Hilbert direct sums of unitary representations

Assume the Axiom of Choice (The Axiom of Choice). Let $I$ be a set and let $(H_i{)}_{i\in I}$ be a family of complex Hilbert spaces (Hilbert space). A family $v=(v_i{)}_{i\in I}$ with $v_i\in H_i$ for every $i$ is square summable when ${\sum }_{i\in I}\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2<+\mathrm{\infty }$ in the finite-subset-supremum convention of Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$: the sum is the supremum of the finite subsums ${\sum }_{i\in F}\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2$ over finite $F\subseteq I$. The Hilbert direct sum ${\widehat{⨁}}_{i\in I}{H}_i$ is the set of all square-summable families, equipped with componentwise addition and scalar multiplication and with the pairing $⟨v,w⟩:={\sum }_{i\in I}⟨v_i,w_i⟩,$ the scalar family on the right being summed as a finite-subset net in the sense of Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$. By A Hilbert space with a given orthonormal basis is ${\mathrm{\ell }}^2$ of the index set and Square-summable orthogonal families have norm-convergent finite sums the resulting space is a complex Hilbert space whose norm is $\mathrm{\parallel }v\mathrm{\parallel }=({\sum }_{i\in I}\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}$, and the canonical maps $H_j\to {\widehat{⨁}}_i{H}_i$ extending a vector by zero are linear isometries (Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces) with pairwise orthogonal closed images (Orthogonality and the orthogonal complement) whose closed linear span is the whole space.

The pairing is well defined. For square-summable $v,w$ and every finite $F\subseteq I$ the finite Cauchy–Schwarz inequality applied to the scalar lists $(\mathrm{\parallel }{v}_i\mathrm{\parallel }{)}_{i\in F}$, $(\mathrm{\parallel }{w}_i\mathrm{\parallel }{)}_{i\in F}$ gives ${\sum }_{i\in F}\mathrm{\mid }⟨v_i,w_i⟩\mathrm{\mid }\le {\sum }_{i\in F}\mathrm{\parallel }{v}_i\mathrm{\parallel }\mathrm{\parallel }{w}_i\mathrm{\parallel }\le \left({\sum }_{i\in F}\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}\right({\sum }_{i\in F}\mathrm{\parallel }{w}_i{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}\le \mathrm{\parallel }v\mathrm{\parallel }\mathrm{\parallel }w\mathrm{\parallel },$ so the family $(⟨v_i,w_i⟩{)}_{i\in I}$ is absolutely summable and its finite-subset net converges to a scalar $⟨v,w⟩$ with $\mathrm{\mid }⟨v,w⟩\mathrm{\mid }\le \mathrm{\parallel }v\mathrm{\parallel }\mathrm{\parallel }w\mathrm{\parallel }$; this is the scalar summation theory of Square-summable families on an arbitrary index set and the space ${\mathrm{\ell }}^2(I)$. Componentwise sesquilinearity, conjugate symmetry and positive definiteness pass to the finite-subset net by linearity of the scalar sum, so the pairing is an inner product. For completeness let $(v^{(n)}{)}_{n\ge 1}$ be a Cauchy sequence and let $K$ be a bound for it; for each $i$ the components satisfy $\mathrm{\parallel }{v}_i^{(n)}-v_i^{(m)}\mathrm{\parallel }\le \mathrm{\parallel }{v}^{(n)}-v^{(m)}\mathrm{\parallel }$, so they converge to some $v_i\in H_i$, and for every finite $F$ the limit relation ${\sum }_{i\in F}\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2={\lim }_n{\sum }_{i\in F}\mathrm{\parallel }{v}_i^{(n)}{\mathrm{\parallel }}^2\le K^2$ shows that $v=(v_i)$ is square summable; then $\mathrm{\parallel }{v}^{(n)}-v{\mathrm{\parallel }}^2={\sup }_F{\lim }_m{\sum }_{i\in F}\mathrm{\parallel }{v}_i^{(n)}-v_i^{(m)}{\mathrm{\parallel }}^2$ is eventually below any prescribed ${\epsilon }^2$, so $v^{(n)}\to v$ and ${\widehat{⨁}}_i{H}_i$ is complete. Both arguments are choice-free beyond the completeness of the factors.

