Let ${\mathcal G}$ be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let $G= {\mathcal G}(k)$. We prove that if $\gamma\in G$ such that γ is a commutator and $\delta\in G$ such that $\langle \delta\rangle= \langle \gamma\rangle$ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.

A new characterization of $\text {CMO}(\mathbb R^n)$ is established replying upon local mean oscillations. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order commutators.

In 1981, Takeuti introduced quantum set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed linear subspaces of a Hilbert space in a manner analogous to Boolean-valued models of set theory, and showed that appropriate counterparts of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) hold in the model. In this paper, we aim at unifying Takeuti’s model with Boolean-valued models by constructing models based on general complete orthomodular lattices, and generalizing the transfer principle in Boolean-valued models, which asserts that every theorem in ZFC set theory holds in the models, to a general form holding in every orthomodular-valued model. One of the central problems in this program is the well-known arbitrariness in choosing a binary operation for implication. To clarify what properties are required to obtain the generalized transfer principle, we introduce a class of binary operations extending the implication on Boolean logic, called generalized implications, including even nonpolynomially definable operations. We study the properties of those operations in detail and show that all of them admit the generalized transfer principle. Moreover, we determine all the polynomially definable operations for which the generalized transfer principle holds. This result allows us to abandon the Sasaki arrow originally assumed for Takeuti’s model and leads to a much more flexible approach to quantum set theory.

In this paper, we study the integer sequence $(E_{n})_{n\geq 1}$ , where $E_{n}$ counts the number of possible dimensions for centralisers of $n\times n$ matrices. We give an example to show another combinatorial interpretation of $E_{n}$ and present an implicit recurrence formula for $E_{n}$ , which may provide a fast algorithm for computing $E_{n}$ . Based on the recurrence, we obtain the asymptotic formula $E_{n}=\frac{1}{2}n^{2}-\frac{2}{3}\sqrt{2}n^{3/2}+O(n^{5/4})$ .

Let ${{T}_{\Omega }}$ be the singular integral operator with kernel $\left( \Omega \left( x \right) \right)/{{\left| x \right|}^{n}}$, where $\Omega $ is homogeneous of degree zero, has mean value zero, and belongs to ${{L}^{q}}\left( {{S}^{n-1}} \right)$ for some $q\,\in \,\left( 1,\,\infty \right]$. In this paper, the authors establish the compactness on weighted ${{L}^{p}}$ spaces and the Morrey spaces, for the commutator generated by $\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ function and ${{T}_{\Omega }}$. The associated maximal operator and the discrete maximal operator are also considered.

Let $b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$ and ${{T}_{\Omega }}$ be the singular integral operator with kernel $\Omega \left( x \right)/{{\left| x \right|}^{n}}$, where $\Omega$ is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere ${{S}^{n-1}}$. In this paper, using Fourier transform estimates and approximation to the operator ${{T}_{\Omega }}$ by integral operators with smooth kernels, it is proved that if $b\,\in \,\text{CMO}\left( {{\mathbb{R}}^{n}} \right)$ and $\Omega$ satisfies certain minimal size condition, then the commutator generated by $b$ and ${{T}_{\Omega }}$ is a compact operator on ${{L}^{p}}\left( {{\mathbb{R}}^{n}} \right)$ for appropriate index $p$. The associated maximal operator is also considered.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.

Under the assumption that $\mu $ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\,\le \,y$ on bounded operators is our model for a definition of ${{C}^{*}}$-relations being residually finite dimensional.

Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices.

Applications are shown regarding norms of exponentials, the norms of commutators, and “positive” noncommutative $*$-polynomials.

We study Markov measures and p-adic random walks with the use of states on the Cuntz algebras Op. Via the Gelfand–Naimark–Segal construction, these come from families of representations of Op. We prove that these representations reflect selfsimilarity especially well. In this paper, we consider a Cuntz–Krieger type algebra where the adjacency matrix depends on a parameter q ( q=1 is the case of Cuntz–Krieger algebra). This is an ongoing work generalizing a construction of certain measures associated to random walks on graphs.

In this paper, we discuss the H1L-boundedness of commutators of Riesz transforms associated with the Schrödinger operator L=−△+V, where H1L(Rn) is the Hardy space associated with L. We assume that V (x) is a nonzero, nonnegative potential which belongs to Bq for some q>n/2. Let T1=V (x)(−△+V )−1, T2=V1/2(−△+V )−1/2 and T3 =∇(−△+V )−1/2. We prove that, for b∈BMO (Rn) , the commutator [b,T3 ] is not bounded from H1L(Rn) to L1 (Rn) as T3 itself. As an alternative, we obtain that [b,Ti ] , ( i=1,2,3 ) are of (H1L,L1weak) -boundedness.

Call a non-Moufang Bol loop minimally non-Moufang if every proper subloop is Moufang and minimally nonassociative if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to 1. In the process, we derive many commutator and associator identities which hold in such loops.

In this paper we investigate the boundedness on Hardy spaces for the higher order commutator Tb, m generated by the BMO function b and fractional integral type operator Tτ, and establish the boundness theorems for Tτb, m from Hp1.q1.sb, m to Lp2 and to Hp2 (0 < p1 ≤ 1), and from H Ka. p1.sq1, b, m to Ka.p2q2 and to H Ka. p2q2, respectively, for certain ranges of α, p1, q1, p2, q2 and s.

Let $x\,=\,\left( {{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}} \right)\,\in \,{{\mathbb{R}}^{n}}$ and ${{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x\,=\,\left( {{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{1}}}}{{x}_{1}},\,.\,.\,.\,,\,{{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{n}}}}{{x}_{n}} \right)$, where $\text{ }\lambda \,>\text{0}$ and $1\,\le \,{{\alpha }_{1}}\,\le \,\cdot \,\cdot \,\cdot \,\le \,{{\alpha }_{n}}$. Denote $\left| \alpha \right|\,=\,{{\alpha }_{1}}+\,\cdot \,\cdot \,\cdot \,+{{\alpha }_{n}}$. We characterize those functions $A\left( x \right)$ for which the parabolic Calderón commutator

1

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{T}_{A}}f\left( x \right)\equiv \text{p}\text{.v}\text{.}\int_{{{\mathbb{R}}^{n}}}{K\left( x-y \right)\left[ A\left( x \right)-A\left( y \right) \right]}f\left( y \right)dy$$

is bounded on ${{L}^{2}}\left( {{\mathbb{R}}^{n}} \right)$, where $K\left( {{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x \right)\,=\,{{\text{ }\!\!\lambda\!\!\text{ }}^{-\,\left| \alpha \right|\,-\,1}}K\left( x \right)$, $K$ is smooth away fromthe origin and satisfies a certain cancellation property.

We characterize, by means of congruence identities, all varieties having a weak difference term, and all neutral varieties. Our characterization of varieties with a difference term is new even in the particular case of locally finite varieties.