Locally compact quantum group

Definitions

Before we can even begin to properly define a locally compact quantum group, we first need to define a number of preliminary concepts and also state a few theorems.

Definition (weight). Let A be a C*-algebra, and let A_{\geq 0} denote the set of positive elements of A . A weight on A is a function \phi: A_{\geq 0} \to [0,\infty] such that

• \phi(a_{1} + a_{2}) = \phi(a_{1}) + \phi(a_{2}) for all a_{1},a_{2} \in A_{\geq 0} , and
• \phi(r \cdot a) = r \cdot \phi(a) for all r \in 0,\infty) and a \in A_{\geq 0} .

Some notation for weights. Let \phi be a weight on a C*-algebra A . We use the following notation:

• \mathcal{M}{\phi}^{+} := { a \in A{\geq 0} \mid \phi(a) , which is called the set of all positive \phi -integrable elements of A .
• \mathcal{N}_{\phi} := { a \in A \mid \phi(a^{*} a) , which is called the set of all ** \phi -square-integrable elements** of A .
• \mathcal{M}{\phi} := \text{Span} ~ \mathcal{M}{\phi}^{+} = \text{Span} ~ \mathcal{N}{\phi}^{*} \mathcal{N}{\phi} , which is called the set of all ** \phi -integrable** elements of A .

Types of weights. Let \phi be a weight on a C*-algebra A .

• We say that \phi is faithful if and only if \phi(a) \neq 0 for each non-zero a \in A_{\geq 0} .
• We say that \phi is [lower semi-continuous if and only if the set { a \in A_{\geq 0} \mid \phi(a) \leq \lambda } is a closed subset of A for every \lambda \in [0,\infty] .
• We say that \phi is densely defined if and only if \mathcal{M}{\phi}^{+} is a dense subset of A{\geq 0} , or equivalently, if and only if either \mathcal{N}{\phi} or \mathcal{M}{\phi} is a dense subset of A .
• We say that \phi is proper if and only if it is non-zero, lower semi-continuous and densely defined.

Definition (one-parameter group). Let A be a C*-algebra. A one-parameter group on A is a family \alpha = (\alpha_{t}){t \in \mathbb{R}} of *-automorphisms of A that satisfies \alpha{s} \circ \alpha_{t} = \alpha_{s + t} for all s,t \in \mathbb{R} . We say that \alpha is norm-continuous if and only if for every a \in A , the mapping \mathbb{R} \to A defined by t \mapsto {\alpha_{t}}(a) is continuous.

Definition (analytic extension of a one-parameter group). Given a norm-continuous one-parameter group \alpha on a C*-algebra A , we are going to define an analytic extension of \alpha . For each z \in \mathbb{C} , let : I(z) := { y \in \mathbb{C} \mid |\Im(y)| \leq |\Im(z)| } , which is a horizontal strip in the complex plane. We call a function f: I(z) \to A norm-regular if and only if the following conditions hold:

• It is analytic on the interior of I(z) , i.e., for each y_{0} in the interior of I(z) , the limit \displaystyle \lim_{y \to y_{0}} \frac{f(y) - f(y_{0})}{y - y_{0}} exists with respect to the norm topology on A .
• It is norm-bounded on I(z) .
• It is norm-continuous on I(z) . Suppose now that z \in \mathbb{C} \setminus \mathbb{R} , and let : D_{z} := { a \in A \mid \text{There exists a norm-regular} ~ f: I(z) \to A ~ \text{such that} ~ f(t) = {\alpha_{t}}(a) ~ \text{for all} ~ t \in \mathbb{R} }. Define \alpha_{z}: D_{z} \to A by {\alpha_{z}}(a) := f(z) . The function f is uniquely determined (by the theory of complex-analytic functions), so \alpha_{z} is well-defined indeed. The family (\alpha_{z})_{z \in \mathbb{C}} is then called the analytic extension of \alpha .

