Category Archive

You are currently browsing the category archive for the ‘Research’ category.

Research Update 9

January 25, 2011 in Research, Research Updates | Leave a comment

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

I managed to get through two sections last week: Compact Hilbert Space Operators and The Spectral Theorem. I also have 9 of 12 chapter 2 exercises completed. I have been writing my study up here and this is proving fruitful on three counts:

  1. I can put questions in red for my supervisor to see
  2. I am not happy putting up something on this page that I haven’t justified to myself. This means I have to fill in some extra steps (in blue)
  3. I should have a nice set of notes to peruse should I need them

Unfortunately this week will be mostly concerned with preparing lectures for two modules that I will be lecturing in CIT:

MATH6014

MATH6037

C*-Algebras and Operator Theory: 2.5 The Spectral Theorem

January 19, 2011 in C*-Algebras & Operator Theory, Research | 2 comments

Let X be a compact Hausdorff space and H a Hilbert Space. A spectral measure E relative to (X,H) is a map from the \sigma-algebra of all Borel sets of X to the set of projections in B(H) such that

  1. E(\emptyset)=0, E(X)=1;
  2. E(S_1\cap S_2)=E(S_1)E(S_2) for all Borel sets S_1,\,S_2 of X;
  3. for all x,y\in H, the function E_{x,y}:S\mapsto \langle E(S)x,y\rangle, is a regular Borel complex measure on X.

A Borel measure \mu is a measure defined on Borel sets. If every Borel set in X is both outer and inner regular, then \mu is called regular. A measurable A\subset X is inner and outer regular if

\mu(A)=\sup\left\{\mu(F):\text{ closed }F\subset A\right\}, and

\mu(A)=\inf\left\{\mu(G):A\subset G\text{ open }\right\}

Denote by M(X) the Banach space of all regular Borel complex measures on X, and by B_\infty(X) the C*-algebra of all bounded Borel-measurable complex-valued functions on X (I assume with respect to the Borel \sigma-algebra on \mathbb{C}).

Read the rest of this entry »

C*-Algebras and Operator Theory: 2.4 Compact Hilbert Space Operators

January 18, 2011 in C*-Algebras & Operator Theory, Research | 2 comments

This starts on P.58 rather than p.53. I underline my own extra explanations, calculations, etc. The notation varies on occasion from Murphy to notation which I prefer.

H is always a Hilbert space. H^1 is the set of unit vectors.

If P is a finite-rank projection on H, then the C*-algebra A=PB(H)P is finite dimensional. To see this, write P=\sum_{j=1}^ne_j\otimes e_j, where e_1,\dots,e_n\in H. If T\in B(H), then

PTP=\sum_{j,k=1}^n(e_j\otimes e_j)T(e_k\otimes e_k)=\sum_{j,k=1}^n\langle Te_k,e_j\rangle (e_j\otimes e_k)

Hence, A\subset <e_j\otimes e_k>, (j,k=1,\dots,n) (*), and therefore is finite dimensional.

Read the rest of this entry »

Research Update 8

January 17, 2011 in Research, Research Updates | Leave a comment

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

\|a^*a\|=\|a\|^2, \forall\,a\in A

Self-adjoint or Hermitian elements are defined to have the property a^*=a. As a consequence of this, and the C*-equation, the spectral radius of a self-adjoint element, \nu(a), is equal to its norm, \|a\|. 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, A, can be unitised to form a new algebra, \tilde{A}, which contains A as a subspace. In general, the norm got by extending the norm on A to a norm on \tilde{A} does not make \tilde{A} into a C*-algebra. However Theorem 2.1.6 shows that there does exist a (unique) norm on \tilde{A} 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, \tilde{A}, 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 C_0(X), for some locally compact Hausdorff space, X. In fact X is the character space \Phi(A) (as with Belton, this is via the Gelfand transformation). This identification allows the development of the powerful functional calculus. Briefly, if a is a normal element of a C*-algebra A, (a^*a=aa^*), and z is the inclusion map from \sigma(a)\rightarrow \mathbb{C}, then there exists a unique *-homomorphism \varphi:C(\sigma(a))\rightarrow A such that \varphi(z)=a. This unique *-homorphism is called the functional calculus at a. This particular section ended with the Belton result that if X is a compact Hausdorff space, \Phi(C(X))\cong X (via x\mapsto \delta_x).

