weak Hopf C*-algebra

Definition 0.1   A weak Hopf $ C^*$ -algebra is defined as a weak Hopf algebra which admits a faithful $ *$ -representation on a Hilbert space. The weak C*-Hopf algebra is therefore much more likely to be closely related to a `quantum groupoid' than the weak Hopf algebra. However, one can argue that locally compact groupoids equipped with a Haar measure are even closer to defining quantum groupoids. There are already several, significant examples that motivate the consideration of weak C*-Hopf algebras which also deserve mentioning in the context of `standard' quantum theories. Furthermore, notions such as (proper) weak C*-algebroids can provide the main framework for symmetry breaking and quantum gravity that we are considering here. Thus, one may consider the quasi-group symmetries constructed by means of special transformations of the “coordinate space” $ M$ .

Remark: Recall that the weak Hopf algebra is defined as the extension of a Hopf algebra by weakening the definining axioms of a Hopf algebra as follows :

(1)
The comultiplication is not necessarily unit-preserving.

(2)
The counit $ \varepsilon $ is not necessarily a homomorphism of algebras.

(3)
The axioms for the antipode map $ S : A {\longrightarrow}A$ with respect to the counit are as follows. For all $ h \in H$ ,

\begin{equation*}\begin{aligned}m({\rm id}\otimes S) \Delta (h) &= (\varepsilon ...
...lta(1)) \\ S(h) &= S(h_{(1)}) h_{(2)} S(h_{(3)}) ~. \end{aligned}\end{equation*}

These axioms may be appended by the following commutative diagrams

$\displaystyle {\begin{CD}A \otimes A @> S\otimes {\rm id}>> A \otimes A \\ @A \...
...A \otimes A \\ @A \Delta AA @VV m V \\ A @ > u \circ \varepsilon >> A \end{CD}}$ (0.2)

along with the counit axiom:

$\displaystyle \xymatrix@C=3pc@R=3pc{ A \otimes A \ar[d]_{\varepsilon \otimes 1}...
...{\rm id}_A} \ar[d]^{\Delta} \\ A & A \otimes A \ar[l]^{1 \otimes \varepsilon }}$ (0.3)

Some authors substitute the term quantum `groupoid' for a weak Hopf algebra.

Examples of weak Hopf C*-algebra.

(1)

In Nikshych and Vainerman (2000) quantum groupoids were considered as weak C*-Hopf algebras and were studied in relationship to the noncommutative symmetries of depth 2 von Neumann subfactors. If

$\displaystyle A \subset B \subset B_1 \subset B_2 \subset \ldots$ (0.4)

is the Jones extension induced by a finite index depth $ 2$ inclusion $ A \subset B$ of $ II_1$ factors, then $ Q= A' \cap B_2$ admits a quantum groupoid structure and acts on $ B_1$ , so that $ B
= B_1^{Q}$ and $ B_2 = B_1 \rtimes Q$  . Similarly, in Rehren (1997) `paragroups' (derived from weak C*-Hopf algebras) comprise (quantum) groupoids of equivalence classes such as associated with 6j-symmetry groups (relative to a fusion rules algebra). They correspond to type $ II$ von Neumann algebras in quantum mechanics, and arise as symmetries where the local subfactors (in the sense of containment of observables within fields) have depth $ 2$ in the Jones extension. Related is how a von Neumann algebra $ N$ , such as of finite index depth $ 2$ , sits inside a weak Hopf algebra formed as the crossed product $ N \rtimes A$ (Böhm et al. 1999).

(2)
In Mack and Schomerus (1992) using a more general notion of the Drinfeld construction, develop the notion of a quasi triangular quasi-Hopf algebra (QTQHA) is developed with the aim of studying a range of essential symmetries with special properties, such the quantum group algebra $ {\rm U}_q (\rm {sl}_2)$ with $ \vert q \vert =1$  . If $ q^p=1$ , then it is shown that a QTQHA is canonically associated with $ {\rm U}_q (\rm {sl}_2)$ . Such QTQHAs are claimed as the true symmetries of minimal conformal field theories.

Von Neumann Algebras (or $ W^*$ -algebras).

Let $ \H$ denote a complex (separable) Hilbert space. A von Neumann algebra $ \mathcal A$ acting on $ \H$ is a subset of the $ *$ -algebra of all bounded operators $ \mathcal L(\H )$ such that:

(1)
$ \mathcal A$ is closed under the adjoint operation (with the adjoint of an element $ T$ denoted by $ T^*$ ).

(2)
$ \mathcal A$ equals its bicommutant, namely:

$\displaystyle \mathcal A= \{A \in \mathcal L(\H ) : \forall B \in \mathcal L(\H ), \forall C\in \mathcal A,~ (BC=CB)\Rightarrow (AB=BA)\}~.$ (0.5)

If one calls a commutant of a set $ \mathcal A$ the special set of bounded operators on $ \mathcal L(\H )$ which commute with all elements in $ \mathcal A$ , then this second condition implies that the commutant of the commutant of $ \mathcal A$ is again the set $ \mathcal A$ .

On the other hand, a von Neumann algebra $ \mathcal A$ inherits a unital subalgebra from $ \mathcal L(\H )$ , and according to the first condition in its definition $ \mathcal A$ does indeed inherit a *-subalgebra structure, as further explained in the next section on C*-algebras. Furthermore, we have the notable Bicommutant theorem which states that $ \mathcal A$ is a von Neumann algebra if and only if $ \mathcal A$ is a *-subalgebra of $ \mathcal L(\H )$ , closed for the smallest topology defined by continuous maps $ (\xi,\eta)\longmapsto (A\xi,\eta)$ for all $ <A\xi,\eta)>$ where $ <.,.>$ denotes the inner product defined on $ \H$  . For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994).

Commutative and noncommutative Hopf algebras form the backbone of quantum `groups' and are essential to the generalizations of symmetry. Indeed, in most respects a quantum `group' is identifiable with a Hopf algebra. When such algebras are actually associated with proper groups of matrices there is considerable scope for their representations on both finite and infinite dimensional Hilbert spaces.

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