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

Before the Christmas break I finished off the chapter 1 exercises.

Chapter 2: C*-Algebras and Hilbert Space Operators.

2.1 C*-Algebras

Initially we defined a C*-algebra, , as a complete normed algebra, together with a conjugate-linear involution * that satisfies the C*-equation:

,

Self-adjoint or Hermitian elements are defined to have the property . As a consequence of this, and the C*-equation, the spectral radius of a self-adjoint element, , is equal to its norm, . As a corollary of this, of all the norms that can be put on the *-algebra, only one makes it into a C*-algebra – i.e. satisfying the C*-equation.

In the previous chapter we have seen that an algebra, , can be unitised to form a new algebra, , which contains as a subspace. In general, the norm got by extending the norm on to a norm on  does not make  into a C*-algebra. However Theorem 2.1.6 shows that there does exist a (unique) norm on  making it a C*-algebra. In many examples we may now assume that a general C*-algebra is unital – replacing it with the unique unitisation, , if necessary.

One such result which depends on this fact is that the the spectrum of a self-adjoint element is real.

A central result in this chapter is that all abelian C*-algebras are , for some locally compact Hausdorff space, . In fact  is the character space  (as with Belton, this is via the Gelfand transformation). This identification allows the development of the powerful functional calculus. Briefly, if is a normal element of a C*-algebra , (), and  is the inclusion map from , then there exists a unique *-homomorphism  such that . This unique *-homorphism is called the functional calculus at . This particular section ended with the Belton result that if  is a compact Hausdorff space,  (via ).

2.2 Positive Elements of C*-Algebras

This section introduces a partial order on (the set of self-adjoint elements of ). Namely, an element  is positive if . The partial order is defined in the obvious way.

As a consequence of the Gelfand transformation and the functional calculus, we can show that positive elements of a C*-algebra possess unique positive square roots.  Another prominent result is that for an arbitrary element ,  is positive.

2.3 Operators and Sesquilinear Forms

As a first move, we prove that bounded operators on Hilbert spaces have adjoints. Next projections are examined and partial isometries are examined. This leads onto the polar decomposition theorem. Namely, if  is a continuous linear operator on a Hilbert space , there exists a unique partial isometry  such that ; where . The rest of the section focusses on the connection between operators and sesquilinear forms.

2.4 Compact Hilbert Space Operators

At first this chapter looks at some of the basic properties of these objects – e.g. if  is compact so are  and . Thus  is self-adjoint and thus a C*-algebra (it is a closed ideal in ). We see that normal compact operators are diagonalisable.

We look at the finite rank operators,  and see that they are dense in . Next the operator  is examined:

These are rank-one, and the are rank-one projections if  is a unit vector. This leads on to the fact that  is linearly spanned by these rank-one projections.

This is a synopsis of what I covered up until recently (up to p.56). As an experiment I am attempting to do my study of Murphy by way of fully presenting the details on this webpage. I am unsure of whether or not this is too time consuming. Presently I am on page 63 and I will have to cover the rest of the chapter material (10 pages) in one day or similar if I am going to consider this tactic feasible.

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