Theorem 1. The set \cap_{z \in \mathbb{C}} D_{z} , called the set of analytic elements of A , is a dense subset of A .

Definition (K.M.S. weight). Let A be a C*-algebra and \phi: A_{\geq 0} \to [0,\infty] a weight on A . We say that \phi is a K.M.S. weight ('K.M.S.' stands for 'Kubo-Martin-Schwinger') on A if and only if \phi is a proper weight on A and there exists a norm-continuous one-parameter group (\sigma_{t})_{t \in \mathbb{R}} on A such that

• \phi is invariant under \sigma , i.e., \phi \circ \sigma_{t} = \phi for all t \in \mathbb{R} , and
• for every a \in \text{Dom}(\sigma_{i / 2}) , we have \phi(a^{} a) = \phi(\sigma_{i / 2}(a) [\sigma_{i / 2}(a)]^{}) .

We denote by M(A) the multiplier algebra of A.

Theorem 2. If A and B are C*-algebras and \pi: A \to M(B) is a non-degenerate *-homomorphism (i.e., \pi[A] B is a dense subset of B ), then we can uniquely extend \pi to a *-homomorphism \overline{\pi}: M(A) \to M(B) .

Theorem 3. If \omega: A \to \mathbb{C} is a state (i.e., a positive linear functional of norm 1 ) on A , then we can uniquely extend \omega to a state \overline{\omega}: M(A) \to \mathbb{C} on M(A) .

Definition (Locally compact quantum group). A (C*-algebraic) locally compact quantum group is an ordered pair \mathcal{G} = (A,\Delta) , where A is a C*-algebra and \Delta: A \to M(A \otimes A) is a non-degenerate *-homomorphism called the co-multiplication, that satisfies the following four conditions:

• The co-multiplication is co-associative, i.e., \overline{\Delta \otimes \iota} \circ \Delta = \overline{\iota \otimes \Delta} \circ \Delta .
• The sets \left{ \overline{\omega \otimes \text{id}}(\Delta(a)) ~ \big| ~ \omega \in A^{}, ~ a \in A \right} and \left{ \overline{\text{id} \otimes \omega}(\Delta(a)) ~ \big| ~ \omega \in A^{}, ~ a \in A \right} are linearly dense subsets of A .
• There exists a faithful K.M.S. weight \phi on A that is left-invariant, i.e., \phi ! \left( \overline{\omega \otimes \text{id}}(\Delta(a)) \right) = \overline{\omega}(1_{M(A)}) \cdot \phi(a) for all \omega \in A^{*} and a \in \mathcal{M}_{\phi}^{+} .
• There exists a K.M.S. weight \psi on A that is right-invariant, i.e., \psi ! \left( \overline{\text{id} \otimes \omega}(\Delta(a)) \right) = \overline{\omega}(1_{M(A)}) \cdot \psi(a) for all \omega \in A^{*} and a \in \mathcal{M}_{\phi}^{+} .

From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S. weight \psi is automatically faithful. Therefore, the faithfulness of \psi is a redundant condition and does not need to be postulated.

Duality

The category of locally compact quantum groups allows for a dual construction with which one can prove that the bi-dual of a locally compact quantum group is isomorphic to the original one. This result gives a far-reaching generalization of Pontryagin duality for locally compact Hausdorff abelian groups.

Alternative formulations

The theory has an equivalent formulation in terms of von Neumann algebras.

References

• Johan Kustermans & Stefaan Vaes. "Locally Compact Quantum Groups." Annales Scientifiques de l’École Normale Supérieure. Vol. 33, No. 6 (2000), pp. 837–934.
• Thomas Timmermann. "An Invitation to Quantum Groups and Duality – From Hopf Algebras to Multiplicative Unitaries and Beyond." EMS Textbooks in Mathematics, European Mathematical Society (2008).

References

1. (23 May 2000). "Locally compact quantum groups in the von Neumann algebraic setting". Mathematica Scandinavica.