Cyclomorphy

Dan Voiculescu^{1}

Department of Mathematics University of California Berkeley, California 94720–3840

Abstract. A main result is that, roughly, a dense set of the infinitesimal trace-preserving deformations of a semicircular system s_{1}, …, s_{n} arise from oneparameter groups of automorphisms of the free-group factor L(F (n)) generated by s_{1}, …, s_{n}. More generally the paper studies cyclic gradients in von Neumann algebras, Lie algebras of noncommutative trace-preserving vector fields and the class of cyclomorphic maps which preserve the orthogonals of spaces of cyclic gradients.

0 Introduction

The word “cyclomorphy” is meant to remind the reader of “holomorphy” and of the “cyclic derivative” of Rota-Sagan-Stein ([8]). The paper is about cyclic derivatives in von Neumann algebras.

Several questions around free probability theory lead to cyclomorphy: moments of noncommutative random variables, free entropy ([12],[14]) and its connection with large deviations for random matrices ([4],[5]), orbits of the equivalence in distribution and the free Wasserstein distance ([1]). Last but not least I was motivated by curiositiy about Aut(L(F (n))), the automorphism group of a free group factor.

In general, if Y_{j} = Y_{j}^{∗}, 1 ≤ j ≤ m, generate a II_{1} factor M, the orbit of Y = (Y_{j})_{1≤j≤n} under Aut(M) provides a parametrization of Aut(M). The derivations in the “Lie algebra of Aut(M)” which have the Y_{j} ’s in their domain of definition, are then parametrized by the “tangent space” of the orbit at Y . This “tangent space” is a more tractable object as we shall see.

1Research at MSRI supported in part by National Science Foundation grant DMS–0079945.

Since automorphisms of M preserve its trace-state τ, the“tangent space” is orthogonal to the space of cyclic gradients of noncommutative polynomials in (Y_{1}, …, Y_{n}). The question then is which of the self-adjoint n-tuples in the orthogonal of cyclic gradients give rise to derivations which exponentiate to one-parameter groups of automorphisms? We show that exponentiation is possible if the n-tuple is polynomial, or more generally is given by noncommutative power series of sufficiently large radius of convergence evaluated at Y .

The polynomial elements in the orthogonal to cyclic gradients form a Lie algebra, which is a natural noncommutative relative of the Lie algebra of vector fields which preserve a volume form.

A natural generator for L(F (n)) is provided by a semicircular system s_{1}, …, s_{n} ([10],[11]). In this case we can give a complete description of the orthogonal to cyclic gradients. In particular we show that polynomial elements are dense in this space. Exponentiation therefore works for the corresponding derivations and we have an “infinitesimally rich” subalgebra of the “Lie algebra” of the automorphism group in this case.

From this study of automorphic orbits it becomes clear that it is important to understand the differential geometric picture arising on M^{n} (or on the hermitian subspace M_{h}^{n}) from the field of subspaces of the cotangent spaces arising from cyclic gradients and their orthogonals which are a field of subspaces in the tangent spaces. In particular there is a natural class of maps, those for which the differentials preserve the orthogonals to cyclic gradients. We look at basic properties of cyclomorphic maps and of a generalization of these to the case of B -von Neumann algebras (B a von Neumann subalgebra).

The paper has eight sections without the introduction. Section 1 deals with preliminaries. Section 2 is about the estimates which will prove that certain derivations can be exponentiated to automorphism groups. Section 3 combines the estimates of section 2 with the condition of orthogonality to cyclic gradients to obtain results about exponentiation of derivations. Section 4 is a brief look at endomorphic orbits and their tangent spaces. In section 5 we study real and complex cyclomorphic maps. In Theorem 5.13 and Proposition 5.14 we point out the role of independence and of free independence in this context. Section 6 is a collection of generalities about the Lie algebras of “noncommutative vector-fields” which occur in this paper. Section 7 deals with the semicircular case. We compute the orthogonal to the cyclic gradients in this case. Section 8 is about B -morphic maps, the extension to von Neumann algebras “over B ” of cyclomorphic maps. We only sketch how this generalization is done and will develop the details elsewhere.

