Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).     

Maps preserving peripheral spectrum of Jordan semi-triple products of operators

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σ_π(A) of A is the set σ_π(A)={z∈σ(A):|z|=max_ω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σ_π(BAB)=σ_π(Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈C with λ³=1 and either there exists an invertible operator T∈B(X,Y) such that Φ(A)=λTAT^-1 for every A∈A; or there exists an invertible operator T∈B(X^∗,Y) such that Φ(A)=λTA^∗T^-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA^∗B are also characterized. Such maps are of the form A?UAU^∗ or A?UA^tU^∗, where U∈B(H,K) is a unitary operator, A^t denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.     

Infinite-dimensional super Lie groups

A super Lie group is a group whose operations are G^∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G^∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators.In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G, then h is the super Lie algebra of a sub-super Lie group of G. Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G. We also show that if H is a closed sub-super Lie group of a super Lie group G, then G→G/H is a principal fiber bundle.We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G^∞ category.     

All-derivable points of operator algebras

Let A be an operator subalgebra in B(H), where H is a Hilbert space. We say that an element Z∈A is an all-derivable point of A for the norm-topology (strongly operator topology, etc.) if, every norm-topology (strongly operator topology, etc.) continuous derivable linear mapping φ at Z (i.e. φ(ST)=φ(S)T+Sφ(T) for any S,T∈A with ST=Z) is a derivation. In this paper, we show that every invertible operator in the nest algebra is an all-derivable point of the nest algebra for the strongly operator topology. We also prove that every nonzero element of the algebra of all 2×2 upper triangular matrixes is an all-derivable point of the algebra.     

Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C^∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C^∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier.     

Maps preserving operator pairs whose products are projections

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dimH?2. It is proved that a surjective map φ on B(H) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U on H such that φ(A)=λU^∗AU for all A in B(H) for some constants λ with λ²=1. Related results for surjective maps preserving operator pairs whose triple Jordan products are nonzero projections in both directions are also obtained. These show that the operator pairs whose products or triple Jordan products are nonzero projections are isometric invariants of B(H).     

Linear mappings derivable at some nontrivial elements

Suppose that A is an algebra and M is an A-bimodule. Let A be any element in A. A linear mapping δ from A into M is said to be derivable at A if δ(ST)=δ(S)T+Sδ(T) for any S,T in A with ST=A. Given an algebra A, such as a non-abelian von Neumann algebra or an irreducible CDCSL algebra on a Hilbert space H with dimH?2, we show that there exists a nontrivial idempotent P in A such that for any Q∈PAP which is invertible in PAP, every linear mapping derivable at Q from A into some unital A-bimodule (for example, A or B(H)) is derivation.     

Separated Lie models and the homotopy Lie algebra

A simply connected topological space X has homotopy Lie algebra π_∗(ΩX)⊗Q. Following Quillen, there is a connected differential graded free Lie algebra (dgL) called a Lie model, which determines the rational homotopy type of X, and whose homology is isomorphic to the homotopy Lie algebra. We show that such a Lie model can be replaced with one that has a special property that we call being separated. The homology of a separated dgL has a particular form which lends itself to calculations.     

Characterizations of derivations of Banach space nest algebras: All-derivable points

Let N be a nest on a complex Banach space X with N∈N complemented in X whenever N_-=N, and let AlgN be the associated nest algebra. We say that an operator Z∈AlgN is an all-derivable point of AlgN if every linear map δ from AlgN into itself derivable at Z (i.e. δ(A)B+Aδ(B)=δ(Z) for any A,B∈A with AB=Z) is a derivation. In this paper, it is shown that if Z∈AlgN is an injective operator or an operator with dense range, or an idempotent operator with ran(Z)∈N, then Z is an all-derivable point of AlgN. Particularly, if N is a nest on a complex Hilbert space, then every idempotent operator with range in N, every injective operator as well as every operator with dense range in AlgN is an all-derivable point of the nest algebra AlgN.     

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra (g,g^∗), we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g^∗. We write down families of examples for the case that g is a classical complex simple Lie algebra.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Norms of certain Jordan elementary operators

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. For A,B∈B(H), the Jordan elementary operator UA,B is defined by UA,B(X)=AXB+BXA, ∀X∈B(H). In this short note, we discuss the norm of UA,B. We show that if dimH=2 and ‖UA,B‖=‖A‖‖B‖, then either AB^∗ or B^∗A is 0. We give some examples of Jordan elementary operators UA,B such that ‖UA,B‖=‖A‖‖B‖ but AB^∗≠0 and B^∗A≠0, which answer negatively a question posed by M. Boumazgour in [M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386-393].     

