M. Rieffel considered the abelian Lie group \(\mathbb B=\mathbb R^n\) with a standard translation invariant symplectic form \(\omega^0(x,y)\) and proved that for any continuous and isometric action \(\alpha\) of \(\mathbb B\) on any Fréchet algebra or a C*-algebra \(A\), the dense subspace \(A^\infty\), consisting of smooth vectors, is deformed with parameter \(\theta\) as the corresponding Moyal product, which can be written in the form of an oscillatory integral \[ a\star^\alpha_\theta b = \int_{\mathbb B \times \mathbb B} K_\theta(x,y) \alpha_x(a)\alpha_y(b)dxdy, \] where \[ K_\theta(x,y) := \theta^{-2n}\exp\{\frac{i}{\theta}\omega^0(x,y)\}. \] He proved also many properties, namely the continuity of the field of deformed C*-algebras, invariance with respect to the \(K\)-theory, etc….
In the Memoir under review, the authors propose some non-abelian generalization of the deformation theory of C*-algebras endowed with an isometric action of a negatively curved Kählerian Lie group \(\mathbb B\). The first candidate of such a non-abelian \(\mathbb B\) is the so-called elementary nornmal \(\mathbf j\)-group \(\tilde{\mathbb S} = (\mathbb R \rtimes \mathbb R^{2d}) \rtimes \mathbb R\). In Chapters 3 and 4, a generalization is constructed for these groups. The groups of this kind have some global parametization coordinates and are diffeomorphic with \(\mathbb R^{2d+2}\). In particular, the authors prove in Theorem 4.5 that there exists a corresponding family of kernel \[ \{K_\theta(x,y)\}=(\pi\theta)^{-2(d+1)}A_{\theta,\tau}(x_1,x_2)\exp\{\frac{2i}{\theta} S^{\mathbb S}_{can}(x_1,x_2) \} \] with the exact amplitude (Definition 3.15) for the point \((x_1,x_2)\), \(x_i = (a_i,v,t_i), i=1,2\), \(\omega^\mathbb S = 2da\wedge dt + \omega^0\), \[ A_{\theta,\tau}(x_1,x_2) = A_{can}^{\mathbb S}{\theta,\tau}(x_1,x_2) \exp\{\tau(\frac{2}{\theta}\sinh 2a_1)+ \tau(-\frac{2}{\theta}\sinh 2a_2) -\tau(\frac{2}{\theta}\sinh (2a_1-2a_2))\}, \]
\[ A_{can}^{\mathbb S}{\theta,\tau}(x_1,x_2)= (\cosh a_1\cosh a_2\cos(a_1-a_2))^d(\cosh2a_1 \cosh 2a_2\cosh(2a_1-2a_2)^{1/2} \] and phase (Definition 3.15) \[ S^{\mathbb S}_{can}(x-1,x_2) = t_2\sinh 2a_1 -t_1\sinh 2a_2 + \omega^0(v_1,v_2)\cosh a_1\cosh a_2. \] Next, the authors considere more general cases of negatively curved Kählerian Lie normal \(\mathbf j\)-algebras and \(\mathbf j\)-groups, which are obtained from this by iterating the semi-direct product (Proposition 3.5): every normal \(\mathbf j\)-algebra \(\mathfrak b\) in the sense of Pyatetskii-Shapiro can be decomposed as a split extension of elementary normal \(\mathbf j\)-algebras \(\mathfrak b = (\dots(\mathfrak s_N \ltimes \mathfrak s_{N-1}) \ltimes \dots \ltimes \mathfrak s_2) \ltimes \mathfrak s_1\) and the same for a nornal \(\mathbf j\)-group, which can be obtained as repeated split extensions of elementary normal \(\mathbf j\)-groups \(\mathbb B = (\dots(\mathbb S_N \ltimes \mathbb S_{N-1}) \ltimes\dots \ltimes \mathbb S_2) \ltimes \mathbb S_1.\) The general theory is therefore constructed in Chapters 5, 6, 7. The technical preparation was done in Chapter 2. Chapter 8 is reserved for an application to the problem of quantization, continuity of deformation and \(K\)-theory invariance. The book is interesting and readable, and the readers can access the subject easily.