LIE IDEALS IN OPERATOR ALGEBRAS HOPENWASSER, ALAN; PAULSEN, VERN
Journal of operator theory,
09/2004, Volume:
52, Issue:
2
Journal Article
Peer reviewed
Let 𝒜 be a Banach algebra for which the group of invertible elements is connected. A subspace 𝓛 ⊆ 𝒜 is a Lie ideal in 𝒜 if and only if it is invariant under inner automorphisms. This applies, in ...particular, to any canonical subalgebra of an AF C*-algebra. The same theorem is also proven for strongly closed subspaces of a totally atomic nest algebra whose atoms are ordered as a subset of the integers and for CSL subalgebras of such nest algebras. We also give a detailed description of the structure of a Lie ideal in any canonical triangular subalgebra of an AF C*-algebra.
In this note, we show that if a bounded linear operator T acting on an infinite dimensional, separable, complex Hilbert space 𝓗 is not of the form scalar plus compact, then every bounded linear ...operator on 𝓗 can be written as a linear combination of 14 or fewer operators unitarily equivalent to T, as a linear combination of 6 or fewer operators similar to T, and as a sum of 8 or fewer operators similar to T. When T is not polynomially compact, the set of all sums of 2 operators similar to T is dense in 𝓑(𝓗), while if T is polynomially compact, but not of the form scalar plus compact, then the set of sums of 3 operators similar to T is dense in 𝓑(𝓗).
A result on derivations Lee, Tsiu-Kwen; Lin, Jer-Shyong
Proceedings of the American Mathematical Society,
06/1996, Volume:
124, Issue:
6
Journal Article
Peer reviewed
Open access
Let R be a semiprime ring with a derivation d and let U be a Lie ideal of R, a\in R. Suppose that ad(u)^n=0 for all u\in U, where n is a fixed positive integer. Then ad(I)=0 for I the ideal of R ...generated by U,U and if R is 2-torsion free, then ad(U)=0. Furthermore, R is a subdirect sum of semiprime homomorphic images R_1 and R_2 with derivations d_1 and d_2, induced canonically by d, respectively such that \overline ad_1(R_1)=0 and the image of U in R_2 is commutative (central if R is 2-torsion free), where \overline a denotes the image of a in R_1. Moreover, if U=R, then ad(R)=0. This gives Bresar's theorem without the (n-1)!-torsion free assumption on R.
Let R be a prime ring of characteristic 6= 2 with right quotient ring
U and extended centroid C, g 6= 0 a generalized derivation of R, L
a non-central Lie ideal of R and n ≥ 1 such that g(u), un = 0, ...for
all u 2 L. We prove that there exists an element a 2 C such that
g(x) = ax, for all x 2 R, unless when R satisfies s4 and there exists an
element b 2 U such that g(x) = bx + xb, for all x 2 R.
Let R be a 2-torsion free σ-prime ring having a σ-square closed Lie ideal U and an automorphism T centralizing on U. We prove that if there exists u0 in Saσ(R) with Ru0 ⊂ U and if T commutes with σ ...on U, then U is contained in the center of R. This result is then applied to generalize the result of J. Mayne for centralizing automorphisms to σ-prime rings. Finally, for a 2-torsion free σ-prime ring possessing a nonzero derivation, we give suitable conditions under which the ring must be commutative.
Let R be a prime ring with center Z and let U be a noncentral Lie ideal of R. Suppose that d is a derivation of R such that dn(u) ∈ Z for all u ∈ U, where n is a fixed integer. It is shown that ...either dn(R) = 0 or R is an order of a 4-dimensional simple algebra over a field of characteristic 2.
In a ring R with involution whose symmetric elements S are central, the skew‐symmetric elements K form a Lie algebra over the commutative ring S . The classification of such rings which are 2 ...‐torsion free is equivalent to the classification of Lie algebras K over S equipped with a bilinear form f that is symmetric, invariant and satisfies x , y , z = f ( y , z ) x − f ( z , x ) y . If S is a field of char ≠2, f ≠ 0 and dim K > 1 then K is a semisimple Lie algebra if and only if f is nondegenerate. Moreover, the derived algebra K ′ is either the pure quaternions over S or a direct sum of mutually orthogonal abelian Lie ideals of dim≤2.
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