compact quantum groupoids related to C*-algebras
1 Compact quantum groupoids (CGQs) and C*-algebras
1.1 Introduction: von Neumann and C*-algebras. Quantum operator algebras in quantum theories
C*-algebra has evolved as a key concept in quantum operator algebra
(QOA) after the introduction of thevon Neumann algebra
for the mathematical foundation
of quantum mechanics. The von Neumann algebra classification is simpler and studied in greater depth than that of general C*-algebra classification theory. The importance ofC*-algebras for understanding the geometry of quantum state spaces
(viz. Alfsen and Schultz, 2003 [1]) cannot be overestimated. Moreover, the introduction of non-commutative C*-algebras in noncommutative geometry
has already played important roles in expanding the Hilbert space
perspective of quantum mechanics developed by von Neumann. Furthermore, extended quantum symmetries are currently being approached in terms of groupoid
C*- convolution algebra and their representations
; the latter also enter into the construction of compact
quantum groupoids as developed in the Bibliography cited, and also briefly outlined here in the third section
. The fundamental connections
that exist between categories
of -algebras and those of von Neumann and other quantum operator algebras, such as JB- or JBL- algebras are yet to be completed and are the subjectof in depth studies [1].
1.2 Basic definitions
Let us recall first the basic definitions of C*-algebra and involution on a complex algebra.Further details can be found in a separate entry focused on -algebras (http://planetmath.org/CAlgebra).
A C*-algebra is simultaneously a *–algebra and a Banach space -with additional conditions- as defined next.
Let us consider first the definition of an involution on a complex algebra .
Definition 1.1.
An involution on a complex algebra is a real–linear map such that for all and , we have
A *-algebra is said to be a complex associative algebra together with an involution .
Definition 1.2.
A C*-algebra is simultaneously a *-algebra and a Banach space ,satisfying for all the following conditions:
One can easily verify that .
By the above axioms a C*–algebra is a special case of a Banach algebra where the latter requires the above C*-norm property, but not the involution () property.
Given Banach spaces the space of (bounded) linear operators from to forms a Banach space, where for , the space is a Banach algebra with respect to the norm
In quantum field theory one may start with a Hilbert space , and consider the Banachalgebra of bounded linear operators which given to be closed under the usualalgebraic operations and taking adjoints
, forms a –algebra of bounded operators
, where theadjoint operation functions as the involution, and for we have :
and
By a morphism between C*-algebras we mean a linear map , such that for all , the following hold :
where a bijective morphism is said to be an isomorphism
(in which case it is then anisometry). A fundamental relation
is that any norm-closed -algebra in is a C*-algebra (http://planetmath.org/CAlgebra3), and conversely, any C*-algebra (http://planetmath.org/CAlgebra3) is isomorphic to a norm–closed -algebra in for some Hilbert space .One can thus also define the category of C*-algebras and morphisms between C*-algebras.
For a C*-algebra (http://planetmath.org/CAlgebra3) , we say that is self–adjoint if . Accordingly, the self–adjoint part of is a realvector space since we can decompose as :
A commutative C* -algebra is one for which the associative multiplication iscommutative. Given a commutative C* -algebra , we have ,the algebra of continuous functions
on a compact Hausdorff space .
The classification of -algebras is far more complex than that of von Neumann algebras that providethe fundamental algebraic content of quantum state and operator spaces in quantum theories.
1.3 Quantum groupoids and the groupoid C*-algebra
Quantum groupoid (or their dual, weak Hopf coalgebras) and algebroid symmetries figure prominently both in the theory of dynamical deformations
of quantum groups
(or their dual Hopf algebras
) and the quantum Yang–Baxter equations (Etingof et al., 1999, 2001; [12, E2k]). On the other hand, one can also consider the naturalextension
of locally compact (quantum) groups to locally compact(proper) groupoids equipped with a Haar measure and a corresponding groupoid representation
theory(Buneci, 2003,[MB2k3]) as a major, potentially interesting source for locally compact (butgenerally non-Abelian
) quantum groupoids. The corresponding quantum groupoid representations on bundles ofHilbert spaces extend quantum symmetries well beyond those of quantum groups and their dual Hopf algebras, and also beyond the simpler operator algebra representations, and are also consistent
with the locally compact quantum group representations. The latter quantum groups are neither Hopf algebras, nor are they equivalent
to Hopf algebras or their dual coalgebras. As pointed out in the previous section, quantum groupoid representations are, however, the next important step towards unifying quantum field theories with General Relativity in a locally covariant and quantized form. Such representations need not however be restricted to weak Hopf algebra representations, as the latter have no known connection to any type of GR theory and also appear to be inconsistent with GR.
