Operator algebra

About: Operator algebra is a research topic. Over the lifetime, 5783 publications have been published within this topic receiving 165303 citations.

Isomorphisms of standard operator algebras

Abstract: Let X and Y be Banach spaces, dim X = oo, and let v and g be standard operator algebras on X and Y, respectively. Assume that q$: vW -q is abijective mapping satisfying II0(AB)-0(A)0(B)II < e, A, B E , where e is a given positive real number (no linearity or continuity of q is assumed). Then q is a spatially implemented linear or conjugate linear algebra isomorphism. In particular, q is continuous. Let X be a Banach space. By R(X) we mean the algebra of all bounded linear operators on X. We denote by F(X) the subalgebra of bounded finite rank operators. We shall call a subalgebra v of R(X) standard provided v contains F(X) (X need not be closed). For any x E X and f E X' we denote by x? f the bounded linear operator on X defined by (x? f)y = f(y)x for y E X. Note that every operator of rank one can be written in this form. The operator x 0 f is a projection if and only if f(x) = 1 . Let X and Y be Banach spaces, and let v and 7 be standard operator algebras on X and Y, respectively. It is a classical result [4] that every algebra isomorphism q: v 7 is spatial, i.e., there is a linear topological isomorphism T: X -Y such that q(A) = TAT-1 for all A E ,v. When discussing isomorphisms of algebras one usually assumes that these mappings are linear. A more general approach is to consider the algebra only as a ring. It seems that the first step in this direction was made by Rickart [9, Theorem 3.2], who treated isomorphisms of primitive real Banach algebras which are not assumed to be linear, i.e., they are isomorphisms merely in the ring sense. The famous result of Kaplansky [6, 7] decomposes a ring isomorphism between two semisimple complex Banach algebras into a linear part, a conjugate linear part, and a nonreal linear part on a finite-dimensional ideal. Let R be a ring. Recall that R is called prime if aRb = 0 implies a = 0 or b = 0. Assume that a prime ring R contains an idempotent e # 0, 1 (R need not have an identity). Then every multiplicative bijective mapping of R onto an arbitrary ring S is additive [8]. It is an easy consequence of the Hahn-Banach theorem that R(X) is a prime ring. Thus, the above-mentioned results imply that if dimX = ox, then every multiplicative bijective mapping q of R (X) onto R (Y) is of the form +(A) = TAT-1 , where T: X -Y is Received by the editors May 11, 1993 and, in revised form, October 18, 1993. 1991 Mathematics Subject Classification. Primary 47D30. Supported by a grant from the Ministry of Science of Slovenia. ? 1995 American Mathematical Society 0002-9939/95 $1.00 + $.25 per page

Quantum Kac-Moody Algebras and Vertex Representations

Abstract: We introduce an affinization of the quantum Kac-Moody algebra associated to a symmet- ric generalized Cartan matrix. Based on the affinization, we construct a representation of the quantum Kac-Moody algebra by vertex operators from bosonic fields. We also obtain a combinatorial identity about Hall-Littlewood polynomials. Mathematics Subject Classifications ( 1991): Primary: 17Bxx; Secondary: 05Exx.

Orthogonal functions generalizing Jack polynomials

Abstract: The rational Cherednik algebra ℍ is a certain algebra of differential-reflection operators attached to a complex reflection group W and depending on a set of central parameters. Each irreducible representation S λ of W corresponds to a standard module M(λ) for ℍ. This paper deals with the infinite family G(r, 1, n) of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra t of H discovered by Dunkl and Opdam. In this case, the irreducible W-modules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on ℂ n with values in the representation S λ . For G(1,1, n) with λ = (n) these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of M(λ) in the case in which the orthogonal functions are all well-defined. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of ℍ so that the rational Cherednik algebra for G(r,p, n) is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for G(r, p, n) by Clifford theory.

Crystal base and q-vertex operators

Abstract: Theq-deformed vertex operators of Frenkel and Reshetikhin are studied in the framework of Kashiwara's crystal base theory. It is shown that the vertex operators preserve the crystal structure, and are naturally labeled by the global crystal base. As an application the one point functions are calculated for the associated elliptic RSOS models, following the scheme of Kang et al. developed for the trigonometric vertex models.