1 Preliminaries

1.1 Noncommutative polynomials

If X_{1}, …, X_{n} are noncommuting indeterminates, the ring of noncommutative polynomials will be denoted by ℂ_{⟨n⟩} = ℂ⟨X_{1}, …, X_{n}⟩. If a_{1}, …, a_{n} are elements in an unital algebra over ℂ and P ∈ ℂ_{⟨n⟩} then P (a_{1}, …, a_{n}) or ε_{(a_{1},...,a_{n})}P denotes the evaluation of P at a_{1}, …, a_{n}.

We shall also use the notation ℂ⟨a_{1}, …, a_{n}⟩ for ε_{(a_{1},...,a_{n})}ℂ⟨X_{1}, …, X_{n}⟩, i.e. for the algebra generated by 1, a_{1}, …, a_{n}. Clearly this algebra is isomorphic to the algebra of noncommutative polynomials only if a_{1}, …, a_{n} are algebraically free. When we will use the notation it will be clear from the context whether a_{1}, …, a_{n} are algebraically free or not.

1.2 Cyclic gradients and free difference quotients

The partial free difference quotients are the derivations

∂_{j} : ℂ_{⟨n⟩} → ℂ_{⟨n⟩} ⊗ ℂ_{⟨n⟩}

such that ∂_{j}X_{k} = 0 if j ≠ k and ∂_{j}X_{j} = 1 ⊗ 1, where j, k ∈ {1, …, n}. The ℂ_{⟨n⟩}-bimodule structure on ℂ_{⟨n⟩} ⊗ ℂ_{⟨n⟩} is the obvious one a_{1}(b ⊗ c)a_{2} = a_{1}b ⊗ ca_{2}.

The partial cyclic derivatives are then δ_{j} = μ̃ ∘ ∂_{j} : ℂ_{⟨n⟩} → ℂ_{⟨n⟩} where μ̃(a ⊗ b) = ba. Thus

δ_{j}X_{i_{1}} … X_{i_{p}} = ∑_{1k|1≤k≤p, i_{k}=j} X_{i_{k+1}} … X_{i_{p}}X_{i_{1}} … X_{i_{k−1}} .

The map δ : ℂ_{⟨n⟩} → (ℂ_{⟨n⟩})^{n} = ℂ_{⟨n⟩} ⊕ ⋯ ⊕ ℂ_{⟨n⟩} given by

δP = (δ_{j}P)_{1≤j≤n} = δ_{1}P ⊕ ⋯ ⊕ δ_{n}P

is the cyclic gradient.

Similarly ∂ : ℂ_{⟨n⟩} → (ℂ_{⟨n⟩} ⊗ ℂ_{⟨n⟩})^{n} denotes the free difference quotient gradient.

In case a_{1}, …, a_{n} are algebraically free the corresponding maps for ℂ_{⟨n⟩}⟨a_{1}, …, a_{n}⟩ will be denoted

∂_{j}^{(a_{1},...,a_{n})}, ∂^{(a_{1},...,a_{n})}, δ_{j}^{(a_{1},...,a_{n})}, δ^{(a_{1},...,a_{n})}

or ∂_{j}^{α}, ∂^{α}, δ_{j}^{α}, δ^{α} where α = (a_{1}, …, a_{n}).

The notation for cyclic derivatives and gradients should not be confused with Kronecker symbols δ_{ij} which have two lower indices.

If a, a_{1}, …, a_{n} are elements in some unital algebra A over ℂ let m_{a} : ℂ⟨a_{1}, …, a_{n}⟩ ⊗ ℂ⟨a_{1}, …, a_{n}⟩ → A be the map given by

m_{a}(P_{1} ⊗ P_{2}) = P_{1}aP_{2} .

In particular if a_{1}, …, a_{n} are algebraically free, then a derivation of ℂ⟨a_{1}, …, a_{n}⟩ into A can always be written

P ⇝ ∑_{1≤j≤n} m_{b_{j}} ∂_{j}^{α}P

where b_{1}, …, b_{n} ∈ A. The elements b_{1}, …, b_{n} are uniquely determined, being the values of the derivation on a_{1}, …, a_{n}.

