30.1 Involutions

Definition 30.1.1 (Involution).label Let $A$ be an associative algebra over $\complex$, and $*: A \to A$ be a $\real$-linear map, then $*$ is an involution if:

1. (C1)

   For each $\lambda \in \complex$, $(\lambda x)^{*} = \ol \lambda x^{*}$.

2. (C2)

   For each $x \in A$, $x^{**}= x$.

3. (I)

   For every $x, y \in A$, $(xy)^{*} = y^{*}x^{*}$.

The space $A$ equipped with an involution is an involutive algebra over $\complex$.

Definition 30.1.2 ($C^{*}$-Algebra).label Let $A$ be an involutive Banach algebra over $\complex$, then $A$ is a $C^{*}$-algebra if for every $x \in A$, $\normn{x^*x}_{A} = \norm{x}_{A}^{2}$.

Proposition 30.1.3.label Let $A$ be a $C^{*}$ algebra, then:

1. (1)

   For each $x \in A$, $\norm{x}_{A} = \normn{x^*}_{A}\norm{x}_{A}$.

If $A$ is unital, then

1. (2)

   For each $\lambda \in \complex$, $\lambda^{*} = \ol \lambda$.

2. (3)

   For any $x \in A$, $x \in G(A)$ if and only if $x^{*} \in G(A)$.

3. (4)

   For every $x \in A$, $\sigma_{A}(x^{*}) = \bracsn{\ol \lambda| \lambda \in \sigma_A(x)}$.

4. (5)

   For each $x \in A$, $[x]_{sp}= [x^{*}]_{sp}$.

Definition 30.1.4 (*-Homomorphism).label Let $A, B$ be $C^{*}$-algebras and $\phi: A \to B$, then $\phi$ is a *-homomorphism if:

1. (1)

   $\phi$ is a homomorphism of Banach algebras.

2. (2)

   For every $x \in A$, $\phi(x^{*}) = \phi(x)^{*}$.

If in addition, $\phi(1) = 1$, then $\phi$ is a unital *-homomorphism.