In this paper we characterize when the semi-commutator TfTg − Tfg of two Toeplitz operators Tf and Tg on the Hardy space of the bidisc is zero. We also show that there is no nonzero finite rank semi-commutator on the bidisc. Furthermore explicit examples of compact semi-commutators with symbols continuous on the bitorus T 2 are given.

We discuss bipermutive cellular automata from a combinatorial and topological perspective. We prove a type of topological randomizing property for bipermutive CA, show that the commutator of a bipermutive CA is always small and that bipermutive affine CA have only affine CA in their commutator. We show the last result also in the multidimensional case, proving a conjecture of [Moore-Boykett, 97].

These brief remarks have been prepared in connection with a conference in honor of my thesis advisor, Richard Rochberg.

It is shown that there exists a finitely generated infinite simple group of infinite commutator width and infinite square width in which every conjugation-invariant norm is stably bounded, and in particular stable commutator length vanishes. Moreover, a recursive presentation of such a group with decidable word and conjugacy problems is constructed.

We consider the vector-valued extension of the Fefferman–Stein– Strömberg sharp maximal inequality under growth condition. As an application we obtain a vector-valued extension of the boundedness of the commutator. Furthermore, we prove the boundedness of the commutator. 2000 Mathematics Subject Classification: Primary 42B35; Secondary 42B25.

The basic higher order commutator theorem is formulated for the real interpolation methods associated with the quasi-power parameters, that is, the function spaces on which Hardy inequalities are valid. This theorem unifies and extends various results given by Cwikel, Jawerth, Milman, Rochberg, and others, and incorporates some results of Kalton to the context of commutator estimates for the re...

6 Lie Derivatives and the Commutator Revisited 21 6.1 Integral Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 21 6.2 Congruence of Curves . . . . . . . . . . . . . . . . . . . . . . . 21 6.3 The Commutator Revisited: A Geometric Interpretation . . . 22 6.4 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6.5 Lie Derivatives of a Function . . . . . . . . . . . ....

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I , and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis. Let A be a finite set and let K be a f...