1.3 The exact sequence for cyclic gradients

In [13] we described the set of cyclic gradients by an exact sequence

0 → ℂ1 + [ℂ_{⟨n⟩}, ℂ_{⟨n⟩}] → ℂ_{⟨n⟩} →^{δ} (ℂ_{⟨n⟩})^{n} →^{θ} ℂ_{⟨n⟩}

where θ(P_{1} ⊕ ⋯ ⊕ P_{n}) = ∑_{j}[X_{j}, P_{j}]. Moreover

ℂ1 + [ℂ_{⟨n⟩}, ℂ_{⟨n⟩}] = ℂ1 + ∑_{1≤k≤n}[X_{k}, ℂ_{⟨n⟩}] = Ker C

where C : ℂ_{⟨n⟩} → ℂ_{⟨n⟩} is the cyclic symmetrization map C1 = 0 and

C X_{i_{1}} … X_{i_{p}} = ∑_{1≤j≤p} X_{i_{j+1}} … X_{i_{p}}X_{i_{1}} … X_{i_{j}}

if p ≥ 1.

If instead of X_{1}, …, X_{n} we have algebraically free a_{1}, …, a_{n} we shall use the notations

θ^{α} , C^{α}

where α = (a_{1}, …, a_{n}).

Also, if a_{1}, …, a_{n} are not algebraically free, we have the set of cyclic gradients evaluated at α = (a_{1}, …, a_{n})

(ε_{α})^{n}(δℂ_{⟨n⟩}) .

1.4 Semicircular systems

Recall that in a C^{∗}-probability space (A, φ) a semicircular system is an n-tuple (S_{1}, …, S_{n}) of selfadjoint noncommutative random variables which are freely independent and have (0,1) semicircle distributions. The von Neumann algebra of (S_{1}, …, S_{n}) in the GNS representation associated with the restriction of φ is isomorphic to the free group factor L(F (n)) and the restriction of φ is the trace-state.

There is a natural realization of a semicircular system on the full Fock space

T (ℂ^{n}) = ⨁_{k≥0}(ℂ^{n})^{⊗k}

where (ℂ^{n})^{⊗0} = ℂ1 with 1 the vacuum vector. If e_{j}, 1 ≤ j ≤ n are the standard orthonormal basis vectors in ℂ^{n}, let

ℓ_{j}ξ = e_{j} ⊗ ξ , r_{j}ξ = ξ ⊗ e_{j}

be the left and right creation operators. Then s_{j} = ℓ_{j} + ℓ_{j}^{∗} (1 ≤ j ≤ n) is a semicircular system w.r.t. the vacuum expectation ⟨ ·1, 1⟩ and similarly d_{j} = r_{j} + r_{j}^{∗} (1 ≤ j ≤ n).

In particular T (ℂ^{n}) identifies with L^{2}(W^{∗}(s_{1}, …, s_{n})) and denoting by P_{0}, P_{1}, … the orthonormal polynomials w.r.t. a semicircular distribution we have

e_{i_{1}}^{⊗k_{1}} ⊗ … ⊗ e_{i_{p}}^{⊗k_{p}} = P_{k_{1}}(s_{i_{1}}) … P_{k_{p}}(s_{i_{p}})1 = ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{p}}^{k_{p}}1

where i_{j} ≠ i_{j+1} (1 ≤ j ≤ p − 1) and k_{j} > 0 (1 ≤ j ≤ p).

The polynomials P_{k}(x), k ≥ 0 are equal to C_{k}^{1}(x/2) where C_{k}^{1}(t) are Gegenbauer polynomials, which satisfy the generating function relation

(1 − 2rt + r^{2})^{−1} = ∑_{n≥0} C_{n}^{1}(t)r^{n}

(see [9, ch.IX, §4, sec.12]).

2 Exponentiating noncommutative vector fields

2.1 If E is a ℂ_{⟨n⟩}-bimodule, let Vect E denote E^{n}. If K = (K_{j})_{1≤j≤n} ∈ Vect E let D_{K} : ℂ_{⟨n⟩} → E be the derivation such that D_{K}X_{j} = K_{j}, i.e.,

D_{K} = ∑_{1≤j≤n} m_{K_{j}}∂_{j} .

In particular if E is a unital Banach-algebra and the bimodule structure arises from a unital homomorphism of ℂ_{⟨n⟩} into E which takes X_{j} to T_{j}, then

\frac{d}{dε} P (T_{1} + εK_{1}, …, T_{n} + εK_{n})|_{ε=0} = D_{K}P

where P ∈ ℂ_{⟨n⟩}.

2.2 On ℂ_{⟨n⟩} we define seminorms

|∑_{p≥0} ∑_{1≤j_{1},...,j_{p}≤n} c_{j_{1},...,j_{p}}X_{j_{1}} … X_{j_{p}} |_{R,k} = ∑_{p≥k} ∑_{1≤j_{1},...,j_{p}≤n} |c_{j_{1},...,j_{p}}|R^{p−k}p!((p − k)!)^{−1} .

