$C^*$-algebra

(from wikipedia

Definition

In Abstract Algebra, it is a Banach algebra over the complex numbers together with an involution $x\to x^*$ such that

• $(x+y)^*=x^*+y^*$

• $(xy)^*=x^*y^*$

• $(\lambda x)^*=\overline{\lambda}x^*$, for $\lambda \in \mathbb C$

• $\left\|x^* x \right\|=\|x\| \|x^*\|$ or equivalently $\left\|x^* x \right\|=\|x\|^2$.

$\blacksquare$

According to @strocchi2008introduction, any physical system is modelled over a $C^*$-algebra $\mathcal A$.

By the Gelfand-Naimark theorem, it correspond to the algebra of complex continuous functions of a certain topological space $X$, which can be interpreted as the "states" of a physical system. This topological space, called the Gelfand spectrum, has a bijection to the linear functionals defined on $\mathcal A$. Think of the c-star algebra of continuous functions on $X$, every point $\omega \in X$ is a linear functional on $C(X)$...

Moreover, the $*$-invariant elements, also called self adjoint elements, correspond to the observables of the physical system.

An important subset are the positive observables:

• Idea: observables which only take positive values when they are measured.

• Definition: observables $A$ such that $A=C^2$ with $C=C^*$. Equivalently, $A=B^*B$ for any $B$. See @strocchi2008introduction page 31.

Important particular case: the von Neumann algebras.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es