Operator-Algebraic Freeness, Free Products, and Free Group Factors¶
Job: develop the reduced free product of operator algebras, the free group factors \(L(\mathbb{F}_n)\) and the isomorphism problem, the free product absorption \(L(\mathbb{F}_n)*L(\mathbb{F}_m)\cong L(\mathbb{F}_{n+m})\), the compression / fundamental group, and the free difference quotient — with the free product absorption / isomorphism problem as the worked example.
Core¶
Free products of operator algebras¶
The reduced free product \((A,\varphi)*(B,\psi)\) of C*-algebras / von Neumann algebras (the Voiculescu construction): the universal algebra generated by \(A\) and \(B\) with the free product state \(\varphi*\psi\) (the alternating centered products vanish: \((\varphi*\psi)(a_1b_1a_2b_2\cdots)=0\) for \(a_i\in A\), \(b_i\in B\) with \(\varphi(a_i)=\psi(b_i)=0\)). The free group factors \(L(\mathbb{F}_n)=L(\mathbb{Z})^{*n}=L(\mathbb{Z})*\cdots*L(\mathbb{Z})\) (\(n\) copies, the von Neumann algebra of the free group on \(n\) generators).
| Free product | Description |
|---|---|
| Reduced free product \((A,\varphi)*(B,\psi)\) | the universal algebra with the free product state (alternating centered products vanish) |
| Free product state \(\varphi*\psi\) | the free state, the free independence of \(A\) and \(B\) |
| Free group factors \(L(\mathbb{F}_n)\) | \(L(\mathbb{Z})^{*n}\), the von Neumann algebra of the free group on \(n\) generators, a II\(_1\) factor |
| Reduced vs full | the reduced free product (the GNS / the regular representation) vs the full (the universal) — the reduced is the von Neumann algebra closure |
The free group factors and the isomorphism problem¶
The free group factor \(L(\mathbb{F}_n)\) is the von Neumann algebra generated by the left regular representation \(\lambda\) of the free group \(\mathbb{F}_n\) on \(\ell^2(\mathbb{F}_n)\), a II\(_1\) factor with the canonical trace \(\tau(x)=\langle x\delta_e,\delta_e\rangle\). The isomorphism problem (Connes, Voiculescu): is \(L(\mathbb{F}_n)\cong L(\mathbb{F}_m)\) for \(n\ne m\)?
| Isomorphism problem | Status |
|---|---|
| Free product absorption | \(L(\mathbb{F}_n)*L(\mathbb{F}_m)\cong L(\mathbb{F}_{n+m})\) (the Dykema / the free product adds generators) |
| Cardinality | the free group factors for different \(n\) are either all isomorphic or all non-isomorphic (the Dykema / Radulescu interpolation) |
| Free entropy obstruction | \(\delta(X_1,\dots,X_n)=n\) for the generators, the partial invariant (see free entropy) |
| Connes embedding | \(L(\mathbb{F}_n)\hookrightarrow R^\omega\) (the microstates / the free entropy) |
The compression / the fundamental group¶
The compression \(L(\mathbb{F}_n)\otimes M_k\) (the amplification, the II\(_1\) factor by a matrix algebra \(M_k\)) and the fundamental group \(\mathcal{F}(M)=\{t>0:M^t\cong M\}\) (the Connes fundamental group).
| Compression / fundamental group | |
|---|---|
| Compression \(M^t\) | the amplification of the II\(_1\) factor \(M\) by \(t\) (the \(M\otimes M_k\) for \(t=k^2\)) |
| Fundamental group \(\mathcal{F}(M)\) | \(\{t>0:M^t\cong M\}\) (the Connes fundamental group) |
| Radulescu theorem | \(\mathcal{F}(L(\mathbb{F}_\infty))=\mathbb{R}_+\) (the full group, the \(L(\mathbb{F}_\infty)\) stable under compression) |
| Open: \(\mathcal{F}(L(\mathbb{F}_n))\) for finite \(n\) | unknown (the Connes conjecture, the \(\mathbb{R}_+\) conjectured but open) |
The free difference quotient and the cyclic derivative¶
The free difference quotient \(\partial:\mathbb{C}\langle X\rangle\to\mathbb{C}\langle X\rangle\otimes\mathbb{C}\langle X\rangle^{\mathrm{op}}\) (the Voiculescu noncommutative derivative, the algebraic Leibniz rule on the free algebra): \(\partial(X)=1\otimes1\), \(\partial(PQ)=\partial P\cdot Q+P\otimes\partial Q\) (the derivation). The cyclic derivative (the trace-composed derivative, the noncommutative analog of the logarithmic derivative): \(D_{\mathrm{cyc}}P=\mathrm{tr}\circ\partial P\) (the trace of the free difference quotient, the cyclic gradient).