The completion of ℂ_{⟨n⟩} w.r.t. | · |_{R,j}, 0≤j ≤k will be denoted ℂ_{⟨n⟩,R,k} or ℂ⟨X_{1}, …, X_{n}⟩_{R,k} and identifies with a subalgebra of the algebra of noncommutative formal power series denoted ℂ_{⟨⟨n⟩⟩} or ℂ⟨⟨X_{1}, …, X_{n}⟩⟩.

On Vect ℂ⟨X_{1}, …, X_{n}⟩ we have corresponding seminorms

|(K_{1}, …, K_{n})|_{R,k} = max_{1≤s≤n} |K_{s}|_{R,k} .

The completions coincide with Vect ℂ_{⟨n⟩,R,k}.

If K ∈ Vect ℂ_{⟨n⟩} and P ∈ ℂ_{⟨n⟩}, then

|D_{K}P |_{R,0} ≤ |K|_{R,0} |P |_{R,1} .

In particular D_{K}P can be defined for K ∈ Vect ℂ_{⟨n⟩,R,0} and P ∈ ℂ_{⟨n⟩,R,1} as an element in ℂ_{⟨n⟩,R,0}. Also if 0 < R < R′ then |P |_{R,1} ≤ C|P |_{R′,0′} for some constant independent of P, so that D_{K}P ∈ ℂ_{⟨n⟩,R,0} if P ∈ ℂ_{⟨n⟩,R′,0} and K ∈ Vect ℂ_{⟨n⟩,R,0}. Iterating, we have D_{K}^{m}P ∈ ℂ_{⟨n⟩,R,0} if P ∈ ℂ_{⟨n⟩,R′,0} and K ∈ Vect ℂ_{⟨n⟩,R′,0} for some R′ > R.

2.3 The map Φ_{n} : ℂ_{⟨n⟩} → ℂ_{⟨1⟩} is defined by

Φ_{n} (∑_{p≥0} ∑_{1≤j_{1},...,j_{p}≤n} c_{j_{1},...,j_{p}}X_{j_{1}} … X_{j_{p}} ) = ∑_{p≥0} ∑_{1≤j_{1},...,j_{p}≤n} |c_{j_{1},...,j_{p}}|X_{1}^{p} .

Clearly |Φ_{n}(P)|_{R,k} = |P |_{R,k} = (D_{X_{1}}^{k}Φ_{n}(P))(R) where R > 0, k ≥ 0.

Obviously Φ_{n} extends to a map of ℂ_{⟨n⟩,R,k} to ℂ_{⟨1⟩,R,k}.

2.4 If P ∈ ℂ_{⟨n⟩} let |P | be the noncommutative polynomial with coefficients the absolute values of the coefficients of P. Also, if P, Q ∈ ℂ_{⟨n⟩} we shall write |P | ≤ |Q| if the inequality holds among the coefficients of |P | and |Q| which are nonnegative numbers).

We extend this definition to Vect ℂ_{⟨n⟩} by putting |(K_{j})_{1≤j≤n}| = (|K_{j}|)_{1≤j≤n} and |K| ≤ |K′| if |K_{j}| ≤ |K′_{j}| (1 ≤ j ≤ n) where K = (K_{j})_{1≤j≤n}, K′ = (K′_{j})_{1≤j≤n}.

We shall also write |P | ∨ |Q| for the coefficientwise maximum.

2.5 The analogue of Φ_{n} on Vect ℂ_{⟨n⟩} is the map Ψ_{n} : Vect ℂ_{⟨n⟩} → Vect ℂ_{⟨1⟩} defined by

Ψ_{n}((K_{j})_{1≤j≤n}) = Φ_{n}(K_{1}) ∨ ⋯ ∨ Φ_{n}(K_{n}) .

We have

|K|_{R,k} ≤ (D_{X_{1}}^{k} Ψ_{n}(K))(R)

with equality if k = 0.

2.6 Lemma. Let K, K′ ∈ Vect ℂ_{⟨n⟩}, and P, P′ ∈ ℂ_{⟨n⟩}. Then |D_{K}P | ≤ D_{|K|}|P |. If |K| ≤ |K′| and |P | ≤ |P′|, then

D_{|K|}|P | ≤ D_{|K′|}|P′| andΦ_{n}(P) ≤ Φ_{n}(P′) , Ψ_{n}(K) ≤ Ψ_{n}(K′) .

