nLab H-star-algebra

Definition

Definition

An $H^*$-algebra is a (not necessarily unital) Banach 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

$$\langle a b, c \rangle_A = \langle b, a^{\dagger} c \rangle_A = \langle a , c b^{\dagger} \rangle_A \,, \;\;\; \text{for all}\; a,b,c \in A.$$

(Ambrose 1945 Def. 1.2)

Definition

An $H^\ast$-algebra $A$ (Def. ) is called proper if

• The only element $a \in A$ such that $a \cdot (-)$ is the zero map $A \to A$ is $a = 0$.

or equivalently

• The only element $a \in A$ such that $(-) \cdot a$ is the zero map $A \to A$ is $a = 0$.

(Ambrose 1945 Def. 2.1)

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 proper $H^*$-algebra (Def. ), where the the involution is defined by sending an element $a \in A$ to

(cf. Abramsky & Heunen 2012)

Related concepts

• $C^\ast$-algebra

References

The original article:

• Warren Ambrose: Structure Theorems for a special class of Banach Algebras, Transactions of the American Mathematical Society 57 (1945) 364–386 [doi:10.2307/1990182, jstor:1990182, pdf]

Discussion in relation to Frobenius algebra-structures on finite-dimensional Hilbert spaces:

• Samson Abramsky, Chris Heunen: $H^\ast$-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics, in Clifford Lectures, AMS Proceedings of Symposia in Applied Mathematics 71 (2012) 1–24 [arXiv:1011.6123]

• 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]