Hyponormal operators with rank-two self-commutators

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 a rank-one self-commutator then T is subnormal and if instead T_|ker[T^∗,T] has a rank-two self-commutator then T is either a subnormal operator or the kth minimal partially normal extension, , of a (k+1)-hyponormal operator Tk which has a rank-two self-commutator for any k∈Z₊. Hence, in particular, every weakly subnormal (or 2-hyponormal) operator with a rank-two self-commutator is either a subnormal operator or a finite rank perturbation of a k-hyponormal operator for any k∈Z₊.     

Representability and Specht problem for G-graded algebras

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idG(W) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z/2Z×G-graded algebra A over K such that idG(W)=idG(A^∗) where A^∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal idG(W) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that idG(W)=idG(A).     

Module homomorphisms and multipliers on locally compact quantum groups

For a Banach algebra A with a bounded approximate identity, we investigate the A-module homomorphisms of certain introverted subspaces of A^∗, and show that all A-module homomorphisms of A^∗ are normal if and only if A is an ideal of A∗∗. We obtain some characterizations of compactness and discreteness for a locally compact quantum group G. Furthermore, in the co-amenable case we prove that the multiplier algebra of L¹(G) can be identified with M(G). As a consequence, we prove that G is compact if and only if LUC(G)=WAP(G) and M(G)≅Z(LUC^∗(G)); which partially answer a problem raised by Volker Runde.     

Functional calculus and ∗-regularity of a class of Banach algebras II

In this article, we define a natural Banach ∗-algebra for a C^∗-dynamical system (A,G,α) which is slightly bigger than L¹(G;A) (they are the same if A is finite-dimensional). We will show that this algebra is ∗-regular if G has polynomial growth. The main result in this article extends the two main results in [C.W. Leung, C.K. Ng, Functional calculus and ∗-regularity of a class of Banach algebras, Proc. Amer. Math. Soc., in press].     

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let Z be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g^∗ with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G₂.     

On ideals in the bidual of the Fourier algebra and related algebras

Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G×H) the Fourier algebra of G×H, and UC₂(G×H) the space of uniformly continuous functionals in VN(G×H)=A^∗(G×H). We use weak factorization of operators in the group von Neumann algebra VN(G×H) to prove that there exist at least ²b(G)² left ideals of dimensions at least ²b(G)² in A(G×H)^∗∗ and in UC₂^∗(G×H). We show that every nontrivial right ideal in A(G×H)^∗∗ and in UC₂^∗(G×H) has dimension at least ²b(G)².     

On the essential commutant of analytic Toeplitz operators associated with spherical isometries

Let T∈Bn(H) be an essentially normal spherical isometry with empty point spectrum on a separable complex Hilbert space H, and let AT⊂B(H) be the unital dual operator algebra generated by T. In this note we show that every operator S∈B(H) in the essential commutant of AT has the form S=X+K with a T-Toeplitz operator X and a compact operator K. Our proof actually covers a larger class of subnormal operator tuples, called A-isometries, which includes for example the tuple T=(Mz₁,…,Mzn)∈B(H²n(σ)) consisting of the multiplication operators with the coordinate functions on the Hardy space H²(σ) associated with the normalized surface measure σ on the boundary ∂D of a strictly pseudoconvex domain D⊂Cn. As an application we determine the essential commutant of the set of all analytic Toeplitz operators on H²(σ) and thus extend results proved by Davidson (1977) [6] for the unit disc and Ding and Sun (1997) [11] for the unit ball.     

Vertex coalgebras, comodules, cocommutativity and coassociativity

We introduce the notion of vertex coalgebra, a generalization of vertex operator coalgebras. Next we investigate forms of cocommutativity, coassociativity, skew-symmetry, and an endomorphism, D^∗, which hold on vertex coalgebras. The former two properties require grading. We then discuss comodule structure. We conclude by discussing instances where graded vertex coalgebras appear, particularly as related to Primc’s vertex Lie algebra and (universal) enveloping vertex algebras.