Strongly continuous unitary representations. Suppose now that $K$ is a topological group and that each $H_i$ carries a strongly continuous unitary representation ${\pi }_i$ of $K$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners). The Hilbert direct sum of the representations, still written ${\widehat{⨁}}_i{\pi }_i$, acts componentwise, $\left({\widehat{⨁}}_i{\pi }_i\right)(k)(v_i{)}_{i\in I}:=({\pi }_i(k)v_i{)}_{i\in I},$ which is again a square-summable family because every ${\pi }_i(k)$ is isometric, and which is a group homomorphism into the unitary group of the sum by componentwise computation. To see strong continuity let $v\in \widehat{⨁}{H}_i$ and $\epsilon >0$; choose a finite $F\subseteq I$ with ${\sum }_{i\mathrm{\notin }F}\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2<{\epsilon }^2\mathrm{/}16$, possible by the tail-control property of the finite-subset-supremum convention, and, if $F=\mathrm{\varnothing }$, take $U=K$, since the tail estimate alone is less than $\epsilon \mathrm{/}2$. Otherwise, for each $i\in F$ choose a neighbourhood $U_i$ of the identity with $\mathrm{\parallel }{\pi }_i(k)v_i-v_i\mathrm{\parallel }<\epsilon \mathrm{/}(2\mathrm{\mid }F\mathrm{\mid })$ for $k\in U_i$, which is possible by the finitely many strong continuity assumptions; then $U={\bigcap }_{i\in F}{U}_i$ is an identity neighbourhood and, using unitarity of each ${\pi }_i(k)$ to bound the tail of $\pi (k)v-v$ by twice the square root of the tail of $v$, $\parallel \left({\widehat{⨁}}_i{\pi }_i\right)(k)v-v\parallel \le {\sum }_{i\in F}\mathrm{\parallel }{\pi }_i(k)v_i-v_i\mathrm{\parallel }+2({\sum }_{i\mathrm{\notin }F}\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2{)}^{1\mathrm{/}2}<\epsilon$ for every $k\in U$. Hence ${\widehat{⨁}}_i{\pi }_i$ is a strongly continuous unitary representation of $K$.

Direct sums of subrepresentations. A strongly continuous unitary representation $\pi$ of $K$ on a complex Hilbert space $H$ is the Hilbert direct sum of a family of subrepresentations $(H_i,{\pi }_i{)}_{i\in I}$ when the $H_i$ are pairwise orthogonal closed $\pi (K)$-invariant subspaces of $H$ whose closed linear span is $H$ and $\pi {\mathrm{\mid }}_{H_i}={\pi }_i$ for every $i$. In that case the canonical map ${\widehat{⨁}}_{i\in I}{H}_i⟶H,(v_i{)}_{i\in I}\mapsto {\sum }_{i\in I}{v}_i,$ is well defined by Square-summable orthogonal families have norm-convergent finite sums, which makes the finite-subset net of the partial sums converge with squared norm ${\sum }_i\mathrm{\parallel }{v}_i{\mathrm{\parallel }}^2$; it is a linear isometry by orthogonality, its image is closed because ${\widehat{⨁}}_i{H}_i$ is complete, and the image contains every $H_i$ and hence has closed linear span $H$, so the map is a unitary intertwiner. Conversely, if this canonical map is a unitary intertwiner for some pairwise orthogonal closed subspaces $H_i$ with closed linear span $H$ that are $\pi (K)$-invariant, then $\pi$ is the Hilbert direct sum of the subrepresentations $(H_i,\pi {\mathrm{\mid }}_{H_i})$. The Axiom of Choice (The Axiom of Choice) is declared for this development because the decomposition theorems select representatives in unitary-equivalence classes and apply the cited Hilbert-space suppliers; the construction of the direct sum and the verifications above use no choice beyond their inputs.