Almost commuting unitaries with spectral gap are near commuting unitaries

Almost Commuting Unitaries with Spectral Gap Are near Commuting Unitaries

Let Mn be the collection of n×n complex matrices equipped with operator norm. Suppose U, V ∈ Mn are two unitary matrices, each possessing a gap larger than ∆ in their spectrum, which satisfy ‖UV −V U‖ ≤ ǫ. Then it is shown that there are two unitary operators X and Y satisfying XY −Y X = 0 and ‖U−X‖+‖V −Y ‖ ≤ E(∆/ǫ) “ ǫ ∆2 ” 1 6 , where E(x) is a function growing slower than x 1 k for any posit...

Almost-commuting matrices are almost jointly diagonalizable

We study the relation between approximate joint diagonalization of self-adjoint matrices and the norm of their commutator, and show that almost commuting self-adjoint matrices are almost jointly diagonalizable by a unitary matrix.

All Unitaries Having Operator Schmidt Rank 2 are Controlled Unitaries

Almost commuting elements in compact Lie groups

C∗-pseudo-multiplicative unitaries

We introduce C∗-pseudo-multiplicative unitaries and (concrete) Hopf C∗-bimodules, which are C∗-algebraic variants of the pseudo-multiplicative unitaries on Hilbert spaces and the Hopf-von Neumann-bimodules studied by Enock, Lesieur, and Vallin [5, 6, 4, 10, 19, 20]. Moreover, we associate to every regular C∗-pseudo-multiplicative unitary two Hopf-C∗bimodules and discuss examples related to loca...