Operator algebra

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

Operator-algebraic superridigity for SL n (ℤ), n ≥3

Abstract: For n≥3, let Γ=SL n (ℤ). We prove the following superridigity result for Γ in the context of operator algebras. Let L(Γ) be the von Neumann algebra generated by the left regular representation of Γ. Let M be a finite factor and let U(M) be its unitary group. Let π:Γ→U(M) be a group homomorphism such that π(Γ)”=M. Then either (i) M is finite dimensional, or (ii) there exists a subgroup of finite index Λ of Γ such that π|Λ extends to a homomorphism U(L(Λ))→U(M). This answers, in the special case of SL n (ℤ), a question of A. Connes discussed in [Jone00, p. 86]. The result is deduced from a complete description of the tracial states on the full C *–algebra of Γ. As another application, we show that the full C *–algebra of Γ has no faithful tracial state, thus answering a question of E. Kirchberg.

Banach KK-theory and the Baum-Connes conjecture

Abstract: The report below describes the applications of Banach KK-theory to a conjecture of P. Baum and A. Connes about the K-theory of group C*-algebras.

Perturbation of operator algebras

Abstract: The Banach-Stone theorem says, in the language we shall use, that if two commutative C*-algebras are isometrically isomorphic, then they are isomorphic as C*-algebras. In this formulation there are two natural question: whether the theorem holds for noncommutative C*-algebras, and whether it is possible to replace "isometrically isomorphic" with "nearly isometrically isomorphic". A complete answer to the first question was given by Kadison in [10], and M. Cambern settled the second in the affirmative in [5]. In Section 3 we show that Cambern's result may be partially extended to noncommutative von Neumann algebras having the property P ([9, 18, 19]). When we have the result of Section 3 that a nearly isometric completely positive map of a von Neumann algebra having property P is close to a * isomorphism between the algebras, we have nearly proved that von Neumann algebras whose unit balls are close in the Hausdorff metric are unitarily equivalent by a unitary close to the identity. The only thing we have to prove then is that if q) is an isomorphism of a v o n Neumann algebra A having property P onto a v o n Neumann algebra B acting on the same Hilbert space, and 4~ satisfies, for some positive k less than one and any x in A,

The construction of the q-analogues of the harmonic oscillator operators from ordinary oscillator operators

Abstract: The q-analogues of boson operators are constructed from ordinary boson operators by an embedding approach. Accordingly the q-deformed quantum algebras SU(n)q can be obtained straightforwardly from corresponding universal enveloping algebras of Lie algebra SU(n).

Pure point spectrum under 1-parameter perturbations and instability of Anderson localization

Abstract: We consider a selfadjoint operator,A, and a selfadjoint rank-one projection,P, onto a vector, φ, which is cyclic forA. We study the set of all eigenvalues of the operatorAt=A+tP (t∈∝) that belong to its essential spectrum (which does not depend on the parametert). We prove that this set is empty for a dense set of values oft. Then we apply this result or its idea to questions of Anderson localization for 1-dimensional Schrodinger operators (discrete and continuous).