**Memoirs of the American Mathematical Society**

1996;
68 pp;
Softcover

MSC: Primary 43; 46; 47;

Print ISBN: 978-0-8218-0539-8

Product Code: MEMO/124/593

List Price: $42.00

Individual Member Price: $25.20

**Electronic ISBN: 978-1-4704-0178-8
Product Code: MEMO/124/593.E**

List Price: $42.00

Individual Member Price: $25.20

# Completely Positive Hypergroup Actions

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*Ajit Iqbal Singh*

It is now well known that the measure algebra \(M(G)\) of a locally compact group can be regarded as a subalgebra of the operator algebra \(B(B(L^2(G)))\) of the operator algebra \(B(L^2(G))\) of the Hilbert space \(L^2(G)\). In this memoir, the author studies the situation in hypergroups and finds that, in general, the analogous map for them is neither an isometry nor a homomorphism. However, it is completely positive and completely bounded in certain ways. This work presents the related general theory and special examples.

#### Table of Contents

# Table of Contents

## Completely Positive Hypergroup Actions

- Contents vii8 free
- Introduction xi12 free
- Chapter 1. Presentations 114 free
- Chapter 2. Complete Positivity and Other Properties for Presentations and Opresentations 821
- 2.1. The C*–algebra M[sub(n)](A) and completely positive maps 821
- 2.2. Matrix ordered and matricially normed spaces 922
- 2.3. Matrix ordered and matricially normed spaces of linear maps 1124
- 2.4. Interconnections amongst different notions of positivity and complete positivity 1427
- 2.5. Matricially order or norm admissible and compatible pairs 1730
- 2.6. Examples of matricially norm admissible and compatible pairs 1831
- 2.7. Matricially order or norm admissible and compatible pairs of spaces of linear maps 1932
- 2.8. Completely positive and completely bounded presentations and opresentations 2134
- 2.9. Topological structures on spaces of presentations 2336
- 2.10. Properties of the dual presentation and opresentation 2437
- 2.11. Completely positive and completely bounded actions and opactions 2538
- 2.12. Examples and remarks 3043

- Chapter 3. Presentations of Hypergroups and Associated Actions 3144
- 3.1. (M(K), C[sub(b)]K)) as a natural matricially order compatible pair 3245
- 3.2. Matricially order compatible structures on (M(K), C[sub(b)]K)) through representations 3447
- 3.3. Special matrix orders on M(K) for commutative K 3649
- 3.4. Role of conjugate representations in Fourier transform 3750
- 3.5. Positive definite presentations and completely positive opresentations of hypergroups 3851
- 3.6. The spectral subspaces of a presentation of a hypergroup 4154
- 3.7. Quantized positive–definite presentations and opresentations of hypergroups 4457
- 3.8. Completely positive instruments with values in K and their characteristic functions 4962
- 3.9. Completely positive hypergroup actions and opactions 5265

- Chapter 4. Some Concrete Presentations and Actions of Hypergroups 5568
- 4.1. Presentations and opresentations arising from the left regular representation 5568
- 4.2. The situation in 2–fold absolutely continuous hypergroups 5669
- 4.3. Amenability for hypergroups 5669
- 4.4. Folner hypergroups 5770
- 4.5. Isometry condition on Λ(μ) 5972
- 4.6. Actions and opactions arising from the left regular representation 6073
- 4.7. The special case X = L[sup(2)](K) = X[sub(*)] 6073

- References 6477

#### Readership

Graduate students and research mathematicians interested in abstract harmonic analysis, functional analysis, and operator theory.