*-algebras of unbounded operators affiliated with a von Neumann algebra

In the paper, the *-algebras of measurable operators, locally measurable operators, and τ-measurable operators associated with a von Neumann algebra M are considered. Conditions under which some of these algebras coincide are given. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 183–197.     

Irreducible * representations of realC*-algebras

In this paper, we show that a topologically irreducible * representation of a realC*-algebra is also algebraically irreducible. Moreover, the properties of pure real states on a realC*-algebra and their left kernels are discussed.Partially supported by the National Natural Science Foundation of China.     

Integral representations of unbounded operators by infinitely smooth kernels

In this paper, we prove that every unbounded linear operator satisfying the Korotkov-Weidmann characterization is unitarily equivalent to an integral operator in L ₂(R), with a bounded and infinitely smooth Carleman kernel. The established unitary equivalence is implemented by explicitly definable unitary operators.     

On the complexity of the description of -algebra representations by unbounded operators

We study the complexity of the problem to describe, up to unitary equivalence, representations of -algebras by unbounded operators on a Hilbert space. A number of examples are developed in detail.

     

WQ*-algebras of measurable operators

Every C*-algebra $\mathfrak{A}$ has a faithful *-representation π in a Hilbert space $\mathcal{H}$ . Consequently it is natural to pose the following question: under which conditions, the completion of a C*-algebra in a weaker than the given one topology, can be realized as a quasi *-algebra of operators? The present paper presents the possibility of extending the well known Gelfand — Naimark representation of C*-algebras to certain Banach C*-modules.     

Spectral reciprocity and matrix representations of unbounded operators

We study a family of unbounded Hermitian operators in Hilbert space which generalize the usual graph-theoretic discrete Laplacian. For an infinite discrete set X, we consider operators acting on Hilbert spaces of functions on X, and their representations as infinite matrices; the focus is on ?²(X), and the energy space HE. In particular, we prove that these operators are always essentially self-adjoint on ?²(X), but may fail to be essentially self-adjoint on HE. In the general case, we examine the von Neumann deficiency indices of these operators and explore their relevance in mathematical physics. Finally we study the spectra of the HE operators with the use of a new approximation scheme.     

Elementary operators on antiliminal C^*-algebras

We establish the equivalence of the following three properties of a -algebra A. (a) Every positive elementary operator on A is completely positive. (b) The norm and the cb-norm coincide for every elementary operator on A. (c) A is an extension of an antiliminal -algebra by an abelian one. Received: 15 July 1998 / in revised form: 22 September 1998     

Spectral representations for unbounded operators with real spectrum

The author would like to thank Prof. S. Shatz and the University of Pennsylvania for their hospitality     

On unbounded Bergman operators

This paper explores properties of the Bergman operator on unbounded open subsets of the plane. In addition to the characterization of the bounded commutant of such operators it proves the Berger-Shaw theorem and gives some general criteria under which the operator and its self-commutator are densely defined.     

On unbounded subnormal operators

A minimal normal extension of unbounded subnormal operators is established and characterized and spectral inclusion theorem is proved. An inverse Cayley transform is constructed to obtain a closed unbounded subnormal operator from a bounded one. Two classes of unbounded subnormals viz analytic Toeplitz operators and Bergman operators are exhibited.     

Elementary operators on primeC *-algebras. I

The results in this paper are based on results in the author's doctoral thesis at the University of Tübingen, FRG     

Maximal Op*-algebras containing only bounded operators

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 44, pp. 55–57, 1985.