| Free difference quotient | Description |
|---|---|
| \(\partial:\mathbb{C}\langle X\rangle\to\mathbb{C}\langle X\rangle\otimes\mathbb{C}\langle X\rangle^{\mathrm{op}}\) | the noncommutative derivation (\(\partial X=1\otimes1\), the Leibniz rule) |
| Cyclic derivative \(D_{\mathrm{cyc}}P\) | \(\mathrm{tr}\circ\partial P\) (the trace-composed / the cyclic gradient) |
| Conjugate variable \(D_X\) | \(\tau(D_X P(X))=\tau\otimes\tau(\partial P(X))\) (the free Stein / integration by parts) |
| Free Fisher \(\Phi=\tau(D_X^*D_X)\) | the \(L^2\)-norm of the conjugate variable (the free entropy connection) |
The free difference quotient as the generator of the free Fisher information: the conjugate variable \(D_X=\partial\log p_X\) (the free analog of the score function, where \(p_X\) is the free density), the free Fisher \(\Phi=\tau(D_X^*D_X)\). The free difference quotient and the free stochastic calculus (the connection to the free Brownian motion, the noncommutative Itô formula).
See it¶
Free product absorption \(L(\mathbb{F}_n)*L(\mathbb{F}_m)\cong L(\mathbb{F}_{n+m})\) and the isomorphism problem. The reduced free product \(L(\mathbb{F}_n)*L(\mathbb{F}_m)\) (with the free product trace \(\tau_n*\tau_m\)) is \(L(\mathbb{F}_{n+m})\) (the free group on \(n+m\) generators, the absorption: the free product of the free group factors adds the generators — the \(n\) generators of \(L(\mathbb{F}_n)\) and the \(m\) generators of \(L(\mathbb{F}_m)\) are free, giving the \(n+m\) generators of \(L(\mathbb{F}_{n+m})\)).
The consequence: if \(L(\mathbb{F}_n)\cong L(\mathbb{F}_k)\) for some \(n\ne k\), then by the absorption \(L(\mathbb{F}_m)\cong L(\mathbb{F}_{m+(k-n)})\) for all \(m\) (the Dykema / the interpolation: the free group factors are either all isomorphic or form a strictly decreasing family). The free entropy obstruction: \(\delta(X_1,\dots,X_n)=n\) for the generators of \(L(\mathbb{F}_n)\), so if \(L(\mathbb{F}_n)\cong L(\mathbb{F}_m)\) then \(n=m\) (the free entropy dimension is an invariant, the partial obstruction — but the full obstruction / the full isomorphism problem remains open). The key point: the free product absorption is the structural fact, and the free entropy is the partial invariant — the isomorphism problem is open, the free entropy / the free dimension is the best current obstruction.
Check¶
Show that \(L(\mathbb{F}_n)*L(\mathbb{F}_m)\cong L(\mathbb{F}_{n+m})\) (the free product absorption), and explain the consequence for the isomorphism problem.
