The inner product on the space bounded linear operators over a Hilbert space

Question

I am reading book "A short course on Spectral Theory", written by William Averson and I got some stucks.

Lemma 2.4.4 (page 54) claimed that:

Let $A$ be a normal operator on a Hilbert space $H$ and assume that the $*-$ algebra generated by $A$ and the identity has a cyclic vector. Then $A$ is diagonalizable.

In order to proof it, the author stated that

Recalling that the functional calculus for normal operators provides a $*-$ homomorphism $f \in C(X) \rightarrow f(A) \in B(H)$.

In here, $X$ is the spectrum of $A$.

If we denote $ \varphi : f \in C(X) \rightarrow f(A) \in B(H)$, what is the adjoint $\varphi ^*$ of $\varphi$?

Furthermore, how to prove that $\varphi$ is a $*-$ homomorphism?

Answer 1

Let $A$ and $B$ be two (Banach) *-algebras. Then a map $\varphi:A\to B$ is called a *-homomorphism if:

1. It is linear.
2. It is multiplicative, i.e. $\varphi(xy)=\varphi(x)\varphi(y)$.
3. It is continuous.
4. It preserves involutions, i.e. $\varphi(x^*)=\varphi(x)^*$.

The functional calculus for normal operators consists in a *-homomorphism $\varphi: C(X)\to B(H)$ between the $*$-algebras $C(X)$ and $B(H)$.

Here, $C(X)$ is a *-algebra with pointwise function sum and multiplication, and the * is given by complex conjugation, whereas in $B(H)$ the operations are operator sum and composition, and the * is given by the adjoint operator.

Therefore, if we denote $\varphi(f)$ as $f(A)$, we get that $\varphi(f)^*=f(A)^*=\bar{f}(A)$. In particular, $f(A)$ is self-adjoint iff $f$ is a real valued function.

To prove that there exist a $*$-homomorphism $\varphi: C(X)\to B(H)$, you construct it as follows. You first set $\varphi(1\mapsto x)=I$, $\varphi(x\mapsto x)=A$, $\varphi(x\mapsto \bar{x})=A^*$, and then extend it first to all polynomials $p(x,x^*)$ by linearity and multiplicativity, and then to all continuous functions in $C(X)$ by density. It is not hard to check that the resulting extended map is still a $*$-homomorphism, in fact an isometric one. Its range is the sub-*-algebra of $B(H)$ generated by $A$ (i.e. the smallest *-algebra containing $A$).