Marco Thill | Searcharxiv

arXiv2019

A Monotone Selection Principle in C*-algebras

We apply to operator algebra theory a monotone selection principle which apparently escaped attention (of operator algebra theorists) so far. This principle relates to the basic order theoretic characterisation of von Neumann algebras given by Kadison, and the simplified form this result takes in separable Hilbert spaces. In the separable case we need only consider increasing sequences rather than increasing nets. We apply an argument of Klaus Floret to show that, within the realm of commutativity, there exists a general monotone selection principle providing for this simplification. Thereby we obtain a valuable shortcut and a handy tool for related purposes. Actually, a more general selection principle is proved within the framework of vector lattices.

math.FA↗

arXiv2010

Introduction to Normed *-Algebras and their Representations, 7th ed

This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C^*-algebra of a general normed *-algebra.

math.OA↗

arXiv2009

Representations of Hermitian Commutative *-Algebras by Unbounded Operators

We give a spectral theorem for unital representations of Hermitian commutative unital *-algebras by possibly unbounded operators in a pre-Hilbert space. A better result is known for the case in which the *-algebra is countably generated.

math.OA↗

arXiv2008

Introduction to Normed *-Algebras and their Representations, 6th ed

This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C*-algebra of a general normed *-algebra.

math.OA↗

arXiv2008

Simplified proof of the Theorem of Varopoulos in the commutative case

We give continuity properties of bitraces on (possibly non-commutative) Banach *-algebras based on the Closed Graph Theorem, leading to a simplified proof of the Theorem of Varopoulos in the commutative case.

math.RT↗

arXiv2007

Introduction to Normed *-Algebras and their Representations, 5th ed

This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C*-algebra of a general normed *-algebra.

math.RT↗

arXiv2004

Subspaces discerning nullcontinuity

Given positive linear functional l on a vector lattice L of real functions, and a vector subspace M of L, we construct a vector subspace P(M) of M in such a way that 1) l is nullcontinuous on P(M), and 2) if l is nullcontinuous on M then P(M) is all of M. We mention here that this result continues to hold for quite general modes of convergence, including tau-continuity. Our construction uses a new method involving the "kernel" of a seminorm.

math.FA↗

arXiv2003

A Non-commutative Monotone Selection Principle

We give an elementary proof of a monotone selection principle which allows to pass from increasing nets to increasing sequences in the Hermitian part of a $σ$-finite von Neumann algebra. This is to be seen as a ``monotone version'' of first countability.

math.FA↗