Consequently, every sub-Laplacian (see Definition 4.1.6) on a stratified Lie group is hypoelliptic on the entire group, since every basis of the first stratum satisfies the H¨ormander condition. The operator A defined just above is called the infinitesimal generator of the semi-group {Q(t)}t>0. In this case, the operators of the semigroup {Q(t)}t>0 generated by A are denoted by.
A vector xis in the domain of the operator AB whenever is in the domain of B and Bxis in the domain of A. This is the subject of the theory of singular integrals onRn, especially when the power kaboben is equal. Let us present here the main lines of the generalization of the theory of singular integrals to the setting of 'spaces of homogeneous type' where there is no (apparent) trace of a group structure.
The constants K in definition A.4.1 and N in definition A.4.2 are called constants of a space of homogeneous type X. In the statement of the fundamental theorem of singular integrals on spaces of homogeneous types cf. However, the proof only requires that the kernel κ is locally integrable off the diagonal, in addition to the L2-boundedness of the operator T .
As explained at the beginning of this section, we are interested in "nice" kernels κo(x, y) with shape control.
Almost orthogonality
Here, the constant C is the unitary limit of the operator norms Tj and is independent of vorN. The same proof shows that the sequence ( . |j|≤NTjv)N∈Nis Cauchy when (kerTj)⊥ is finite. When working with groups, we sometimes have to deal with operators that map the L2 space on the group to the L2 space on its unitary dual.
This requires one to use the version of Cotlar's lemma for operators mapping between two different Hilbert spaces. In this case, the theorem of Theorem A.5.2 still holds, for an operator T : H → G, provided we take the operator norms Ti∗Tj and TiTj∗ in appropriate spaces. For example, (A.9) can be replaced by This condition often occurs when considering dyadic decomposition.) Indeed, by applying Proposition A.5.3 to {T2k+1}k∈Z and to {T2k}k∈Z, we obtain that the series.
Interpolation of analytic families of operators
From the proof it appears that if N =M =Rn is equipped with the usual Borel structure and the Lebesgue measures, one can require that the assumptions and the conclusion be on simple functions f with compact support. We also refer to Definition 6.4.17 for the concept of the complex interpolation (which requires stronger estimates). Open Access. This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you gives appropriate credit to the original author(s) and source, a link to the Creative Commons license is provided, and any changes are indicated.
The images or other third-party material in this chapter are included in the work's Creative Commons license, unless otherwise indicated in the credit line; if such material is not included in the work's Creative Commons license and the respective action is not permitted by statutory regulations, users will need permission from the licensee to duplicate, adapt or reproduce the material.
Appendix B
Direct integral of Hilbert spaces
- Convention: Hilbert spaces are assumed separable
- Measurable fields of vectors
- Direct integral of tensor products of Hilbert spaces
- Separability of a direct integral of Hilbert spaces
- Measurable fields of operators
- Integral of representations
GroupC∗and von Neumann algebras Furthermore, a separable Hilbert space of infinite dimension is unitarily equivalent to the Hilbert space of square integrable complex sequences: that is, to . Here we recall the definitions of measurable fields of Hilbert spaces, of vectors and of operators. In practice, for the set Γ in the following definition, we can also choose Γ⊂B. Let Z be a measurable space and a positive sigmafinite measure on Z. A μ-measurable field of Hilbert spacesoverZ is a pair E=. Hζ)ζ∈Z,Γ where (Hζ)ζ∈Z is a family of (separable).
Hilbert spaces indexed byZ and where Γ⊂B. ζ∈ZHz satisfies the following conditions:. ii) there exists a sequence (x)∈N of elements of Γ such that for each ζ ∈Z, the sequence (x(ζ))∈NspansHζ (in the sense that the subspace formed by the finite linear combination of thex(ζ) , ∈N, is dense in Hζ);. iii) for each x∈Γ, the functionζ→ x(ζ)Hζ is μ-measurable;. The set of square integrable vector fields form a (possibly indivisible) Hilbert space denoted by . After a brief recollection of the definitions of tensor products, we will be able to analyze the direct integral of tensor products of Hilbert spaces, as well as their decomposable operators.
