# weak Hopf C*-algebra

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 $M$.

• (1)

The comultiplication is not necessarily unit-preserving.

• (2)
• (3)

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

 $\displaystyle m({\rm id}\otimes S)\Delta(h)$ $\displaystyle=(\varepsilon\otimes{\rm id})(\Delta(1)(h\otimes 1))$ (0.1) $\displaystyle m(S\otimes{\rm id})\Delta(h)$ $\displaystyle=({\rm id}\otimes\varepsilon)((1\otimes h)\Delta(1))$ $\displaystyle S(h)$ $\displaystyle=S(h_{(1)})h_{(2)}S(h_{(3)})~{}.$
 ${\begin{CD}A\otimes A@>{S\otimes{\rm id}}>{}>A\otimes A\\ @A{\Delta}A{}A@V{}V{m}V\\ A@ >u\circ\varepsilon>>A\end{CD}}\qquad{\begin{CD}A\otimes A@>{{\rm id}\otimes S% }>{}>A\otimes A\\ @A{\Delta}A{}A@V{}V{m}V\\ A@ >u\circ\varepsilon>>A\end{CD}}$ (0.2)

along with the counit axiom:

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

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

## 0.2 Von Neumann Algebras (or $W^{*}$-algebras).

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

If one calls a commutant of a set $\mathcal{A}$ the special set of bounded operators on $\mathcal{L}(\mathcal{H})$ 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 $\mathcal{L}(\mathcal{H})$, 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 $\mathcal{L}(\mathcal{H})$, closed for the smallest topology  defined by continuous maps $(\xi,\eta)\longmapsto(A\xi,\eta)$ for all $$ where $<.,.>$ denotes the inner product defined on $\mathcal{H}$ . For further instruction on this subject, see e.g. Aflsen and Schultz (2003), Connes (1994).

## References

 Title weak Hopf C*-algebra Canonical name WeakHopfCalgebra Date of creation 2013-03-22 18:12:47 Last modified on 2013-03-22 18:12:47 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 62 Author bci1 (20947) Entry type Topic Classification msc 08C99 Classification msc 16S40 Classification msc 81R15 Classification msc 81R50 Classification msc 16W30 Classification msc 57T05 Synonym quantum groupoids Related topic WeakHopfAlgebra Related topic VonNeumannAlgebra Related topic TopologicalAlgebra Related topic QuantumGroupoids2 Related topic LocallyCompactQuantumGroup Related topic HopfAlgebra Related topic LocallyCompactGroupoids Related topic QuantumGroupoids2 Related topic GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries Related topic GrassmanHopfAlgebrasAndTheirDualCoAlge Defines weak Hopf algebra Defines weak Hopf C*-algebra Defines weak bialgebra Defines quantum group Defines quantum groupoid Defines von Neumann algebra Defines $W^{*}$–algebra