Using the Freese-McKenzie commutator theory for congruence modular varieties as the starting point, we develop commutator theory for the variety of loops. The fundamental theorem of congruence commutators for loops relates generators of the congruence commutator to generators of the total inner mapping group. We specialize the fundamental theorem into several varieties of loops, and also discus...

In a free group no nontrivial commutator is a square. And in the free group F2 = F (x1, x2) freely generated by x1, x2 the commutator [x1, x2], is never the product of two squares in F2, although it is always the product of three squares. Let F2,3 = 〈x1, x2〉 be a free nilpotent group of rank 2 and class 3 freely generated by x1, x2. We prove that in F2,3 = 〈x1, x2〉, it is possible to write cert...

For countably generated ideals, J , of B(H), geometric stability is necessary for the canonical spectral characterization of sums of (J ; B(H)){commutators to hold. This answers a question raised by Dykema, Figiel, Weiss and Wodzicki. There are some ideals, J , having quasi{nilpotent elements that are not sums of (J ; B(H)){commutators. Also, every trace on every geometrically stable ideal is a...

In this paper, we obtain bounded criteria on certain weighted Morrey spaces for the commutators generalized by some sublinear integral operators and weighted Lipschitz functions. We also present bounded criteria for commutators generalized by such sublinear integral operators and weighted BMO function on the weighted Morrey spaces. As applications, our results yield the same bounded criteria fo...

Let T be the Calderón-Zygmund operator, Coifman, Rochberg and Weiss (see [4]) proves that the commutator [b, T ](f) = bT (f) − T (bf)(where b ∈ BMO(R)) is bounded on L(R) for 1 < p <∞. Chanillo (see [2]) proves a similar result when T is replaced by the fractional operators. In [8, 16], Janson and Paluszynski study these results for the Triebel-Lizorkin spaces and the case b ∈ Lipβ(R), where Li...

In this paper it is shown that if T ∈ L(H) satisfies (i) T is a pure hyponormal operator; (ii) [T ∗, T ] is of rank-two; and (iii) ker [T ∗, T ] is invariant for T , then T is either a subnormal operator or the Putinar’s matricial model of rank two. More precisely, if T |ker [T∗,T ] has the rank-one self-commutator then T is subnormal and if instead T |ker [T∗,T ] has the ranktwo self-commutato...

We combine concepts from random matrix theory and free probability together with ideas from the theory of commutator length in groups and maps from surfaces, and establish new connections between the two. More particularly, we study measures induced by free words on the unitary groups U (n). Every word w in the free group Fr on r generators determines a word map from U (n) to U (n), defined by ...