Compact-group isotypic projection

Assume the Axiom of Choice (The Axiom of Choice). Let $K$ be a compact Hausdorff topological group (Topological group: multiplication and inversion are continuous, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) with normalized Haar probability measure $\mu$ (Normalized Haar probability on a compact group); by AC the Axiom of Countable Choice holds (AC supplies the countable and dependent choices used in Banach integration).

Let $\sigma :K\to U(V_{\sigma })$ be an irreducible strongly continuous unitary representation of $K$ on a nonzero complex Hilbert space $V_{\sigma }$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners). By Irreducible unitary representations of compact groups are finite dimensional the space $V_{\sigma }$ is finite dimensional; write $d_{\sigma }:={\dim }_{\mathbb{C}}{V}_{\sigma }<\mathrm{\infty },d_{\sigma }\ge 1,$ the degree of $\sigma$ (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). Fix an orthonormal basis $e_1,\dots ,e_{d_{\sigma }}$ of $V_{\sigma }$ (Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis). The character of $\sigma$ is ${\chi }_{\sigma }(k):=\mathrm{tr}\sigma (k)(k\in K),$ the trace of the endomorphism $\sigma (k)$ of $V_{\sigma }$ (The basis-independent trace of an endomorphism of a finite-dimensional vector space). It is a class function, ${\chi }_{\sigma }(hk{h}^{-1})={\chi }_{\sigma }(k)$ for all $h,k\in K$, because conjugation by $\sigma (h)$ is a similarity (Similar matrices have the same trace), and it is continuous: for a finite-dimensional space the strong continuity of $\sigma$ makes every matrix coefficient $k\mapsto ⟨\sigma (k)e_i,e_j⟩$ continuous, and the trace is the finite sum ${\sum }_{i=1}^{d_{\sigma }}⟨\sigma (k)e_i,e_i⟩$ of these.

Let $\pi :K\to U(H)$ be a strongly continuous unitary representation of $K$ on a complex Hilbert space $H$, and fix $v\in H$. The integrand is the function $f_v:K\to H,f_v(k):=\overline{{\chi }_{\sigma }(k)}\pi (k)v\mathrm{.}$

Well-definedness of the vector-valued integral. The function $f_v$ is continuous: $k\mapsto \pi (k)v$ is norm-continuous by strong continuity of $\pi$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners), $k\mapsto \overline{{\chi }_{\sigma }(k)}$ is continuous, and products of a continuous scalar function with a continuous vector-valued function are continuous. Its image $f_v[K]$ is therefore a compact subset of the metric space $H$ (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide), hence totally bounded (A compact metric space is complete and totally bounded, and neither implication uses any choice principle, Finite $\epsilon$-net and totally bounded metric space, Open ball, closed ball and sphere in a metric space, Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric): for every $n\ge 1$ there is a finite $F_n\subseteq f_v[K]$ with $f_v[K]\subseteq {\bigcup }_{y\in F_n}B(y,1\mathrm{/}n)$. Choosing one such finite net for each $n$, and enumerating each finite net, uses Countable Choice and finite choice only (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Every natural-number-indexed list of nonempty sets has a choice function on its family of values). Writing $A_j:=f_v^{-1}[B(y_j,1\mathrm{/}n)]\setminus {\bigcup }_{i<j}{A}_i$ for an enumeration $F_n=\{y_0,\dots ,y_m\}$ produces pairwise disjoint Borel sets covering $K$ (A continuous map has Borel preimages of Borel sets, The Borel sigma-algebra of a topological space) and hence a measurable $H$-valued simple function $t_n={\sum }_j{y}_j{\mathbf{1}}_{A_j}$ with $\mathrm{\parallel }{t}_n(k)-f_v(k)\mathrm{\parallel }<1\mathrm{/}n$ for every $k$; thus $f_v$ is the pointwise norm limit of simple functions, that is, strongly measurable (Strongly measurable Banach-valued function, Banach-valued simple function and integral). Moreover $\mathrm{\parallel }\overline{{\chi }_{\sigma }(k)}\pi (k)v\mathrm{\parallel }=\mathrm{\mid }{\chi }_{\sigma }(k)\mathrm{\mid }\mathrm{\parallel }v\mathrm{\parallel }\le M_v$ for all $k$, where $M_v:=({\max }_{k\in K}\mathrm{\mid }{\chi }_{\sigma }(k)\mathrm{\mid })\mathrm{\parallel }v\mathrm{\parallel }<\mathrm{\infty }$: the continuous function $\mathrm{\mid }{\chi }_{\sigma }\mathrm{\mid }$ on the compact space $K$ is bounded (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism), and $\mu$ is a probability. Hence ${\int }_K\mathrm{\parallel }{f}_v\mathrm{\parallel }d\mu \le M_v<\mathrm{\infty }$, so $f_v$ is Bochner integrable by the Bochner integrability criterion (Bochner integrability criterion, Bochner-integrable function, Measure spaces).

The definition. With the Bochner integral just justified, put $P_{\sigma }v:=d_{\sigma }{\int }_K{f}_v(k)d\mu (k)=d_{\sigma }{\int }_K\overline{{\chi }_{\sigma }(k)}\pi (k)vd\mu (k)(v\in H)\mathrm{.}$ This defines a map $P_{\sigma }:H\to H$, the $\sigma$-isotypic projection attached to the irreducible $\sigma$ and the representation $\pi$; the norm inequality for Bochner integrals (Bochner integral norm inequality) gives $\mathrm{\parallel }{P}_{\sigma }v\mathrm{\parallel }\le d_{\sigma }{\int }_K\mathrm{\mid }{\chi }_{\sigma }\mathrm{\mid }d\mu \mathrm{\parallel }v\mathrm{\parallel }$ for every $v$. Only scalar functions are integrated directly in the construction: for each $k$ the integrand is the vector $\overline{{\chi }_{\sigma }(k)}\pi (k)v$, and the choice of nets above is the only place Countable Choice is consumed.

The $\sigma$-isotypic subspace. A closed linear subspace $M\subseteq H$ is $\sigma$-isotypic of type $\sigma$, or simply a $\sigma$-copy, when $\pi (k)M=M$ for every $k$ and there is a unitary intertwiner $U:V_{\sigma }\to M$ with $U\sigma (k)=\pi (k)U$ for every $k$; that is, when $\pi {\mathrm{\mid }}_M$ is unitarily equivalent to $\sigma$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, Linear subspace of a vector space). The $\sigma$-isotypic subspace of $H$ is the closed linear span $H_{\sigma }:=\stackrel{}{\mathrm{span}}\{M\subseteq H:M\text{ is a }\sigma \text{-copy}\},$ the smallest closed invariant subspace of $H$ containing every $\sigma$-copy.

What is not asserted here. No sum over the unitary dual of $K$ is formed, and no equality ${\sum }_{\sigma }{P}_{\sigma }=I_H$ is claimed: completeness of the family of isotypic subspaces is the content of Peter--Weyl theory, which is not available at this point. All that is claimed about $P_{\sigma }$ below is proved in Isotypic projections are mutually orthogonal equivariant projections ↗ without any density or dual-sum assertion: $P_{\sigma }$ is a bounded self-adjoint idempotent commuting with $\pi (K)$, its range is exactly $H_{\sigma }$, and distinct inequivalent types give orthogonal ranges. The boundedness, linearity and self-adjointness of $P_{\sigma }$ are not part of the definition; they are theorems.