Moreover we have

Φ_{n}(D_{|K|}|P |) ≤ D_{Ψ_{n}(K)}Φ_{n}(P)

The proof reduces to the most obvious majorizations and is left to the reader.

2.7 Theorem. Let 0 < R < R′ and let K ∈ Vect ℂ_{⟨n⟩,R′,0} and P ∈ ℂ_{⟨n⟩,R′,0}. Then

|D_{K}^{m}P |_{R,0} ≤ 1 · 3 · 5 · . . . (2m − 1)(R′)^{m+1}(R′ − R)^{−2m−1}|K|_{R′,0}^{m}|P |_{R′,0} .

In particular, D_{K}^{m}P ∈ ℂ_{⟨n⟩,R,0} and if 0 < r < |K|_{R′,0}^{−1}(R′ − R)^{2}(2R′)^{−1}, then

∑_{m≥0} |D_{K}^{m}P |_{R,0} \frac{r^{m}}{m!} < ∞

i.e., P is an analytic vector for D_{K} in ℂ_{⟨n⟩,R,0}.

Proof. Clearly all assertions are an easy consequence of the estimate for |D_{K}^{m}P |_{R,0} and it suffices to prove it when K ∈ Vect ℂ_{⟨n⟩} and P ∈ ℂ_{⟨n⟩}.

If |P |_{R′,0} = M then (Φ_{n}(P))(R′) = M and the coefficient of X_{1}^{k} of Φ_{n}(P) is then majorized by M(R′)^{−k} so that

Φ_{n}(P) ≤ M(1 − X_{1}/R′)^{−1} .

Similarly, if |K|_{R′,0} = N we have

Ψ_{n}(K) ≤ N(1 − X_{1}/R′)^{−1} .

Using Lemma 2.6 we get

D_{Ψ_{n}(K)}^{m}Φ_{n}(P) ≤ |K|_{R′,0}^{m} |P |_{R′,0} D_{Λ}^{m}Λ

where Λ = (1 − X_{1}/R′)^{−1}. By induction we easily get

D_{Λ}^{m}Λ = 1 · 3 · 5 · . . . (2m − 1)(R′)^{−m}Λ^{2m+1} .

It follows that

|D_{K}^{m}P |_{R,0} ≤ |K|_{R′,0}^{m} |P |_{R′,0} D_{Λ}^{m}Λ(R)
= (2m)!(m!)^{−1} 2^{−m}(R′)^{m+1}(R′ − R)^{−2m−1}|K|_{R′,0}^{m} |P |_{R′,0}
□

3 Cyclic gradients and exponentiation

3.1 The basic property of cyclic gradients

Let (M, τ) be a von Neumann algebra with a normal faithful trace state, i.e., a tracial W^{∗}-probability space, and let T_{j} ∈ M, K_{j} ∈ M, 1 ≤ j ≤ n. If P ∈ ℂ_{⟨n⟩}, then

\frac{d}{dε} τ(P (T_{1} + εK_{1}, …, T_{n} + εK_{n}))|_{ε=0} = ∑_{1≤j≤n} τ((δ_{j}P)(T_{1}, …, T_{n})K_{j}) .

Thus the cyclic gradient (δP)(T_{1}, …, T_{n}) is precisely the gradient at (T_{1}, …, T_{n}) of the function

M^{n} ∋ (T_{1}, …, T_{n}) → τ(P (T_{1}, …, T_{n})) ∈ ℂ

when we use the bilinear scalar product given by ∑_{1≤j≤n} τ(a_{j}b_{j}) on M^{n} = Vect M (w.r.t. the bimodule structure induced by ε_{(T_{1},...,T_{n})}). Under the sesquilinear scalar product ∑_{1≤j≤n} τ(a_{j}b_{j}^{∗}) the gradient would be (δP (T_{1}, …, T_{n}))^{∗} = ((δ_{j}P (T_{1}, …, T_{n}))^{∗})_{1≤j≤n}.

3.2 On ℂ_{⟨n⟩} we consider the involution P → P^{∗} such that X_{j} = X_{j}^{∗}, i.e., (cX_{i_{1}} … X_{i_{p}})^{∗} = c̄X_{i_{p}} … X_{i_{1}} . If T_{j} ∈ M then

(P (T_{1}, …, T_{n}))^{∗} = (P^{∗}(T_{1}^{∗}, …, T_{n}^{∗})

and if T_{j} = T_{j}^{∗}, then

(P (T_{1}, …, T_{n}))^{∗} = (P^{∗}(T_{1}, …, T_{n}) .

