$C^{\ast }$-algebra

(from wikipedia)
Definition
In Abstract Algebra, it is a Banach algebra over the complex numbers together with an involution $x\to x^{\ast }$ such that

• $(x+y{)}^{\ast }=x^{\ast }+y^{\ast }$
• $(xy{)}^{\ast }=x^{\ast }{y}^{\ast }$
• $(\lambda x{)}^{\ast }=\bar{\lambda }{x}^{\ast }$, for $\lambda \in \mathbb{C}$
• $\Vert x^{\ast }x\Vert =\parallel x\parallel \parallel x^{\ast }\parallel$ or equivalently $\Vert x^{\ast }x\Vert =\parallel x{\parallel }^2$.
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According to @strocchi2008introduction, any physical system is modeled over a $C^{\ast }$-algebra $\mathcal{A}$.
By the Gelfand-Naimark theorem, it corresponds 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 $\ast$-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 $C=C^{\ast }$. Equivalently, $A=B^{\ast }B$ for any $B$. See @strocchi2008introduction page 31.

Important particular case: the von Neumann algebras.