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Direct sums of irreducible operators

Jun Shen Fang, Chun-Lan Jiang, Pei Yuan Wu (2003)

Studia Mathematica

It is known that every operator on a (separable) Hilbert space is the direct integral of irreducible operators, but not every one is the direct sum of irreducible ones. We show that an operator can have either finitely or uncountably many reducing subspaces, and the former holds if and only if the operator is the direct sum of finitely many irreducible operators no two of which are unitarily equivalent. We also characterize operators T which are direct sums of irreducible operators in terms of the...

Discontinuity of the Fuglede-Kadison determinant on a group von Neumann algebra

Benjamin Küter (2014)

Communications in Mathematics

We show that in contrast to the case of the operator norm topology on the set of regular operators, the Fuglede-Kadison determinant is not continuous on isomorphisms in the group von Neumann algebra 𝒩 ( ) with respect to the strong operator topology. Moreover, in the weak operator topology the determinant is not even continuous on isomorphisms given by multiplication with elements of [ ] . Finally, we define T 𝒩 ( ) such that for each λ the operator T + λ · id l 2 ( ) is a self-adjoint weak isomorphism of determinant class but...

On strong generation of B(ℋ) by two commutative C*-algebras

R. Berntzen, A. Sołtysiak (1997)

Studia Mathematica

The algebra B(ℋ) of all bounded operators on a Hilbert space ℋ is generated in the strong operator topology by a single one-dimensional projection and a family of commuting unitary operators with cardinality not exceeding dim ℋ. This answers Problem 8 posed by W. Żelazko in [6].

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