It is also easily seen that δ_{j}P^{∗} = (δ_{j}P)^{∗}. Similarly ∂_{j}P^{∗} = { \atop (∂_{j}P)}^{∗} where on ℂ_{⟨n⟩} ⊗ ℂ_{⟨n⟩} we

define (ξ ⊗ η)^{∗} = ξ^{∗} ⊗ η^{∗} and { \atop ξ ⊗ η} = η ⊗ ξ.

3.3 Let Y_{j} = Y_{j}^{∗} ∈ M, 1 ≤ j ≤ n be algebraically free so that ℂ_{⟨n⟩} and ℂ⟨Y_{1}, …, Y_{n}⟩ are isomorphic, via ε_{(Y_{1},...,Y_{n})}. This turns L^{2}(M, τ) into a ℂ_{⟨n⟩} bimodule. Assume also {Y_{1}, …, Y_{n}} generates M. If K ∈ Vect L^{2}(M, τ) we shall consider D_{K}^{ε} = D_{K} ∘ ε_{(Y_{1},...,Y_{n})}^{−1} which is a derivation of ℂ⟨Y_{1}, …, Y_{n}⟩ into L^{2}(M, τ). We shall view D_{K}^{ε} as an unbounded densely defined operator L^{2}(M) with domain of definition ℂ⟨Y_{1}, …, Y_{n}⟩.

3.4 Proposition. The following are equivalent conditions on K.

(i) ∑_{1≤j≤n} τ((δ_{j}P)(Y_{1}, …, Y_{n})K_{j}) = 0 for all P ∈ ℂ_{⟨n⟩}.

(ii) τ(D_{K}^{ε}Q) = 0 for all Q ∈ ℂ⟨Y_{1}, …, Y_{n}⟩

(iii) If P_{1}, P_{2} ∈ ℂ⟨Y_{1}, …, Y_{n}⟩ then in L^{2}(M, τ), ⟨D_{K}^{ε}P_{1}, P_{2}⟩ = −⟨P_{1}, D_{K^{∗}}^{ε}P_{2}⟩ where K^{∗} = (K_{1}^{∗}, …, K_{n}^{∗}).

Proof. (i) ⇔ (ii) follows from 2.1 and 3.1. We have

⟨D_{K}^{ε}P_{1}, P_{2}⟩ + ⟨P_{1}, D_{K^{∗}}^{ε}P_{2}⟩ = τ((D_{K}^{ε}P_{1})P_{2}^{∗} + P_{1}(D_{K^{∗}}^{ε}P_{2})^{∗})
= τ((D_{K}^{ε}P_{1})P_{2}^{∗} + P_{1}(D_{K}^{ε}P_{2}^{∗}))
= τ(D_{K}^{ε}(P_{1}P_{2}^{∗})) .

Clearly τ(D_{K}(P_{1}P_{2}^{∗})) = 0 for all P_{1}, P_{2} ∈ ℂ⟨Y_{1}, …, Y_{n}⟩ is equivalent to (ii) so that (ii) ⇒ (iii). □

3.4 Corollary. Assume K satisfies the equivalent conditions of Proposition 3.3. Then:

a) D_{K}^{ε} is closable

b) If K = K^{∗}, the densely defined operator D_{K}^{ε} is antisymmetric.

3.5 If K satisfies the equivalent conditions of Proposition 3.3, we shall say K is a tracepreserving or τ -preserving (if we want to specify the trace) noncommutative vector field.

Note also that the equivalence in Proposition 3.3 actually holds more generally for K ∈ Vect L^{1}(M, τ) while (iii) can be adapted to this case in terms of the duality of L^{1}(M, τ) and M.

3.6 If K = K^{∗} ∈ Vect M then D_{K}^{ε}P ∈ M if P ∈ ℂ⟨Y_{1}, …, Y_{n}⟩ and D_{K}^{ε}P^{∗} = (D_{K}^{ε}P)^{∗} so that D_{K}^{ε} is a symmetric derivation (see 3.2.21 in [3]) of C^{∗}(Y_{1}, …, Y_{n}).

