions in a C*-algebra A, with x normal and with spectrum within the domain of f. In particular we discuss [x^2, y] and [x^(1/2), y] for 0 <=, x <=, 1. For larger values of \delta = \|[x; y]\| we can rigorous calculate the best possible upper bound \|[f(x), y]\| for many f. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.Comment: We are posting the next version of this paper at : http://repository.unm.edu/handle/1928/23462. Also posted at http://repository.unm.edu is theMatlab code used to generate example