On the symbolic calculus in Banach algebra 1. Symbolic calculus in general Banach algebra

Rudin's This "functional analysis" is really not very good to read, in general, because the books in the details of the place is indeed too much, not very good grasp. I look at this book with the purpose of learning operator algebra, so I don't want to fall into the function space, especially the Fourier transformation problem associated with it, so

After studying the relevant knowledge about the topological and space, the local convex space and the completeness, I studied the last three chapters directly, that is, the subject of Banach algebra and c*-algebra.

Reading this book, I harvested the largest place in Banach algebra, and B (H) in the context of the symbolic calculus.

Let's briefly explain what a symbolic calculus is. The most essential purpose of symbolic calculus is to grasp the relationship of elements in Banach algebra in a deeper level, and in a sense, we want to translate the related properties of C, the complex plane into Banach algebra, and then get the properties of Banach algebra. The method of symbolic computation is to establish the legality of f (a), where f is any complex function, A is any element of Banach algebra B, in fact a^2 is obviously definable, generally, when f is a complex coefficient polynomial, according to B is a linear space on C and the multiplication defined on it, f (a) Definition is also clear, so when F is more complex, we also want to define an F (a).

To give a simple example, for an arbitrary A to be B, we use a power series to define the e^a=∑a^n/n!, and to discuss whether the definition is meaningful only needs to discuss whether the power series converges to the point in B in accordance with the norm, which is obvious, because B is complete.

For some other function f, of course we can define it in terms of the power series, but it will not eventually make us go farther, and it is difficult to grasp their characteristics in such a manual way.

As for the symbolic calculus in Banach algebra, I will be divided into two aspects, one is general Banach algebra B, the other is c*-algebra B (H), the steps to establish the symbolic calculus and some interesting ideas.

1. Establish symbolic calculus for individual elements based on point spectrum

The establishment of Banach algebraic symbolic calculus is derived from three heuristic points, one is the establishment of the vector value integral, the other is the whole pure function theory, especially the Cauchy integral formula. The third is the wide existence of strong all-pure B-valued function in Banach algebra. The foundation of the establishment is the perfect spectrum theory.

Let's start with a brief discussion of vector-valued integrals. To a compact Hausdorff space (x,m,μ), m,μ respectively is the Borel set of X and Borel measure, a topological space Y, considering f:x→y is a continuous mapping. Define the so-called F about measure μ Integral is a y belongs to y,y satisfies to any Y

The continuous linear functional lambda on the ∫λfdμ=λy. Here λf is a continuous function on X, which is naturally Borel measurable, and we certainly define its integral on X in the way of general measure theory. However, not for any one topological linear space y, so that Y is present and unique

, it is generally guaranteed that Y is Frechet space (the topology is induced by a complete invariant metric), or that the integral is always present and unique when Banach space.

What we are going to use here is to consider the case of a simple closed curve on a complex plane to Banach algebra B. Here the simple closed curve is the compact Hausdorff space, and its topology is induced by the complex plane, and the measure is dz=dx+idy. This is a special complex measure, and the measure of an arc length is only

Endpoint decision. F is a continuous function from X to B, and we hope that this is a function of the so-called strong pure. The strong fully pure function here refers to the norm satisfying//{f (x)-F (y)/x-y}-a (y)//tending to 0 when x tends to each y. This is a natural generalization of all-pure functions. The Chang function is naturally continuous, so it can be used as integral. Such strong all-pure B-valued functions are always present, such as the pure function f on each ω, which defines the F (λ) e:x→b. This is a very ordinary example.

Next talk about the Cauchy integral formula, which is the foundation of the complex changes in the basic content, if only from the formal view of the theorem, it means that all pure function f has integral representation, which is the starting point of the symbolic calculus system, we also want to use the integral of f (a) as a pure vector value integral form. For example

F (a) =∫f (λ) (λe-a) ^-1dλ.

Formally, such a definition is exactly like the Cauchy integral formula! But here's the first thing we need to figure out: first, because (λe-a) ^-1 doesn't always exist, so the final question is, what is the closed-curve integral to C, and secondly, we'll figure out

F (λ) (λe-a) ^-1 is not strong all-pure, or (ΛE-A) ^-1 is not strong all-pure. Thirdly, the definition is not broadly significant, that is, when F (λ) is a polynomial, it should be the same as the usual result.

After we have solved these three problems, the symbolic calculus problem of general Banach algebra is basically set up.

The first question points directly to the spectral theory in Banach algebra, or, when λ takes what value (λe-a) ^-1 exists. The result of spectral theory is that such λ constitutes an unbounded open set (in fact, the spectrum of X is a compact set), which is the most basic and most beautiful theorem in spectral theory, from which we can turn the integral curve into a simple closed curve around σ (x) in Ω if f (λ) (λe-a) ^-1 is strongly pure in Ω ( This function is at least meaningful in Ω, similar to the complex function, so that the integral has homotopy invariance and will not affect the integral value with the deformation of the curve.

The second question is whether (λe-a) ^-1 in its defined open concentration is strong and pure, the answer is certainly, as to prove that will use a power series of an inequality, relatively simple.

The third question, when f is polynomial, whether the calculated integral value is the usual definition, that is, whether it is the generalization of the usual definition, this proof is not very complicated, can be done by inductive method.

In this way, we define the symbolic calculus problem of an element, and we can take it further.

2. Holistic Processing

By using the topological properties of spectral theory and Banach algebra, we can know that the elements in Ω in the ω,b of a given complex plane are composed of an open set O of B, and any given F is an all-pure function in Ω, defining a continuous function t:o→b:

T (x) =∫f (λ) (Λe-x) ^-1dλ., it should be noted that for different x, the different curves of the bypass σ (x) are selected in Ω. All the continuous functions on the o that are constructed are represented by H (O).

Here we are actually processing many of the x in B to get the O to B function.

The most important theorem here is that H (O) forms an commutative algebra according to addition, scalar multiplication and function multiplication, and is an algebraic isomorphism of all pure functions under f→t and Ω.

I'll explain the meaning of the theorem a little bit. Here the H (O) and H (Ω) are made into algebra. H (Ω) is made into an algebra as follows: First it is made into a C-linear space, the key is to define the multiplication: FG (x) =f (x) g (x), because FG is still completely pure, so this definition is clear. But H (O) is not so obvious, if the multiplication UV (x) =u (x) v (x) is defined as H (Ω), the resulting uvs may not even be induced by an F-h (Ω), in other words, it does not necessarily form an algebra by multiplication. But this theorem guarantees this, and induces an algebraic isomorphism under the action of nature, so we can even know that H (0) can not only be made into an algebra, but also an commutative algebra!

This allows us to equate the operation in H (Ω) with H (O).

Summary: From the above process, we can see that the symbolic operation in Banach algebra, in fact, the function of the complex plane into B, the complex plane of the function operation into the operation of the function in H (O), intuitively, from an element, we will be able to construct many elements,

at a deeper level, this is an analogy between the complex plane and B, and the intrinsic nature of Banach algebra is dug out in a never-trivial known (fully pure function theory) . In the symbolic calculus in B (H), we will be able to see this clearly.

On the symbolic calculus in Banach algebra 1. Symbolic calculus in general Banach algebra