If additionally K is in ℂ⟨Y_{1}, …, Y_{n}⟩ and is trace-preserving then by Corollary 3.4, H = −iD_{K}^{ε} is a symmetric unbounded operator defined on ℂ⟨Y_{1}, …, Y_{n}⟩ which is a dense subspace in L^{2}(M, τ). Since D_{K}^{ε} is a derivation, we have

D_{K}^{ε}P = i[H, P]

if P ∈ ℂ⟨Y_{1}, …, Y_{n}⟩. This makes D_{K}^{ε} a spatial derivation of C^{∗}(Y_{1}, …, Y_{n}) implemented by H (see Definition 3.2.54 in [3]). Summarizing, we have proved

Corollary. Assume K = K^{∗} ∈ Vect ℂ⟨Y_{1}, …, Y_{n}⟩ is trace-preserving, then D_{K}^{ε} is a spatial derivation of C^{∗}(Y_{1}, …, Y_{n}) implemented on L^{2}(M, τ).

3.7 Combining the preceding corollary with results on exponentiation of derivations to automorphisms, one gets results about when D_{K}^{ε} can be exponentiated. For instance, using Proposition 3.2.58 in [3] we get the following.

Corollary. Assume K = K^{∗} ∈ Vect ℂ⟨Y_{1}, …, Y_{n}⟩ is trace-preserving. If ℂ⟨Y_{1}, …, Y_{n}⟩ ⊂ L^{2}(M, τ) consists of analytic vectors for D_{K}^{ε}, viewed as an unbounded operator in L^{2}(M, τ), then D_{K}^{ε} exponentiates to a one-parameter automorphism group exp(tD_{K}^{ε}) of M which is trace-preserving τ ∘ exp(tD_{K}) = τ.

3.8 Let R ≥ ∥Y_{j}∥, 1 ≤ j ≤ n. Then |P (Y_{1}, …, Y_{n})|_{2} ≤ ∥P (Y_{1}, …, Y_{n})∥ ≤ |P (X_{1}, …, X_{n})|_{R,0}. Then Theorem 2.7 implies that for some r > 0 we have

∑_{m≥0} |(D_{K}^{ε})^{m}P |_{2} \frac{r^{m}}{m!} < ∞

if P ∈ ℂ⟨Y_{1}, …, Y_{n}⟩, i.e., P is an analytic vector in L^{2}(M). Thus we have actually proved:

Theorem. Assume Y_{j} = Y_{j}^{∗} ∈ M, 1 ≤ j ≤ n, and K = K^{∗} ∈ Vect ℂ⟨Y_{1}, …, Y_{n}⟩. Assume {Y_{1}, …, Y_{n}} generates M and the Y_{j} ’s are algebraically free. If K is trace-preserving, then D_{K}^{ε} exponentiates to a one-parameter group of trace-preserving automorphism of M.

3.9 Let R ≥ ∥Y_{j}∥, 1 ≤ j ≤ n. We could have replaced during all steps the ring ℂ⟨Y_{1}, …, Y_{n}⟩ with the ring

ε_{(Y_{1},...,Y_{n})}ℂ_{⟨n⟩,>R}

of power-series of radius of convergence > R. The only additional assumption is that algebraic freeness has to be replaced with the stronger requirement that

ε_{(Y_{1},...,Y_{n})}ℂ_{⟨n⟩,>R} → M

is one-to-one. With these amendments trace-preserving self-adjoint K ∈ Vect ε_{(Y_{1},...,Y_{n})}ℂ_{⟨n⟩,>R} also “exponentiate” to one-parameter groups of automorphisms of M.

Of course exponentiate both here and in 3.7, 3.8 means either that we take a weak closure of the exponential or use the automorphism group implemented by the unitary group exp(it\overline{H}) where \overline{H} is the closure of H.

4 Endomorphic orbits

4.1 Let Y_{j} = Y_{j}^{∗} ∈ (M, τ), 1 ≤ j ≤ n. We shall denote by M_{h} = {a ∈ M | a = a^{∗}} the hermitian elements of M and by L^{p}(M_{h}, τ) the corresponding L^{p}-space. The scalar product ∑_{1≤j≤n} τ(a_{j}b_{j}^{∗}) on (L^{2}(M, τ))^{n} restricts to a real scalar product on (L^{2}(M_{h}, τ))^{n}. Note also that the set of cyclic gradients is self-adjoint when evaluated at Y_{1}, …, Y_{n}. The orthogonal to cyclic gradients at Y_{1}, …, Y_{n} is:

(δℂ_{⟨n⟩}^{⊥})(Y_{1}, …, Y_{n}) = {Z ∈ (L^{2}(M_{h}, τ))^{n} | ⟨Z, δP (Y_{1}, …, Y_{n})⟩ = 0 if P ∈ ℂ_{⟨n⟩}} .