Quantum groupoids were recently considered as weak C* -Hopf algebras, and were studied in relationship to the non- commutative symmetries of depth 2 von Neumann subfactors. If
(1.1) |
is the Jones extension induced by a finite index depth inclusion of factors, then admits a quantum groupoid structure and acts on , so that and . Similarly, ‘paragroups’ derived from weak C* -Hopf algebras comprise (quantum) groupoids of equivalence classes
such as those associated with -symmetry groups (relative to a fusion rules algebra). They correspond to type von Neumann algebras in quantum mechanics, and arise as symmetries where the local subfactors (in the sense of containment of quantum observables within fields) have depth 2 in theJones extension. A related question is how a von Neumann algebra , such asof finite index depth 2, sits inside a weak Hopf algebra formed as the crossed product.
1.4 Quantum compact groupoids
Compact quantum groupoids were introduced in Landsman (1998; ref. [L98]) as asimultaneous generalization of a compact groupoid and a quantum group. Since this construction is relevant to the definition of locally compact quantum groupoids and their representations investigated here, its exposition is required before we can step up to the next level of generality. Firstly, let and denote C*–algebras equipped with a *–homomorphism
, and a *–antihomomorphism whose images in commute. A non–commutative Haar measure is defined as a completelypositive map which satisfies . Alternatively, the composition is a faithful conditional expectation.
Next consider to be a (topological) groupoid, and let us denote by the space of smooth complex–valued functions with compact support on . In particular, for all , thefunction defined via convolution
(1.2) |
is again an element of , where the convolution product defines the composition law on . We can turn into a -algebra once we have defined the involution, and this is done by specifying .
We recall that following Landsman (1998) a representation of a groupoid , consists of afamily (or field) of Hilbert spaces indexed by , along with a collection of maps , satisfying:
- 1.
,is unitary
.
- 2.
, whenever (the set of arrows).
- 3.
, for all .
Suppose now is a Lie groupoid. Then the isotropy group is a Lie group
, and for a (left or right) Haarmeasure on , we can consider the Hilbertspaces as exemplifying theabove sense of a representation. Putting aside some technicaldetails which can be found in Connes (1994) and Landsman (2006), theoverall idea is to define an operator of Hilbert spaces
(1.3) |
given by
(1.4) |
for all , and . For each , defines an involutive representation . We can define a norm on given by
(1.5) |
whereby the completion of in this norm, definesthe reduced
C*–algebra of . It isperhaps the most commonly used C*–algebra for Lie groupoids(groups) in noncommutative geometry.
The next step requires a little familiarity with the theory ofHilbert modules. We define a left–action and a right –action on by and . For the sake of localization of theintended Hilbert module, we implant a –valued innerproduct on given by . Let us recall that is defined as a completely positive map.Since is faithful, we fit a new norm on given by . The completion of in this newnorm is denoted by leading then to a Hilbertmodule over .
The tensor product can be shown to be a Hilbert bimodule over ,which for , leads to *–homorphisms . Next is to define the (unital) C*–algebra as the C*–algebra contained in that is generated by and . The last stage of the recipe for defining acompact quantum groupoid entails considering a certain coproduct
operation , together with a coinverse that it is both an algebra andbimodule antihomomorphism. Finally, the following axiomaticrelationships are observed :
(1.6) | ||||
where is a flip map : .
There is a natural extension of the above definition of quantum compact groupoidsto locally compact quantum groupoids by taking to be a locally compact groupoid (instead of a compact groupoid), and then following the steps in the above construction with the topological groupoid being replaced by . Additional integrability and Haar measure system conditions need however be also satisfied as in the general case of locally compact groupoid representations (for further details, see for example the monograph by Buneci (2003).
1.4.1 Reduced C*–algebra
Consider to be a topological groupoid. We denote by the space of smooth complex–valued functions with compact support on . In particular, for all , thefunction defined via convolution
(1.7) |
is again an element of , where the convolution productdefines the composition law on . We can turn into a *–algebra once we have defined the involution, and this is done by specifying .
We recall that following Landsman (1998) a representation of a groupoid , consists of afamily (or field) of Hilbert spaces indexed by , along with a collection of maps , satisfying:
- 1.
,is unitary.
- 2.
, whenever (the set of arrows).
- 3.
, for all .
Suppose now is a Lie groupoid. Then the isotropy group is a Lie group, and for a (left or right) Haarmeasure on , we can consider the Hilbertspaces as exemplifying theabove sense of a representation. Putting aside some technicaldetails which can be found in Connes (1994) and Landsman (2006), theoverall idea is to define an operator of Hilbert spaces
(1.8) |
given by
(1.9) |
for all , and . For each , defines an involutive representation . We can define a norm on given by
(1.10) |
whereby the completion of in this norm, definesthe reduced C*–algebra of .
It is perhaps the most commonly used C*–algebra for Lie groupoids(groups) in noncommutative geometry.
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