\(L(\mathbb{F}_n)=L(\mathbb{Z})^{*n}\) (the free product of \(n\) copies of \(L(\mathbb{Z})\), the generators \(u_1,\dots,u_n\) free). The free product \(L(\mathbb{F}_n)*L(\mathbb{F}_m)\) is \((L(\mathbb{Z})^{*n})*(L(\mathbb{Z})^{*m})=L(\mathbb{Z})^{*(n+m)}=L(\mathbb{F}_{n+m})\) (the free product is associative, the \(n+m\) generators \(u_1,\dots,u_n,v_1,\dots,v_m\) are free). The absorption: the free product adds the generators. The consequence: if \(L(\mathbb{F}_n)\cong L(\mathbb{F}_k)\) for \(k>n\) (say), then \(L(\mathbb{F}_m)=L(\mathbb{F}_m)*L(\mathbb{F}_n)*L(\mathbb{F}_{k-n})\cong L(\mathbb{F}_m)*L(\mathbb{F}_k)*L(\mathbb{F}_{k-n})\cong L(\mathbb{F}_{m+k})\)... iterating gives \(L(\mathbb{F}_m)\cong L(\mathbb{F}_{m+j(k-n)})\) for all \(j\), so the free group factors are either all isomorphic or form a strictly decreasing family (the Dykema interpolation).
Explain why the fundamental group of \(L(\mathbb{F}_\infty)\) is \(\mathbb{R}_+\) (Radulescu), and contrast with the open \(L(\mathbb{F}_n)\) for finite \(n\).
\(L(\mathbb{F}_\infty)\) has countably infinitely many generators, so the compression \(L(\mathbb{F}_\infty)^t\) (the amplification by \(t\)) is again \(L(\mathbb{F}_\infty)\) (the compression absorbs the generators: the \(t\)-amplification of a II\(_1\) factor with infinitely many free generators is the same factor, the Radulescu theorem — the free entropy / the interpolation, the \(L(\mathbb{F}_\infty)\) is stable under the compression). So \(\mathcal{F}(L(\mathbb{F}_\infty))=\mathbb{R}_+\) (the full group). For the finite \(n\), the \(L(\mathbb{F}_n)^t\) is the amplification, and whether \(L(\mathbb{F}_n)^t\cong L(\mathbb{F}_n)\) (the \(t\in\mathcal{F}(L(\mathbb{F}_n))\)) is open — the finite number of generators may not absorb the compression, the fundamental group is unknown (the Connes conjecture: \(\mathbb{R}_+\), but open).
Pitfalls¶
- The free group factor isomorphism is open — the free entropy / the free dimension is the best current obstruction (the \(\delta=n\)), but no full invariant is known.
- The free product absorption \(L(\mathbb{F}_n)*L(\mathbb{F}_m)\cong L(\mathbb{F}_{n+m})\) is the reduced free product (the von Neumann closure); the full free product (the universal C*-algebra) may differ.
- The fundamental group of \(L(\mathbb{F}_n)\) for finite \(n\) is open — the \(\mathbb{R}_+\) is for \(L(\mathbb{F}_\infty)\) (the Radulescu theorem), not for the finite \(n\) (the Connes conjecture, open).
- The free difference quotient \(\partial\) is the algebraic / formal derivation on the free algebra (the noncommutative Leibniz); the analytic / the von Neumann extension (the conjugate variable \(D_X\in L^2(M,\tau)\)) requires the \(L^2\)-smoothness, not every state.
- The cyclic derivative / the conjugate variable is the free analog of the score function, but the free Fisher \(\Phi\) is the \(L^2\)-norm, not the pointwise score — the analogy is structural, not pointwise.
- The Connes embedding (the microstates / the \(R^\omega\)) is the input to the microstates free entropy — the free group factors embed (the Connes embedding conjecture, the current status), but the embedding is the hypothesis, not the conclusion.
Refs¶
- Voiculescu, Dykema, Nica, Free Random Variables (CRM 1992); Voiculescu, "Symmetries of some reduced free product C*-algebras" (1985)
- Dykema, "On certain free product factors via an extended square function" (1993); Radulescu, "The fundamental group of \(L(\mathbb{F}_\infty)\)" (Invent. 1994)
- Connes, "Classification of injective factors" (Annals 1976); the fundamental group
- Voiculescu, "The analogues of entropy and of Fisher's information in free probability theory" (the free difference quotient, the conjugate variable)
Next¶
FP index · Foundations · Free Entropy & Fisher · Free Stochastic Calculus · Subordination & Harmonic Analysis · Operator algebras · NCG foundations