scholarly journals Putnam-Fuglede theorem and the range-kernel orthogonality of derivations

2001 ◽  
Vol 27 (9) ◽  
pp. 573-582 ◽  
Author(s):  
B. P. Duggal
Keyword(s):  
Hilbert Space ◽  
Generalized Derivation ◽  
Elementary Operator ◽  
Algebra Of Operators

Letℬ(H)denote the algebra of operators on a Hilbert spaceHinto itself. Letd=δorΔ, whereδAB:ℬ(H)→ℬ(H)is the generalized derivationδAB(S)=AS−SBandΔAB:ℬ(H)→ℬ(H)is the elementary operatorΔAB(S)=ASB−S. GivenA,B,S∈ℬ(H), we say that the pair(A,B)has the propertyPF(d(S))ifdAB(S)=0impliesdA∗B∗(S)=0. This paper characterizes operatorsA,B, andSfor which the pair(A,B)has propertyPF(d(S)), and establishes a relationship between thePF(d(S))-property of the pair(A,B)and the range-kernel orthogonality of the operatordAB.

scholarly journals A generalized commutativity theorem for pk-quasihyponormal operators

Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 77-83
Author(s):  
B.P. Duggal
Keyword(s):  
Hilbert Space ◽  
Generalized Derivation ◽  
Elementary Operator ◽  
Commutativity Theorem ◽  
Hilbert Space Operators

For Hilbert space operators A and B, let ?AB denote the generalized derivation ?AB(X) = AX - XB and let /\AB denote the elementary operator rAB(X) = AXB-X. If A is a pk-quasihyponormal operator, A ? pk - QH, and B*is an either p-hyponormal or injective dominant or injective pk - QH operator (resp., B*is an either p-hyponormal or dominant or pk - QH operator), then ?AB(X) = 0 =? SA*B*(X) = 0 (resp., rAB(X) = 0 =? rA*B*(X) = 0). .

scholarly journals On an elementary operator with 2-isometric operator entries

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5083-5088 ◽  
Author(s):  
Junli Shen ◽  
Guoxing Ji
Keyword(s):  
Hilbert Space ◽  
Generalized Derivation ◽  
Space Operator ◽  
Connected Component ◽  
Elementary Operator ◽  
Isometric Operator ◽  
Hilbert Space Operator

A Hilbert space operator T is said to be a 2-isometric operator if T*2T2- 2T*T + I = 0. Let dAB ? B(B(H)) denote either the generalized derivation ?AB = LA-RB or the elementary operator ?= LARB-I, we show that if A and B* are 2-isometric operators, then, for all complex ?, (dAB-?)-1(0)? (d*AB-?)-1(0), the ascent of (dAB-?) ? 1, and dis polaroid. Let H(?(dAB)) denote the space of functions which are analytic on ?(dAB), and let Hc(?(dAB)) denote the space of f ? H(?(dAB)) which are non-constant on every connected component of ?(dAB), it is proved that if A and B* are 2-isometric operators, then f(dAB) satisfies the generalized Weyl?s theorem and f(d*AB) satisfies the generalized a-Weyl?s theorem.

scholarly journals Periodicity of functionals and representations of normed algebras on reflexive spaces

1976 ◽  
Vol 20 (2) ◽  
pp. 99-120 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Hilbert Space ◽  
Continuous Homomorphism ◽  
Linear Operators ◽  
The Other ◽  
Continuous Linear ◽  
Normed Algebra ◽  
Reflexive Space ◽  
To Come ◽  
Algebra Of Operators ◽  
Continuous Linear Operators

It is a well-known fact that any normed algebra can be represented isometrically as an algebra of operators with the operator norm. As might be expected from the very universality of this property, it is little used in the study of the structure of an algebra. Far more helpful are representations on Hilbert space, though these are correspondingly hard to come by: isometric representations on Hilbert space are not to be expected in general, and even continuous nontrivial representations may fail to exist. The purpose of this paper is to examine a class of representations intermediate in both availability and utility to those already mentioned—namely, representations on reflexive spaces. There certainly are normed algebras which admit isometric representations of the latter type but have not even faithful representations on Hilbert space: the most natural example is the algebra of all continuous linear operators on E where E = lp with 1 < p ≠ 2 < ∞, for Berkson and Porta proved in (2) that if E, F are taken from the spaces lp with 1 < p < ∞ and E ≠ F then the only continuous homomorphism from into is the zero mapping. On the other hand there are also algebras which have no continuous nontrivial representation on any reflexive space—for example the algebra of finite-rank operators on an irreflexive Banach space (see Berkson and Porta (2) or Barnes (1) or Theorem 3, Corollary 1 below).

