nLab H-star-algebra

Definition

An $H^*$-algebra is a $\mathbb{C}$-algebra $A$ that is simultaneously a Hilbert space $(A, \langle \cdot, \cdot \rangle_A)$ with a compatible star-algebra structure, namely with an anti-linear involution $\dagger \colon A \to A$ such that

$$\langle a b, c \rangle_A = \langle b, a^{\dagger} c \rangle_A = \langle a , c b^{\dagger} \rangle_A$$

for all $a,b,c \in A$.

Properties

Classifying Frobenius Algebras

Frobenius structures in the category of finite-dimensional Hilbert spaces can be classified via $H^*$-algebras.

Theorem

A monoid $(A, \mu, \eta)$ internal to $\mathrm{FdHilb}$ is a symmetric dagger Frobenius monoid if and only if it is a finite-dimensional $H^*$-algebra, where the the involution is defined by sending an element $a \in A$ to

References

An exposition of the classification of Frobenius structures in finite-dimensional Hilbert spaces as $H^*$-algebras:

• Chris Heunen, Jamie Vicary: Categories for Quantum Theory, Oxford University Press (2019) [ISBN:9780198739616]

  based on:

  Chris Heunen, Jamie Vicary, Lectures on categorical quantum mechanics (2012) [pdf, pdf]