Here we define firstly the algebraic tensor product of two vector spaces and secondly the tensor products of Hilbert spaces. The tensor product of two Hilbert spaces can be identified with the space of Hilbert Schmidt operators as follows. The Hilbert space given by the tensor product H ⊗ H∗ of Hilbert spaces is isomorphic to HS(H)via.
Let μ be a positive sigma-finite measure on a measurable space Z and E = (Hζ)ζ∈Z,Γ. A μ-measurable field of Hilbert spaces overZ. Hζ⊗ H∗ζ)ζ∈Z,Γ⊗Γ∗ is a μ-measurable field of Hilbert spacesoverZ. A measurable spaceZis a standard Borel space ifZis a Polish space (i.e. a separable complete metrizable topological space) and the sigma algebra considered is the Borel sigma algebra of Z (i.e. the smallest sigma algebra that covers the open contains sets Z). A μ-measurable field of operators over Z is a collection of operators (T(ζ))ζ∈Z such that T(ζ)∈L(Hζ) and for anyx∈Γ, the field (T(ζ)x( ζ) )ζ∈Z is measurable.
In the following definition, μ is a positive sigmafinite measure on a measurable space Z, A is a separable C∗-algebra, and G is a (Hausdorff) locally compact separable group. μ-measurable field representation A or G is a μ-measurable field of the operator (T(ζ))ζ∈Z (see definition B.1.14), so that for every ζ ∈ Z, T(ζ) = πζ is the representation of A, or Let Hζ(jj))ζj∈Zj,Γj be the μj-measurable field of Hilbert spaces over Zj and let (πζ(jj)) be the measurable field of representations of A, or
C ∗ - and von Neumann algebras
- Generalities on algebras
- C ∗ -algebras
- Group C ∗ -algebras
- Von Neumann algebras
- Group von Neumann algebra
- Decomposition of group von Neumann algebras and abstract Plancherel theorem
GroupC∗and von Neumann algebras If we follow the notation of definition B.2.1, we can easily check whether a commutant M is a sub-algebra of A. Group C∗and von Neumann algebra's algebra M(G) always admits the Dirac measureδe on the neutral element of the group as unity. Group C∗ and von Neumann algebras The C∗ algebra of a group is the one-to-one correspondence between the representation theories of G,L1(G) and C∗(G).
Although we do not use it in this monograph, let us briefly recall the Pontryagin duality, since this can be seen as one of the historical motivations to develop the theory of (noncommutative) C∗ algebras. Its spectrum G can be identified with the set of continuous Gand characters is naturally endowed with the structure of a locally compact (Hausdorf) abelian group. Recall that, by the Riemann-Lebesgue Theorem (see e.g. [RT10a, Theorem 1.1.8]), the Euclidean Fourier transform FRn maps L1(Rn) to Co(Rn), and we can show that.
A von Neumann algebra in His a ∗-subalgebraM of L(H) satisfying any of the equivalent properties (i), (ii), or (iii) in Theorem B.2.27. Thus, a von Neumann algebra on Its a∗-subalgebra ofL(H) closed for the operator norm topology is therefore aC∗-subalgebra ofL(H) and itself aC∗-algebra. Among C∗ algebras, von Neumann algebras are C∗ algebras which are realized as a closed ∗-subalgebra of L(H) and furthermore satisfy any of the equivalent properties (i), (ii), or (iii) in Theorem B.2.27.
It is also possible to define von Neumann algebras abstractly as C∗-algebras that have a predual, see e.g. Conversely any abelian von Neumann algebra in a separable Hilbert space can be realized in the manner described in Example B.2.30, see Dixmier [Dix96,. The main example of von Neumann algebras of interest to us is that associated with a group.
Now, let us first define the left and right von Neumann (isomorphic) algebras of a locally compact (Hausdorf) group G. This means that VNL(G) is the smallest von Neumann algebra containing all operatorsπL(x),x∈G, where πL is defined in (B.4), i.e. and 'von Neumann algebra group'.
According to our hypotheses, it is possible to describe the set of the von Neumann algebra as a space of convolution operators, see Eymard [Eym72, Theorem 3.10 and Proposition 3.27]. C∗ and von Neumann group algebras that the representation act is assumed to be separable, see Section B.1.1, while the divisibility of Hilbert spaces is not mentioned in Chapter 1.
Schr¨ odinger representations and Weyl quantization
Explicit symbolic calculus on the Heisenberg group
List of quantizations
