$C^\star$-algebra and Finite ring extensions

First we need to know what a field extension is. The definition is quite straightforward: if you can construct a new field $K$ from old field $k$ so that $k$ is a subset of $K$, then $K$ is an extension field over $k$, denoted as $K/k$.

If $K/k$ is a field, and element $y$ in K is said to be algebraic over $k$ if $y$ is the solution of some equation $k[x]=0$.

A polynomial $a_n y^n + \cdots + a_1 y + a_0$ is said to be monic if $a_n = 1$.

We have two different dependence relations:

  • algebraic dependence relation: $f(y) = a_n y^n + \cdots + a_1 y + a_0 = 0,\quad f\in k[Y]$, and
  • integral dependence relation: $f(y) = y^n + a_{n-1} y^{n-1}\cdots + a_1 y + a_0 = 0,\quad f\in A[Y]$, where $A$ is a ring.

So, similar to saying “y is algebraic over k”, we can also say things like “y is integral over S, for $S\subset R$ is ring extension”.

The following definition will be important.

  • A Banach space is both a vector space and a measure space.

Being a vector space means that one can add two elements in a Banach space together, but in general not multiply them. A vector space with a multiplicative structure is called an algebra, and if the space is a Banach space, and the multiplication satisfies $\lVert xy \rVert \le \lVert x \rVert \lVert y \rVert$ for any two elements, then it is called a Banach algebra.

A $C^\star$-algebra is a Banach algebra with an involution, which means that for any elements x in the space, it is associated with another element $x^*$ that satisfies

  • $x^\star =x$,
  • $\lVert x^\star \rVert = \lVert x \rVert$,
  • $(x+y)^\star =x^\star +y^\star$,
  • $(xy)^\star = y^\star x^\star$,
  • the $C^\star$-identity, $x^\star x = \lVert x^2 \rVert$.

An example of $C^\star$-algebra is the space of all the continuous linear maps on a Hilbert space, $B(H)$. For an element $T\in B(H)$, $T^\star$ is defined to be $\langle x,Ty \rangle = \langle T^\star x,y \rangle$, and the norm is defined to be the smallest const M such that $\lVert Tx \rVert\leq M \lVert x \rVert$.

A fundamental theorem of Gelfand and Naimark states that every $C^\star$-algebra can be represented as a subalgebra of $B(H)$ for some Hilbert space $H$. I don’t know either how to prove it or the significance of it.

Let A be a ring, by definition, an A-algebra is a ring B with a given ring homomorphism $\phi:A\to B$. Let B be an A-algebra,

  • B is a finite A-algebra (or is finite over A) if it is finite as an A-module. Recall that we say B is a finite A-module, or finitely generated A-module if there exists finite $b_1,\dots,b_n, n\in N$ such that every element of B is an A-linear combination of the $b_i$, $b = x_1 r_1 + \dots x_n r_n, \, r_i \in A$.
  • An element $y\in B$ is integer over A if there exists a monic polynomial $f(Y)\in A’[Y]$ such that $f(y)=0$. The A-algebra B is said to be integral over A if, as you can guess, all the elements are integral over A. Note that the terminology integral domain and integral extension are unrelated.

If B is an A-algebra, then we can naturally define $A \cdot B$, so B is automatically an A-module.

As an example, $k[X^2]$ is integral over $k[X]$.

If B is an A-algebra and $A \subset B$, then B is a ring extension of A. We have the following proposition.

  • The subset of B, defined by $\tilde{A} = {y\in B | y\text{ is integral over A}}$, is a subring. Moreover, $\tilde{\tilde{A}}=\tilde{A}$. $\tilde{A}$ is called the integral closure of A in B. If $A = \tilde{A}$ then A is said to be integral closed in B.

An integral domain is normal if it is integrally closed in its field of fractions. That is, an integral domain A is normal if $A = \tilde{A} \subset F=\text{Frac} A$. For any integral domain A, the integral closure of A in $\text{Frac} A$ is also called the normalization of A.

As an important example, in algebraic number theory, a number field is a finite extension $\mathbb{Q}\subset K$ of the rational field, that is, $[K:\mathbb{Q}] = dim_{\mathbb{Q}}K < \infty$. Recall that notation $[K:F]$ denotes the extension field degree (or relative degree, or index) of K over F, namely the dimension of K as a vector space over F.