FACULTY OF ARTS
https://dspace.mic.ul.ie/handle/10395/21
Tue, 08 Oct 2024 11:42:14 GMT2024-10-08T11:42:14ZPseudospectra of elements of reduced Banach algebras (Pre-published version)
https://dspace.mic.ul.ie/handle/10395/3335
Pseudospectra of elements of reduced Banach algebras (Pre-published version)
Let A be a Banach algebra with identity 1 and p ∈ A be a non-trivial idempotent. Then q = 1−p is also an idempotent. The subalgebras pAp and qAq are Banach algebras, called reduced Banach algebras, with identities p and q respectively. For a ∈ A and ε > 0, we examine the relationship between the ε-pseudospectrum Λε(A, a) of a ∈ A, and ε-pseudospectra of pap ∈ pAp
and qaq ∈ qAq. We also extend this study by considering a finite number of idempotents p1, · · · , pn, as well as an arbitrary family of idempotents satisfying certain conditions.
Sun, 01 Jan 2017 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/33352017-01-01T00:00:00ZPseudospectra of elements of reduced Banach algebras II (Pre-published version)
https://dspace.mic.ul.ie/handle/10395/3334
Pseudospectra of elements of reduced Banach algebras II (Pre-published version)
Let A be a Banach algebra with identity 1 and p ∈ A be a non-trivial idempotent. Then q = 1 − p is also an idempotent. The subalgebras pAp and qAq are Banach algebras, called reduced Banach algebras, with identities p and q respectively. Let x ∈ A be such that pxp = xp, and ε > 0. We examine the relationship between the spectrum of x ∈ A, σ(A, x), and the spectra of pxp ∈ pAp, σ(pAp, pxp) and qxq ∈ qAq, σ(qAq, qxq). Similarly, we examine the relationship betweeen the ε-pseudospectrum of x ∈ A, Λε(A, x) and ε-pseudospectra of pxp ∈ pAp, Λε(pAp, pxp) and of qxq ∈ qAq, Λε(qAq, qxq).
Fri, 01 Jun 2018 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/33342018-06-01T00:00:00ZMarkovianity and the Thompson monoid F+ (Pre-published version)
https://dspace.mic.ul.ie/handle/10395/3333
Markovianity and the Thompson monoid F+ (Pre-published version)
We introduce a new distributional invariance principle, called `partial spreadability',
which emerges from the representation theory of the Thompson monoid F+ in noncommutative
probability spaces. We show that a partially spreadable sequence of noncommutative random
variables is adapted to a local Markov filtration. Conversely we show that a large class of
noncommutative stationary Markov sequences provides representations of the Thompson monoid
F+. In the particular case of a classical probability space, we arrive at a de Finetti
theorem for stationary Markov sequences with values in a standard Borel space.
Mon, 13 Mar 2023 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/33332023-03-13T00:00:00ZWittgenstein looking at Wittgenstein: a critical analysis of the self reflexive logical evolution of Wittgenstein's work
https://dspace.mic.ul.ie/handle/10395/3332
Wittgenstein looking at Wittgenstein: a critical analysis of the self reflexive logical evolution of Wittgenstein's work
This thesis traces the evolution of Wittgenstein‟s work through the set theoretic concept of the infinite and the generated problems of Russell‟s paradox and self reference, which I argue develops through three distinct phases as presented in his early, middle and later work. I contend that considering Wittgenstein‟s work through the lens of these specific concepts is pivotal in identifying and understanding the logical evolution from his early to later work, which I argue represents a logical evolution from a closed to an open logical model wherein the influence of Russell proves critical. I identify these concepts as the key to understanding the crucial self reflexive dynamic operative between Wittgenstein‟s early and later work, which I claim further extends to understanding Wittgenstein‟s diametrically opposed early and later positions on the status of the activity of logical analysis, the referential and non referential use of language and his position on the logical distinction between a primary and secondary language. His position on the latter proves methodologically essential, illustrating how these core concepts are the logical means by which this distinction evolves, ultimately framing the move from a closed to an open logical model. Through the inversion of the primary secondary language distinction, I consider Wittgenstein‟s later work on aspect seeing as structurally and conceptually reflective of Russell‟s open logical model of type theory, arguing that Wittgenstein comes to accept in his later writings aspects of Russell‟s position that he had rejected in his early writings. In Wittgenstein‟s later open logical model of aspect seeing the primary secondary language distinction functions not only as the mechanism by means of which logical analysis operates, but also as a meta-analysis of Wittgenstein earlier work. This allows us to retrospectively engage with Wittgenstein‟s own particular form of linguistic aspect seeing, where we encounter the inherent self reflexive nature of this logical evolution allowing us to linguistically observe Wittgenstein looking at Wittgenstein.
Wed, 25 Sep 2024 00:00:00 GMThttps://dspace.mic.ul.ie/handle/10395/33322024-09-25T00:00:00Z