Additive maps having a generalized derivation expansion

2015 ◽  
Vol 14 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Generalized Derivation ◽  
Elementary Operator ◽  
Additive Map ◽  
Finite Dimensional ◽  
Left And Right ◽  
Central Simple ◽  
Skolem Theorem ◽  
Additive Maps

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

Correlations in a General Theory of Quantum Measurement

2007 ◽  
Vol 14 (04) ◽  
pp. 445-458 ◽  
Author(s):  
Hanna Podsędkowska
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
General Theory ◽  
Quantum Measurement ◽  
Von Neumann Algebra ◽  
Classical Case ◽  
Von Neumann ◽  
Full Algebra ◽  
Algebraic Formalism ◽  
Algebra Of Operators

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states — by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.

scholarly journals Isometric representation of M(G) on B(H)

1982 ◽  
Vol 23 (2) ◽  
pp. 119-122 ◽  
Author(s):  
F. Ghahramani
Keyword(s):  
Hilbert Space ◽  
Abelian Group ◽  
Locally Compact Abelian Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Isometric Isomorphism ◽  
Isometric Representation ◽  
University Of Edinburgh ◽  
The University ◽  
Algebra Of Operators

In a recent paper, E. Størmer, among other things, proves the existence of an isometric isomorphism from the measure algebra M(G) of a locally compact abelian group G into BB(L2(G)), ([6], Proposition 4.6). Here we give another proof for this result which works for non-commutative G as well as commutative G. We also prove that the algebra L1(G, λ), with λ the left (or right) Haar measure, is not isometrically isomorphic with an algebra of operators on a Hilbert space. The proofs of these two results are taken from the author's Ph.D. thesis [4], submitted to the University of Edinburgh before Størmer's paper. The author wishes to thank Dr. A. M. Sinclair for his help and encouragement.

Tensor products which do not preserve operator algebras

1990 ◽  
Vol 108 (2) ◽  
pp. 395-403 ◽  
Author(s):  
David P. Blecher
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Operator Algebras ◽  
Tensor Products ◽  
Preserve Operator ◽  
Infinite Dimensional ◽  
Hilbert Space Operators ◽  
Algebra Of Operators

Of late the link between operator algebras and certain tensor products has been reiterated [5]. We prove here that the projective and Haagerup tensor products of two infinite-dimensional C*-algebras is not even topologically isomorphic to an algebra of operators on a Hilbert space. Estimates are given for the distance of the tensor product from such an algebra. Nonetheless with respect to a natural multiplication the Haagerup tensor product of two algebras of Hilbert space operators is completely isometrically isomorphic to an algebra of operators on some B(ℋ).

Commutators in Factors of Type III

1966 ◽  
Vol 18 ◽  
pp. 1152-1160 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Identity Operator ◽  
Von Neumann ◽  
Type Iii ◽  
Underlying Space ◽  
Special Cases ◽  
Weakly Closed ◽  
Algebra Of Operators

Let denote a separable, complex Hilbert space, and let R be a von Neumann algebra acting on . (A von Neumann algebra is a weakly closed, self-adjoint algebra of operators that contains the identity operator on its underlying space.) An element A of R is a commutator in R if there exist operators B and C in R such that A = BC — CB. The problem of specifying exactly which operators are commutators in R has been solved in certain special cases; e.g. if R is an algebra of type In (n < ∞) (2), and if R is a factor of type I∞ (1). It is the purpose of this note to treat the same problem in case R is a factor of type III. Our main result is the following theorem.

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