Virtual braids and virtual curve diagrams

There is a well-known injective homomorphism [Formula: see text] from the classical braid group [Formula: see text] into the automorphism group of the free group [Formula: see text], first described by Artin [Theory of Braids, Ann. Math. (2) 48(1) (1947) 101–126]. This homomorphism induces an action of [Formula: see text] on [Formula: see text] that can be recovered by considering the braid group as the mapping class group of [Formula: see text] (an upper half plane with [Formula: see text] punctures) acting naturally on the fundamental group of [Formula: see text]. Kauffman introduced virtual links [Virtual knot theory, European J. Combin. 20 (1999) 663–691] as an extension of the classical notion of a link in [Formula: see text]. There is a corresponding notion of a virtual braid, and the set of virtual braids on [Formula: see text] strands forms a group [Formula: see text]. In this paper, we will generalize the Artin action to virtual braids. We will define a set, [Formula: see text], of “virtual curve diagrams” and define an action of [Formula: see text] on [Formula: see text]. Then, we will show that, as in Artin’s case, the action is faithful. This provides a combinatorial solution to the word problem in [Formula: see text]. In the papers [V. G. Bardakov, Virtual and welded links and their invariants, Siberian Electron. Math. Rep. 21 (2005) 196–199; V. O. Manturov, On recognition of virtual braids, Zap. Nauch. Sem. POMI 299 (2003) 267–286], an extension [Formula: see text] of the Artin homomorphism was introduced, and the question of its injectivity was raised. We find that [Formula: see text] is not injective by exhibiting a non-trivial virtual braid in the kernel when [Formula: see text].

Universal Enveloping Algebras of Simple Symplectic Anti-Jordan Triple Systems

In this work we are concerned with the universal associative envelope of a finite-dimensional simple symplectic anti-Jordan triple system (AJTS). We prove that if 𝕋 is a triple system as above, then there exists an associative algebra U(𝕋) and an injective homomorphism ε : 𝕋 → U(𝕋), where U(𝕋) is an AJTS under the triple product defined by (a,b,c) = abc - cba. Moreover, U(𝕋) is a universal object with respect to such homomorphisms. We explicitly determine the PBW-basis of U(𝕋), the center Z(U(𝕋)) and the Gelfand-Kirillov dimension of U(𝕋).

THE CO-HOPFIAN PROPERTY OF SURFACE BRAID GROUPS

Let g and n be integers at least 2, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.

Embedding covariance algebras of flows into $AF$-algebras

Suppose that $\{\alpha_t\}_{t\in \mathbb{R}}$ is a flow on the compact metrizable space $X$. We prove that a necessary and sufficient condition for the existence of an embedding (injective $*$-homomorphism) of the crossed product $C(X)\rtimes_\alpha \mathbb{R}$ into some $AF$-algebra is that every point of $X$ be chain recurrent in the sense of Conley.

DenseQ-subalgebras of Banach andC*-algebras and unbounded derivations of Banach andC*-algebras

The paper studies denseQ-subalgebras of Banach andC*-algebras. It proves that the domainD(δ) of a closed unbounded derivation δ of a Banach unital algebraAautomatically contains the identity and is aQ-subalgebra ofA, so thatSpA(x) =SpD(δ)(x) for allx∈D(δ). The paper shows that every finite-dimensional semisimple representation of aQ-subalgebra is continuous. It also shows that if π is an injective *-homomorphism of a dense locally normalQ*-subalgebraBof aC*-algebra, then ‖x‖≦‖π(x)‖ for allx∈B. The paper studies the link between closed ideals of a Banach algebraAand of its dense subalgebraB. In particular, ifAis aC*-algebra andBis a locally normal *-subalgebra ofA, thenI→I∩Bis a one-to-one mapping of the set of all closed two-sided ideals inAonto the set of all closed two-sided ideals inBand.

A Characterization of Ideals of C* -Algebras

Let A be a C*-algebra and let I be a C*-subalgebra of A. Denote by an extension of a state φ of B to a state of A. It is shown that I is an ideal of A if and only if there exists a homomorphism Q from A** onto I** such that Q is the identity map on I** and for every state φ on I. Furthermore it is also shown that I is an essential ideal of A if and only if there exists an injective homomorphism from A into the multiplier algebra of I which is the identity map on I.