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Polar decomposition inside a von Neumann algebra and nonzero partial isometries between nonzero projections in a factor

Statement

Assume AC (The Axiom of Choice). Let $H$ be a complex Hilbert space and $M \subseteq B (H)$ a concrete von Neumann algebra (Von Neumann algebras and commutants). If $H = {0}$ , part 1 is the trivial zero-operator decomposition and part 2 has no nonzero projection inputs; assume $H \neq {0}$ for the remaining clauses. A projection means a self-adjoint idempotent in $M$ ; for projections $r, p$ , $r \leq p$ means $r p = p r = r$ . Projections $r, s \in M$ are called equivalent when there is a partial isometry $w \in M$ with $w^{*} w = r$ and $w w^{*} = s$ . Write $p M q : = {p m q : m \in M}$ . Then:

Every $x \in M$ has a polar decomposition $x = v ∣ x ∣$ , where $∣ x ∣ : = (x^{*} x)^{1 / 2} \in M$ and $v \in M$ is a partial isometry (Isometry coisometry and partial isometry) with $v^{*} v$ the orthogonal projection onto $(\ker x)^{⊥}$ and $v v^{*}$ the orthogonal projection onto $\overline{ran x}$ .
If $M$ is a factor, meaning $M \cap M^{'} = C I$ , and $p, q \in M$ are nonzero projections, then $p M q \neq {0}$ . In particular, a nonzero partial isometry $v \in M$ exists with $v^{*} v \leq p$ and $v v^{*} \leq q$ ; equivalently, a nonzero subprojection of $p$ is equivalent to a nonzero subprojection of $q$ .

Facts & Assumptions

Given: AC, a concrete von Neumann algebra $M \subseteq B (H)$ , and the factor condition where used.

[F1]

AC is the hypothesis of the bicommutant, continuous-calculus and positive-square-root suppliers; it supplies Countable Choice for the Hilbert projection, orthogonal-decomposition, adjoint, range-orthogonality, partial-isometry, positive-spectrum and generated-C*-algebra interfaces (The Axiom of Choice).

[F2]

A concrete von Neumann algebra is a unital $*$ -subalgebra closed in WOT (Von Neumann algebras and commutants).

[F3]

The reciprocal Archimedean bound gives $1 / (n + 1) \to 0$ for $n \in N$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ). For bounded self-adjoint $b$ , continuous functional calculus is isometric, sends the coordinate function to $b$ , and has range $C^{*} (I, b)$ ; in particular $∥ b ∥ = \max_{t \in σ (b)} ∣ t ∣$ . If $b$ is positive, then $σ (b) \subseteq [0, + \infty)$ (Continuous functional calculus for bounded self adjoint operators, Spectrum of a positive operator is nonnegative).

[F4]

Every closed subspace $K$ has a Hilbert orthogonal projection (The Hilbert orthogonal projection onto a closed subspace).

[F5]

A partial isometry is isometric on the orthogonal complement of its kernel and zero on its kernel (Isometry coisometry and partial isometry).

[F6]

For $x \in B (H)$ , $x^{*} x$ is positive because the adjoint identity gives $⟨ x^{*} x ξ, ξ ⟩ = ∥ x ξ ∥^{2} \geq 0$ ; every bounded positive operator has a unique positive square root (The Hilbert-space adjoint of a bounded operator, Real and complex inner-product spaces and their induced length, Self-adjoint, positive, unitary and normal operators, Positive square root).

[F7]

Norm convergence implies strong-operator convergence by $∥ T ξ ∥ \leq ∥ T ∥ ∥ ξ ∥$ , and $M$ is strongly closed by the double-commutant theorem (The operator norm as the least bound and as the unit-sphere or unit-ball supremum, Strong and weak operator topologies, The double commutant theorem for concrete von Neumann algebras).

[F8]

$C^{*} (I, b)$ is the norm closure of the unital $*$ -polynomials in $b$ (C star algebra generated by a normal operator).

