The uniqueness of the adjoint operation I

1986 ◽  
Vol 9 (5) ◽  
pp. 623-634 ◽  
Author(s):  
Scott H. Hochwald
Keyword(s):  
Adjoint Operation

Banach Algebras With an Adjoint Operation

1946 ◽  
Vol 47 (3) ◽  
pp. 528 ◽  
Author(s):  
C. E. Rickart
Keyword(s):  
Banach Algebras ◽  
Adjoint Operation

scholarly journals Spinors and canonical hermitian forms

1988 ◽  
Vol 30 (3) ◽  
pp. 263-270 ◽  
Author(s):  
P. L. Robinson
Keyword(s):  
Clifford Algebra ◽  
Product Space ◽  
Hermitian Form ◽  
Inner Product Space ◽  
Inner Product ◽  
Spin Representation ◽  
Hermitian Forms ◽  
Real Inner Product Space ◽  
Adjoint Operation

The spaceSof spinors associated to a2m-dimensional real inner product space (V, B) carries a canonical Hermitian form 〈 〉 determined uniquely up to real multiples. This form arises as follows: the complex Clifford algebraC(V) of (V, B) is naturally provided with an antilinear involution; this induces an involution on EndSvia the spin representation; this is the adjoint operation corresponding to 〈 〉.

scholarly journals Nonlinear *-Lie Higher Derivations of Standard Operator Algebras

2018 ◽  
Vol 26 (1) ◽  
pp. 15-29
Author(s):  
Mohammad Ashraf ◽  
Shakir Ali ◽  
Bilal Ahmad Wani
Keyword(s):  
Hilbert Space ◽  
Operator Algebra ◽  
Operator Algebras ◽  
Complex Hilbert Space ◽  
Standard Operator ◽  
Standard Operator Algebra ◽  
Finite Dimensional ◽  
Standard Operator Algebras ◽  
Higher Derivation ◽  
Adjoint Operation

Abstract Let ℌ be an in finite-dimensional complex Hilbert space and A be a standard operator algebra on ℌ which is closed under the adjoint operation. It is shown that every nonlinear *-Lie higher derivation D = {δn}gn∈N of A is automatically an additive higher derivation on A. Moreover, D = {δn}gn∈N is an inner *-higher derivation.

Mutually unbiased bases and orthogonal decompositions of Lie algebras

2007 ◽  
Vol 7 (4) ◽  
pp. 371-382
Author(s):  
P.O. Boykin ◽  
M. Sitharam ◽  
P.H. Tiep ◽  
P. Wocjan
Keyword(s):  
Lie Algebras ◽  
Open Problem ◽  
Orthogonal Decomposition ◽  
Mutually Unbiased Bases ◽  
Killing Form ◽  
Unitary Matrices ◽  
Complete Collection ◽  
Cartan Subalgebras ◽  
Orthogonal Decompositions ◽  
Adjoint Operation

We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of $\mu$ MUBs in $\K^n$ gives rise to a collection of $\mu$ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also $\dag$-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for $n\le 5$ an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for $n=6$ there are at most three monomial MUBs.

scholarly journals MORE ABOUT ALL CURRENT-ALGEBRAIC ORBIFOLDS

2001 ◽  
Vol 16 (01) ◽  
pp. 97-162 ◽  
Author(s):  
M. B. HALPERN ◽  
J. E. WANG
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Current Algebra ◽  
Ground State ◽  
Field Theories ◽  
Conformal Weight ◽  
Conformal Field Theories ◽  
Twisted Sector ◽  
Conformal Field ◽  
Adjoint Operation

Recently a construction was given for the stress tensors of all sectors of the general current-algebraic orbifold A(H)/H, where A(H) is any current-algebraic conformal field theory with a finite symmetry group H. Here we extend and further analyze this construction to obtain the mode formulation of each sector of each orbifold A(H)/H, including the twisted current algebra, the Virasoro generators, the orbifold adjoint operation and the commutator of the Virasoro generators with the modes of the twisted currents. As applications, general expressions are obtained for the twisted current–current correlator and ground state conformal weight of each twisted sector of any permutation orbifold A(H)/H, H⊂SN. Systematics are also outlined for the orbifolds A(Lie h(H))/H of the (H and Lie h)-invariant conformal field theories, which include the general WZW orbifold and the general coset orbifold. Finally, two new large examples are worked out in further detail: the general SNpermutation orbifold A(SN)/SNand the general inner-automorphic orbifold A(H(d))/H(d).

scholarly journals A Review Note on the Applications of Linear Operators in Hilbert Space

2021 ◽  
Author(s):  
Karthic Mohan ◽  
Jananeeswari Narayanamoorthy
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Linear Transformations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Wide Range ◽  
The Subject ◽  
Adjoint Operators ◽  
User Friendly ◽  
Adjoint Operation

Hilbert Spaces are the closest generalization to infinite dimensional spaces of the Euclidean Spaces. We Consider Linear transformations defined in a normed space and we see that all of them are Continuous if the Space is finite Dimensional Hilbert Space Provide a user-friendly framework for the study of a wide range of subjects from Fourier Analysis to Quantum Mechanics. The adjoint of an Operator is defined and the basic properties of the adjoint operation are established. This allows the introduction of self Adjoint Operators are the subject of the section.

Radical classes of commutative rings defined in terms of the adjoint operation

1994 ◽  
Vol 22 (13) ◽  
pp. 5533-5548 ◽  
Author(s):  
N.R. McConnell ◽  
Timothy Stokes
Keyword(s):  
Commutative Rings ◽  
Adjoint Operation

Closed Lie Ideals in Operator Algebras

1981 ◽  
Vol 33 (5) ◽  
pp. 1271-1278 ◽  
Author(s):  
C. Robert Miers
Keyword(s):  
Lie Algebra ◽  
Associative Algebra ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Subspace ◽  
Operator Algebras ◽  
Lie Ideal ◽  
Von Neumann ◽  
Adjoint Operation ◽  
Uniformly Closed

If M is an associative algebra with product xy, M can be made into a Lie algebra by endowing M with a new multiplication [x, y] = xy – yx. The Poincare-Birkoff-Witt Theorem, in part, shows that every Lie algebra is (Lie) isomorphic to a Lie subalgebra of such an associative algebra M. A Lie ideal in M is a linear subspace U ⊆ M such that [x, u] ∊ U for all x £ M, u ∊ U. In [9], as a step in characterizing Lie mappings between von Neumann algebras, Lie ideals which are closed in the ultra-weak topology, and closed under the adjoint operation are characterized when If is a von Neumann algebra. However the restrictions of ultra-weak closure and adjoint closure seemed unnatural, and in this paper we characterize those uniformly closed linear subspaces which can occur as Lie ideals in von Neumann algebras.

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