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.     

Property ∏σ and tensor products of operator algebras

In this paper, we introduce Property ∏σ of operator algebras and prove that nest subalgebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have Property ∏σ.Finally, we show that the tensor product formula alg ML1-(×)algNL2 = algM-(×)N(L1 (×) L2) holds for any two finite-width CSLs L1 and L2 in arbitrary von Neumann algebras M and N, respectively.     

Complemented subspaces isomorphic tol p in tensor products and operator spaces

Tartu. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 115–120, September–October, 1992.     

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C^?-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.     

Reducing subspaces of tensor products of weighted shifts

A unilateral weighted shift A is said to be simple if its weight sequence {α_n} satisfies ▽~3(α_n~2)≠0for all n≥2.We prove that if A and B are two simple unilateral weighted shifts,then AI+IB is reducible if and only if A and B are unitarily equivalent.We also study the reducing subspaces of A~kI+IB~l and give some examples.As an application,we study the reducing subspaces of multiplication operators Mzk+αωl on function spaces.     

Biderivations on tensor products of algebras

Let A be a noncommutative unital prime algebra and let S be a commutative unital algebra over a field 𝔽. We describe the form of biderivations of the algebra A?S. As an application, we determine the form of commuting linear maps of A?S.     

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.     

Multilinear maps on products of operator algebras

Let A₁,A₂,…,A_r be C∗-algebras with second duals A₁″,A₂″,…,A_r″, and let X be an arbitrary Banach space. Let be a bounded r-linear map, and denote by the Johnson-Kadison-Ringrose extension (i.e., the separately to continuous r-linear extension) of Γ. The problem of characterising those Γ for which Γ″ takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford-Pettis property fails for noncommutative C∗-algebras, the ‘obvious’ extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C∗-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra.