Groups and Operator Algebras Seminar
Speaker: Adam Dor-On (Haifa University)
Title: Distance to commuting unitary matrices
Abstract: A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to genuinely commuting matrices of the same kind. It was quickly realized that dimension independent results were far more difficult to obtain, and in work of Lin the result was settled for self-adjoint matrices. However, the precise behavior in terms of the commutator was unknown until the seminal work of Kachkovskiy and Safarov where effective bounds were achieved.
On the other hand, it has long been known that there are obstruction for dimension independent approximately commuting unitary matrices to be close to commuting unitary matrices. Following work of Gong and Lin, in work of Eilers, Loring and Pedersen it was finally shown that when an index obstruction vanishes, approximately commuting unitary matrices are close to commuting unitary matrices. However, effective bounds in terms of the commutator of unitary matrices are still unknown.
In this talk I will report on joint work in progress with Hall and Kachkovskiy where we show that under the vanishing of the same index obstruction, we can find effective bounds for the distance to commuting unitary matrices in terms of the commutator of the original matrices.