2.2 Positive Elements of C*-Algebras

This section introduces a partial order on A_{\text{SA}} (the set of self-adjoint elements of A). Namely, an element a\in A_{\text{SA}} is positive if \sigma(a)\subset \mathbb{R}^+. 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 a\in A, a^*a 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 T is a continuous linear operator on a Hilbert space H, there exists a unique partial isometry S such that T=S|T|; where |T|=(T^*T)^{1/2}. 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 T is compact so are |T| and T^*. Thus K(H) is self-adjoint and thus a C*-algebra (it is a closed ideal in B(H)). We see that normal compact operators are diagonalisable.

We look at the finite rank operators, F(H) and see that they are dense in K(H). Next the operator x\otimes y is examined:

(x\otimes y)(z)=\langle z,y\rangle x

These are rank-one, and the x\otimes x are rank-one projections if x is a unit vector. This leads on to the fact that F(H) 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.

Research Update 7

November 29, 2010 in Research, Research Updates | Leave a comment

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.

Research Update 6

November 22, 2010 in Research, Research Updates | Leave a comment

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 T:X\rightarrow Y between Banach spaces is compact if T(B_1^X[0]) is totally bounded. As a corollary, all linear maps on finite dimensional spaces are compact. The transpose T^*:Y^*\rightarrow X^* has been introduced by Murphy is this chapter, and I have seen that if T is compact, then so is T^*. A linear map T is Fredholm if the T(X) and \text{ker }T are finite dimensional. In terms of progress, I am on p.25 of 265.

Initial Topologies

November 16, 2010 in Research | Leave a comment

As part of my research of http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf I came across a myriad of different topologies that could be put on various spaces associated with a space X. This piece is my attempt to collate and organise them. All are examples of initial topologies. An initial topology is determined by a family of functions.

Initial Topology (Belton 1.24)

Let X be a set and F be collection of functions on X, such that f : X \rightarrow Y_f , where (Y_f,\tau_f) is a topological space, for all f \in F. The initial topology generated by F, denoted by \tau_F , is the coarsest topology such that each function f\in F is continuous.  It is clear that \tau_F is the intersection of all topologies on X that contain

\bigcup_{f\in F} f^{-1}(\tau_f)=\{f^{-1}(U):f\in F,\,U\in \tau_f\}

Product Topology (Belton 3.20)

Let \{(X_a, \tau_a) : a \in A\}be a collection of topological spaces. Their topological product is (X, T), where (\sum is Cartesian product)

X =\sum_{a\in A} X_a:=\{(x_a)_{a\in A}: x_a\in X_a\,,\,\forall\,a\in A\}

is the Cartesian product of the sets X_a and \tau is the initial topology generated by the projection maps, for b\in A :

\pi_b:X\rightarrow X_b;(x_a)_{a\in A}\mapsto x_b

Strong Operator Topology (Belton 2.22)

Let X and Y be normed spaces; the initial topology on the bounded linear operators X\rightarrow Y, B(X, Y) generated by the family of maps \{T \mapsto Tx : x \in X\} (where Y is equipped with its norm topology) is called the strong operator topology.

Weak Topology (Belton 3.4)

Any normed space X gains a natural topology from its dual space, its weak topology. This is the initial topology generated by X^*, i.e., the coarsest topology to make each map \varphi\in X^* continuous. The weak topology on X is denoted by \sigma(X,X^*).

Weak* Topology (Belton 3.19)

Let X be a normed vector space. The weak* topology on X^* is the initial topology generated by the maps \hat{x}: X^* \rightarrow \mathbb{F}; \varphi \mapsto \varphi(x),\,\, (x \in X),i.e., the coarsest topology to make these maps continuous. The weak* topology on X^* is denoted by \sigma(X^*,X).

Research Update 5

November 15, 2010 in Research, Research Updates | Leave a comment

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 \mathbb{C}\backslash \sigma(a)\rightarrow A, \lambda\mapsto (a-\lambda)^{-1} 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.

Research Update 4

November 8, 2010 in Research, Research Updates | Leave a comment

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 A contains no invertible elements and is contained in a maximal ideal. I saw that there is a bijection between the set of characters of A and the set of all maximal ideals.

I saw the links between the characters of A and the spectrum of elements of A. 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

Research Update 3

November 1, 2010 in Research, Research Updates | Leave a comment

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 (M\subset X' is separating if for all x\in X, \exists\,\phi\in M such that \phi(x)\neq 0). 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 B_1^H[0].

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 M_{n}(\mathbb{F}). 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.