Category Archive

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

I have finished off section 1.4 including Atkinson’s Theorem and a first look at the unilateral shift. I have done exercises 1-7. In terms of progress, I am on p.31 of 265, with 13 exercises left in this section. Following discussions with my supervisor, I may be able to leave out sections 3.2, 3.5, 4.4, 5.2-6 and the whole of chapter 7.

I have continued to work through Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

I have finished off my revision of sections 1.2 (The Spectrum and the Spectral Radius) & 1.3 (The Gelfand Representation). Section 1.4 is a new topic for me – Compact and Fredholm Operators. A linear map between Banach spaces is compact if is totally bounded. As a corollary, all linear maps on finite dimensional spaces are compact. The transpose has been introduced by Murphy is this chapter, and I have seen that if is compact, then so is . A linear map is Fredholm if the and are finite dimensional. In terms of progress, I am on p.25 of 265.

With a bout of illness last week I only got to finish off Beltonhttp://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf and start Murphy http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&h

At present I am speedily going through Chapter 1: Elementary Spectral Theory. This has all been done in Belton but I like Murphy’s lucid and no-frills approach. In places Murphy takes a different approach to Belton (e.g. the proof that the spectrum is non-empty establishes the differentiability of the map , without recourse to the resolvent). This quick revision will continue until 1.4 Compact and Fredholm Operators – which is a new topic for me. In terms of progress, this starts on p.18; I’m presently on p.9. The entire book weighs in at 245 pages and realistically I certainly wouldn’t expect to be finished before Christmas.

I have continued my work on Belton http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf.  I finished off exercises 6.7-6.8.

I looked at the section on Characters and Maximal Ideals. Some really nice results in this area. For example, every proper ideal of a commutative, unital complex Banach algebra contains no invertible elements and is contained in a maximal ideal. I saw that there is a bijection between the set of characters of and the set of all maximal ideals.

I saw the links between the characters of and the spectrum of elements of . The Jacobson radical was introduced; and the Gelfand topology was presented. I have done the first three exercises 7.1-3 out of 10.

When this is finished I must present a summary of the different initial topologies and review the various definitions, etc.

When I finish this I will be looking at http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&hl=en&ei=6p_OTIfkJNXt4gaXv5n_Aw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA

I have continued my work on Belton http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf I finished off exercises 5.3-5.17. Primarily these were concerned with topological vector spaces (a Hausdorff topology on a vector space that makes the addition and scalar multiplication functions continuous), locally convex spaces, separating families of linear functionals ( is separating if for all , such that ). Also a number of results were derived that concerned the existence of functionals which were dominated on one set by another (e.g. 5.12). Finally some exercises on extreme points; for example every unit vector in a Hilbert space is an extreme point of the closed unit ball .

This section will be revised when I finish Belton. In particulat need to draw a scheme which relates the canonical topologies. Belton introduces them as initial topologies (generated by a family of functions) – the “old” terminology was the weak topology (generated by a family of functions). Also I will relook at the theorems to get a feeling for why and where particular conditions need to be satisfied (e.g. does the set need to be convex, compact, closed, connected?; does the space have to be locally convex, Hausdorff?, etc).

Having finished that section I began a study of normed algebras (vector spaces with an associative multiplication and submultiplicative norm). I saw that every finite dimensional algebra is isomorphic to a subalgebra of . I saw a number of examples of function spaces… basically it was “An Introduction to Normed Algebras” and it is fairly straightforward with some very nice results such as the Gelfand-Mazur Theorem. I have done exercises 6.1-6.6 of 8.

The final section of Belton is on Characters and Maximal Ideals. When I finish this I will be looking at http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&hl=en&ei=6p_OTIfkJNXt4gaXv5n_Aw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA Alot of the stuff is in Belton so hopefully I can run through this text reasonably quickly.

I have continued my work on Belton http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf. I had done Tychonov’s Theroem for finite collections – here I saw the proof for the general case. I looked at the Banach-Alaoglu Theorem (the closed unit ball in the dual space of a normed vector space is compact in the weak* toplology). I saw that every normed vector space is isometrically isomorphic to for some compact, Hausdorff space . I looked at topological vector spaces and convexity. Finally I looked at the Krien-Milman Theorem (in a locally convex topological space , every non-empty, compact, convex subset of is the closed, convex hull of it’s extreme points: ). I have done the first two questions 5.1, 5.2.

Having completed my MSc on Random Walks on Finite Groups, it’s time to go quantum! Quantum here refers primarily to non-commutative geometry. In studying a group (or some other geometry), often a study of the algebra of complex functions on the group, (or some similar object depending on the class of group), can tell us everything about the group – we can reconstruct the underlying structure from the algebra of functions on it. In particular, the algebra of functions is a commutative algebra that can encodes the group axioms in a certain way.

Read the rest of this entry »

The Cut-Off Phenomenon in Random Walks on Finite Groups