[F9]

Every closed subspace $K$ gives an orthogonal decomposition $H = K \oplus K^{⊥}$ (Orthogonal decomposition by a closed subspace).

[F10]

For bounded $T$ , $\overline{ran T} = (\ker T^{*})^{⊥}$ (Kernel–range orthogonality for Hilbert adjoints).

[F11]

The double-commutant theorem gives $M^{''} = M$ (The double commutant theorem for concrete von Neumann algebras).

[F12]

The Hilbert adjoint satisfies $⟨ T ξ, η ⟩ = ⟨ ξ, T^{*} η ⟩$ (The Hilbert-space adjoint of a bounded operator).

Proof

technique · direct

Given: AC, $M$ , and $x$ and $p, q$ where the corresponding clauses apply.

1.1given

If $H = {0}$ then $M = {0}$ , and the unique operator has the stated zero decomposition; there are no nonzero projections for part 2. Assume $H \neq {0}$ below.

1.2F1F2F4F11

For the factor clause, fix a nonzero projection $q \in M$ and set $L = \overset{}{span} {m q ξ : m \in M, ξ \in H}$ . Since $q \neq 0$ , some $ξ$ has $q ξ \neq 0$ , and $I \in M$ puts $q ξ$ in the generating set, so $L \neq {0}$ . This closed subspace is invariant under $M$ and its adjoints, hence reducing for $M$ . It is also invariant under $M^{'}$ and its adjoints, since for $c \in M^{'}$ and $m \in M$ , $c m q ξ = m c q ξ = m q c ξ$ , hence reducing for $M^{'}$ . Thus its orthogonal projection $P_{L}$ commutes with both $M$ and $M^{'}$ . By [F11], $P_{L} \in M^{'} \cap M^{''} = M^{'} \cap M$ , the center of $M$ .

1.3givenF2F6F7F8F12

For arbitrary $x \in M$ , put $b : = x^{*} x \in M$ . By [F6], $b$ is positive and has a positive square root $a : = b^{1 / 2}$ . Positivity makes $⟨ a ξ, ξ ⟩$ real, and [F12] gives $⟨ (a - a^{*}) ξ, ξ ⟩ = 0$ for every $ξ$ . For $B (u, v) : = ⟨ (a - a^{*}) u, v ⟩$ , the four-term expansion $4 B (u, v) = B (u + v, u + v) - B (u - v, u - v) + i B (u + i v, u + i v) - i B (u - i v, u - i v)$ therefore gives $B (u, v) = 0$ for all $u, v$ , hence $a = a^{*}$ . The theorem puts $a$ in $C^{*} (I, b)$ , which is contained in $M$ by [F2, F7, F8] because its generating $*$ -polynomials lie in $M$ . For every $ξ \in H$ , $∥ a ξ ∥^{2} = ⟨ a^{2} ξ, ξ ⟩ = ⟨ x^{*} x ξ, ξ ⟩ = ∥ x ξ ∥^{2}$ , so $\ker a = \ker x$ .

2.1step 1.2given

Since $q \neq 0$ , $q H \subseteq L$ is nonzero, so $P_{L} \neq 0$ . If $M$ is a factor, its center is $C I$ ; the only nonzero scalar projection is $I$ . Hence $P_{L} = I$ and $L = H$ .

2.2F4F5F9F10F12step 1.3

Define $v_{0}$ on $ran a$ by $v_{0} (a ξ) = x ξ$ . The equality of norms in step 1.3 makes this well-defined and isometric. By [F10], $\overline{ran a} = (\ker a^{*})^{⊥} = (\ker a)^{⊥} = (\ker x)^{⊥} = : K$ . It extends to an isometry from $K$ onto $F : = \overline{ran x}$ ; extend it by zero on $\ker a = K^{⊥}$ . Since $v$ is isometric on $K$ and zero on $K^{⊥}$ , $\ker v = K^{⊥}$ and $v$ is a partial isometry. Also $x = v a$ . For $ξ \in K$ and any $η = η_{K} + η_{K^{⊥}}$ , [F12] gives $⟨ v^{*} v ξ, η ⟩ = ⟨ v ξ, v η_{K} ⟩ = ⟨ ξ, η_{K} ⟩ = ⟨ ξ, η ⟩$ , while $v^{*} v$ vanishes on $K^{⊥}$ ; hence $v^{*} v = P_{K}$ . If $ζ \in F^{⊥}$ , then $⟨ v^{*} ζ, ξ ⟩ = ⟨ ζ, v ξ ⟩ = 0$ for every $ξ$ , so $v^{*} ζ = 0$ ; on $F = v K$ , $v v^{*}$ is the identity because $v^{*} v = P_{K}$ . Thus $v v^{*} = P_{F}$ .

3.1step 2.1

If $p M q = {0}$ , then $p$ annihilates every $m q ξ$ and hence their closed span $L = H$ from step 2.1. This forces $p = 0$ , a contradiction. Therefore $p M q \neq {0}$ .

3.2F1F2F3F7F8step 1.3step 2.2

Let $g_{n} (t) = t / (t + 1 / (n + 1))$ on $σ (a) \subseteq [0, ∥ a ∥]$ , using [F3]. By [F3], $∥ g_{n} (a) ∥ \leq 1$ and $(a + 1 / (n + 1) I)^{- 1} \in C^{*} (I, a) \subseteq M$ by [F2, F7, F8]. On $ran a$ , $(I - g_{n} (a)) a = (1 / (n + 1)) a (a + 1 / (n + 1) I)^{- 1}$ has norm at most $1 / (n + 1)$ ; on $\ker a$ , $g_{n} (a) = 0$ . Since $\overline{ran a} = (\ker a)^{⊥}$ and $∥ g_{n} (a) ∥ \leq 1$ , convergence on the dense subspace $ran a + \ker a$ extends to $g_{n} (a) \to P_{(\ker a)^{⊥}} = v^{*} v$ strongly. Therefore $x (a + 1 / (n + 1) I)^{- 1} = v g_{n} (a) \to v (v^{*} v) = v$ strongly. Each approximant lies in $M$ , so its strong closedness [F7] gives $v \in M$ .

4.1F5step 2.2step 3.1step 3.2∎

Apply step 3.1 with $p$ and $q$ interchanged to obtain a nonzero $x \in q M p$ . Its polar partial isometry from steps 2.2 and 3.2 lies in $M$ . Because $x = x p$ , $(I - p) H \subseteq \ker x$ , so the initial space $(\ker x)^{⊥}$ is contained in $p H$ and $v^{*} v \leq p$ ; the containment gives $p (v^{*} v) = (v^{*} v) p = v^{*} v$ . Because $x = q x$ , $ran x \subseteq q H$ , so $v v^{*} \leq q$ ; likewise $q (v v^{*}) = (v v^{*}) q = v v^{*}$ . Since $v \neq 0$ by $x = v a \neq 0$ , its initial and final projections are nonzero. Thus $v^{*} v$ and $v v^{*}$ are nonzero equivalent subprojections of $p$ and $q$ , respectively.

Source notes

Blackadar I.5.2.1–I.5.2.2, printed pp. 23–24, gives the support-projection and polar-decomposition construction and the strong-limit regularizer. The local proof supplies the positive-square-root membership in $M$ and verifies the strong limit used to place the partial isometry in $M$ . Blackadar III.1.3.10, printed p. 244, concerns abelian projections and their central supports; it does not establish $p M q \neq 0$ for arbitrary nonzero $p, q$ , which is proved locally here. Bekka–de la Harpe Appendix A.K, printed pp. 423–424, gives factor-center and support terminology only. The scaffold's locator to pp. 434–440 points to bibliography and index pages, not Appendix A.K.

Depends on

Used by

A separable type I factor is a multiple of an irreducible representation Lemma

Dependency tree · two levels

64 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Bruce Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras (author-hosted complete text) (standard reference, not scraped)
Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019; author-hosted complete book draft) (standard reference, not scraped)