Popa's AOP in von Neumann Algebras

• Popa's AOP is a structural property for von Neumann algebra inclusions, ensuring the asymptotic vanishing of conditional expectations via unitary conjugations.
• Relative AOP extends to amalgamated free products by adapting asymptotic orthogonality to settings with nontrivial base subalgebras, yielding explicit criteria for maximal amenability.
• The property underpins key proofs in operator algebra theory by combining bimodule techniques and Pimsner–Popa unitaries to enforce strict orthogonality in complex algebra towers.

Popa’s Asymptotic Orthogonality Property (AOP) is a structural property for inclusions of von Neumann algebras that enables the demonstration of maximal amenability in a broad class of operator algebraic settings, especially in the context of amalgamated free product constructions. Originally introduced in the 1980s by Sorin Popa, AOP ensures that certain large subalgebras cannot be properly embedded into even larger amenable subalgebras within a fixed ambient von Neumann algebra. The property uses asymptotic vanishing of conditional expectations under conjugation by sequences of unitaries, and in relative settings, it adapts to amalgamated free products—facilitating explicit criteria for maximal amenability in highly nontrivial von Neumann algebra towers (Leary, 2019).

1. Formal Definition of the Asymptotic Orthogonality Property

Let $(M,\tau)$ be a finite von Neumann algebra equipped with a faithful normal tracial state $\tau$, and let $A\subset M$ be a diffuse von Neumann subalgebra. Define

$M \ominus A = \{x \in M : E_A(x) = 0\}$

where $E_A: M \to A$ is the $\tau$-preserving conditional expectation.

Popa’s Asymptotic Orthogonality Property (AOP):

The inclusion $A\subset M$ has AOP if there exists a sequence of unitaries $u_n \in \mathcal{U}(A)$, $n=1,2,\dots$, such that for any $x, y \in M \ominus A$,

$\tau$0

where $\tau$1.

In the framework of ultrapowers: for any free ultrafilter $\tau$2 on $\tau$3, the inclusion $\tau$4 has AOP if for all $\tau$5 and $\tau$6,

$\tau$7

2. Relative AOP in Amalgamated Free Products

In amalgamated free product settings, AOP must be extended to account for nontrivial amalgamation subalgebras. Consider the tower

$\tau$8

with $\tau$9 and $A\subset M$0 finite von Neumann algebras over a common subalgebra $A\subset M$1, with $A\subset M$2.

Relative Asymptotic Orthogonality Property:

If $A\subset M$3, the inclusion $A\subset M$4 has AOP relative to $A\subset M$5 if for every free ultrafilter $A\subset M$6, all $A\subset M$7, and all $A\subset M$8,

$A\subset M$9

A main structural result is as follows: If $M \ominus A = \{x \in M : E_A(x) = 0\}$0 admits a Pimsner–Popa basis of unitaries $M \ominus A = \{x \in M : E_A(x) = 0\}$1, then in the amalgamated free product $M \ominus A = \{x \in M : E_A(x) = 0\}$2, the inclusion $M \ominus A = \{x \in M : E_A(x) = 0\}$3 satisfies the relative AOP [(Leary, 2019), Theorem 3.3].

3. Key Ingredients and Proof Outline

The proof of relative AOP in amalgamated free products involves several operator-algebraic and bimodule-theoretic techniques:

• Hilbert Space Decomposition:

$M \ominus A = \{x \in M : E_A(x) = 0\}$4 is decomposed as

$M \ominus A = \{x \in M : E_A(x) = 0\}$5

where the summands correspond to reduced words of alternating components $M \ominus A = \{x \in M : E_A(x) = 0\}$6 and $M \ominus A = \{x \in M : E_A(x) = 0\}$7.

• Good and Bad Subspaces:

$M \ominus A = \{x \in M : E_A(x) = 0\}$8 is split into $M \ominus A = \{x \in M : E_A(x) = 0\}$9 and $E_A: M \to A$0, with $E_A: M \to A$1 constructed so that pairings involving left/right multiplications by elements from $E_A: M \to A$2 vanish under conditional expectation to $E_A: M \to A$3.

• Averaging via Pimsner–Popa Unitaries:

Utilizing the Pimsner–Popa unitary basis $E_A: M \to A$4, for any $E_A: M \to A$5, the family $E_A: M \to A$6 becomes asymptotically pairwise orthogonal in $E_A: M \to A$7, ensuring any component in $E_A: M \to A$8 vanishes within the required limits.

• Orthogonality Mechanism:

For $E_A: M \to A$9, the construction ensures all requisite inner products vanish, yielding the orthogonality for the relative AOP.

These steps establish that for any $\tau$0 and $\tau$1,

$\tau$2

which is the relative AOP required.

4. Application to Maximal Amenability

AOP is central to proving maximal amenability in amalgamated free products. The following structural criterion, due to Houdayer [(Leary, 2019), Thm. 3.4], is invoked:

Let $\tau$3 be tracial von Neumann algebras. Assume:

• (i) $\tau$4 is amenable,
• (ii) $\tau$5 is weakly mixing through $\tau$6,
• (iii) $\tau$7 has the asymptotic orthogonality property relative to $\tau$8.

Then, for any intermediate amenable algebra $\tau$9, it must hold that $A\subset M$0; thus, $A\subset M$1 is maximal amenable in $A\subset M$2.

Applied to the amalgamated free product $A\subset M$3, with $A\subset M$4 admitting a Pimsner–Popa unitary basis, $A\subset M$5 diffuse and amenable, and no corner of $A\subset M$6 embedding into $A\subset M$7, $A\subset M$8 is maximal amenable in $A\subset M$9 [(Leary, 2019), Theorem 1.1]. If for every nonzero projection $u_n \in \mathcal{U}(A)$0 the Pimsner–Popa index $u_n \in \mathcal{U}(A)$1 is infinite, maximal amenability persists [(Leary, 2019), Corollary 1.2].

5. Canonical Examples and Historical Origins

Group–von Neumann Algebra Example

Let $u_n \in \mathcal{U}(A)$2 be a countable amenable group, $u_n \in \mathcal{U}(A)$3 an infinite-index subgroup. The coset representatives $u_n \in \mathcal{U}(A)$4 provide a Pimsner–Popa unitary basis for $u_n \in \mathcal{U}(A)$5. For any group $u_n \in \mathcal{U}(A)$6, the amalgamated free product $u_n \in \mathcal{U}(A)$7 yields

$u_n \in \mathcal{U}(A)$8

satisfying the hypotheses for maximal amenability of $u_n \in \mathcal{U}(A)$9 in $n=1,2,\dots$0 (Leary, 2019).

Popa’s 1980s Construction

Popa’s original example concerns the generator masa $n=1,2,\dots$1, where a sequence of “rotating” unitaries in $n=1,2,\dots$2 enforces the asymptotic orthogonality inside $n=1,2,\dots$3. The original AOP was formalized in this setting and underlies subsequent results for amalgamated free products, as shown in (Leary, 2019).

6. The Role of AOP in Modern von Neumann Algebra Theory

AOP functions as an “orthogonality engine”—its key utility is in eliminating the possibility of strictly larger amenable subalgebras containing a given subalgebra, provided weak mixing also holds. While modern deformation/rigidity techniques exist, the explicit orthogonality method traced to Popa remains fundamentally effective for constructions involving amalgamation and group-measure space factors, and can yield new maximal amenability theorems in settings with nontrivial base algebras or Pimsner–Popa indexed inclusions (Leary, 2019).