Affiliated subspaces and infiniteness of von Neumann algebras

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.     

On local representations of von Neumann algebras

We prove that every bounded, linear, 2-local Hilbert space representation of a von Neumann algebra is a representation. In contrast, 1-local representations may fail to be multiplicative, even at the 2 by 2 matrix algebra level.

     

Amenable representations and finite injective von Neumann algebras

Let be the unitary group of a finite, injective von Neumann algebra . We observe that any subrepresentation of a group representation into is amenable in the sense of Bekka; this yields short proofs of two known results-one by Robertson, one by Haagerup-concerning group representations into .

     

Invariant subspaces of operator algebras

The survey is devoted to a circle of problems, grouped around one of the oldest problems of functional analysis, namely the invariant subspace problem. It is shown that the investigation of the algebraic and analytic properties of families of operators touches upon the question of the structure of their invariant subspaces.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 26, pp. 65–145, 1988.     

Spectral representations for abelian automorphism groups of von Neumann algebras

Let $\text{A}$ be a von Neumann algebra, let σ be a strongly continuous representation of the locally compact abelian group G as ¹-automorphisms of $\text{A}$. Let M(σ) be the Banach algebra of bounded linear operators on $\text{A}$ generated by ∝ σ_tdμ(t) (μ?M(G)). Then it is shown that M(σ) is semisimple whenever either (i) $\text{A}$ has a σ-invariant faithful, normal, semifinite, weight (ii) σ is an inner representation or (iii) G is discrete and each σ_t is inner. It is shown that the Banach algebra L(σ) generated by $\text{∝ ?(t)σ}{}_{\mathrm{t}}\text{dt (? ? L}{}^1\text{(G))}$ is semisimple if a is an integrable representation. Furthermore, if σ is an inner representation with compact spectrum, it is shown that L(σ) is embedded in a commutative, semisimple, regular Banach algebra with isometric involution that is generated by projections. This algebra is contained in the ultraweakly continuous linear operators on $\text{A}$. Also the spectral subspaces of σ are given in terms of projections.     

Invariant subspaces of certain subnormal operators

Let T be a subnormal, nonnormal operator on a Hilbert space and suppose that the point spectrum of $\text{T}{}^1$ is empty. Then there exist vectors x ≠ 0 for which $\text{(T}{}^1\text{? zI)}{}^{\mathrm{?1}}\text{x}$ exists and is weakly continuous for all z. It is shown that under certain conditions, the Cauchy integral of this vector function taken around an appropriate contour, not necessarily lying in the resolvent set of $\text{T}{}^1$, leads to a proper (nontrivial) invariant subspace of $\text{T}{}^1$.     

Nest-subalgebras of von Neumann algebras

The structure of a certain class of separably acting reflexive operator algebras is investigated for which the nest algebras of J. Ringrose can be considered prototypes. To a fixed von Neumann algebra and a complete nest of projections contained therein one associates the algebra of all operators in the von Neumann algebra which leave every member of the nest invariant. A generalization of the Ringrose criterion for inclusion in the Jacobson radical of a nest algebra is given for this more general class of algebras. Further properties of the radical are studied.     

Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action

It is proved that a Lie algebra of compact operators with a non-zero Volterra ideal is reducible (has a nontrivial invariant subspace). A number of other criteria of reducibility for collections of operators is obtained. The results are applied to the structure theory of Lie algebras of compact operators and normed Lie algebras with compact adjoint action.