Israel Gelfand (1913-2009)

Mark Naimark (1909-1978)

Irvin Segal (1918-1998)

A particular case of the GNS construction

Unital $\phantom{A}^$$^$$\mathcal{M} \subseteq$ C$^*$-algebra $\mathcal{A}$ equipped with a state $\varphi$

Unital $^*$-algebra $\mathcal{M}$:

• Vector space with addition and scalar multiplication.
• $(xy)z=x(yz)$ (associative multiplication).
• $(\lambda x + y)z= \lambda xy + yz$ (distributivity).
• Involution: $(\alpha x+ \beta y)^= \overline{\alpha} x^ + \overline{\beta}y^$, $(xy)^=y^x^,$ $(x^)^=x$.
• multiplicative unit $1$.

$C^*$-algebra $\mathcal{A}$:

• $\| x^*x \|= \|x \|^2$.
• Complete as a normed space.

State $\varphi$ on $\mathcal{A}$:

• linear functional.
• positive: $\varphi (x^*x) \geq 0.$
• $\varphi (1)=1.$
• faithful on $\mathcal{M}$: $\varphi(x^*x)=0 \Rightarrow x=0$ for every $x \in \mathcal{M}$.

Define an Inner product on $\mathcal{M}$:

$$ \langle x| y \rangle_\varphi = \varphi(x^*y) \quad \text{for all }x,y \in \mathcal{M}. $$

Complete $\mathcal{M}$ to become a Hilbert space $\mathcal{H}_\varphi$: