Structure of Iso-Symmetric Operators

For a Hilbert space operator T∈B(H), let LT and RT∈B(B(H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵T∗,Tm(I)=(LT∗RT−I)m(I) and δT∗,Tn(I)=(LT∗−RT)m(I). The operator T is said to be (m,n)-isosymmetric if ▵T∗,TmδT∗,Tn(I)=0. Power bounded (m,n)-isosymmetric operators T∈B(H) have an upper triangular matrix representation T=T1T30T2∈B(H1⊕H2) such that T1∈B(H1) is a C0.-operator which satisfies δT1∗,T1n(I|H1)=0 and T2∈B(H2) is a C1.-operator which satisfies AT2=(Vu⊕Vb)|H2A, A=limt→∞T2∗tT2t, Vu is a unitary and Vb is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C00-contraction.

Left Demazure–Lusztig Operators on Equivariant (Quantum) Cohomology and K-Theory

We study the Demazure–Lusztig operators induced by the left multiplication on partial flag manifolds $G/P$. We prove that they generate the Chern–Schwartz–MacPherson classes of Schubert cells (in equivariant cohomology), respectively their motivic Chern classes (in equivariant K-theory), in any partial flag manifold. Along the way, we advertise many properties of the left and right divided difference operators in cohomology and K-theory and their actions on Schubert classes. We apply this to construct left divided difference operators in equivariant quantum cohomology, and equivariant quantum K-theory, generating Schubert classes and satisfying a Leibniz rule compatible with the quantum product.

Some Properties of Left Weakly Jointly Prime (R,S)-Submodules

Let R and S be commutative rings with identity. A proper (R,S)submodule P of M is called a left weakly jointly prime if for each element a and b in R and (R,S)-submodule K of M with abKS ⊆ P implies either aKS ⊆ P or bKS ⊆ P. In this paper, we present some properties of left weakly jointly prime (R,S)-submodule. We show some necessary and sufficient condition of left weakly jointly prime (R,S)-submodule. Moreover, we present that every left weakly jointly prime (R,S)-submodule contains a minimal left weakly jointly prime (R,S)submodule. At the end of this paper, we also show that in left multiplication (R,S)-module, every left weakly jointly prime (R,S)-submodule is equal to jointly prime (R,S)-submodules.

Higher-order commutators with power central values on rings and algebras involving generalized derivations

Let {\mathfrak{R}} be a ring with center {Z(\mathfrak{R})}. In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation {\mathfrak{F}} such that {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every {s\in\mathfrak{R}}, then either {\mathfrak{F}(s)=ps} for every {s\in\mathfrak{R}} or {\mathfrak{R}} satisfies {s_{4}} and {\mathfrak{F}(s)=sp} for every {s\in\mathfrak{R}} and {p\in\mathfrak{U}}, the Utumi quotient ring of {\mathfrak{R}}. As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

On torsion-free nilpotent loops

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

A Computation with the Connes–Thom Isomorphism

Let A ∊ Mn(ℝ) be an invertible matrix. Consider the semi-direct product ℝn ⋊ ℤ where the action of ℤ on ℝn is induced by the left multiplication by A. Let (α, τ) be a strongly continuous action of ℝn ⋊ ℤ on a C*-algebra B where α is a strongly continuous action of ℝn and τ is an automorphism. The map τ induces a map . We show that, at the K-theory level, τ commutes with the Connes–Thom map if det(A) > 0 and anticommutes if det(A) < 0. As an application, we recompute the K-groups of the Cuntz–Li algebra associated with an integer dilation matrix.

Pre-jordan Algebras

The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.