Clearly in the previous formula we might consider only P = P^{∗}. Also, we have corresponding p-spaces replacing L^{2} by L^{p}.

4.2 By Aut(M, τ) (resp. End(M, τ)) we denote automorphisms (resp. unital endomorphisms) α of M such that τ ∘α = τ. If M is a II_{1}-factor then preservation of the trace is automatic and we can write Aut(M) (resp. End(M)). If Y ∈ M_{h}^{n} let AO(Y) = Aut(M, τ)·Y (resp. EO(Y) = End(M, τ) · Y ) be the automorphic (resp. endomorphic) orbit of Y .

We define the tangent set to AO(Y) (resp. EO(Y)) at Y to be

TAO(Y) = {ξ ∈ (L^{2}(M_{h}, τ))^{n} | ∃ η_{k} ∈ AO(Y),

∃ μ_{k} ∈ ℝ\{0}, μ_{k} → 0, lim_{k→∞} |μ_{k}^{−1}(η_{k} − Y) − ξ|_{2} = 0}

(resp. T EO(Y) defined similarly using EO instead of AO).

There are several variants of these sets TAO_{p}(Y), wTAO_{p}(Y) (resp. T EO_{p}(Y), wT EO_{p}(Y)_{n}) where we take x ∈ (L^{p}(M_{h}, τ))^{n} and require the limit to hold in p-norm (1 ≤ p ≤ ∞) or for wTAO_{p}(Y) (resp. T EO_{p}(Y)) to hold w.r.t. the weak convergence in the duality with (L^{q}(M_{h}, τ))^{n} (p^{−1} + q^{−1} = 1). If p = 2 we simply write wTAO(Y) (resp. wT EO(Y)). Remark also that wT EO(Y) ⊃ T EO(Y) ∪ TAO(Y) ∪ wTAO(Y).

4.3 Proposition. wT EO(Y) ⊂ (δℂ_{⟨n⟩}^{⊥})(Y).

Proof. Let η_{k} ∈ EO(Y) so that w − lim_{k→∞}(μ_{k}^{−1}(η_{k} − Y) − ξ) = 0 in (L^{2}(M_{h}, τ))^{n}. Since the weak convergence is for a sequence, we have uniform boundedness in L^{2} and hence μ_{k}^{−1} = O(|η_{k} − Y |_{2}^{−1}). On the other hand, since η_{k}, Y are in (M_{h})^{n}, we have

τ(P (η_{k}) − P (Y) − ∑_{j}(η_{k} − Y)_{j}(δ_{j}P)(Y)) = O(|η_{k} − Y |_{2}^{2})

where P ∈ ℂ_{⟨n⟩}. Since

μ_{k}^{−1}O(|η_{k} − Y |_{2}^{2}) = O(|η_{k} − Y |_{2}^{−1}|η_{k} − Y |_{2}^{2})
= O(|η_{k} − Y |_{2}) = o(1)

and τ(P (η_{k})) = τ(P (Y)) we have

lim_{k→∞} τ (∑_{j} μ_{k}^{−1}(η_{k} − Y)_{j}(δ_{j}P)(Y)) = 0 .

We infer ∑_{j} τ(ξ_{j}(δ_{j}P)(Y)) = 0, i.e., ξ ∈ (δℂ_{⟨n⟩}^{⊥})(Y). □

4.4 If Y ∈ (M_{h})^{n} is a generator of M, then EO(Y) parametrizes End(M, τ). It is also easily seen that EO(Y) is a closed set in the topology of the L^{2}-metric. On the other hand, if Y = (Y_{1}, …, Y_{n}) is a generating semicircular system of L(F (n)),

then AO(Y) being L^{2}-closed would imply the non-isomorphism of free group factors.

Indeed L(F (∞)) has a sequence of automorphisms α_{k} so that α_{k}(λ(g_{p})) = λ(g_{q}) if q − p ≡ 1 (mod k) when 1 ≤ p ≤ k and p = q if p > k, which converge pointwise on L^{2} to a non-invertible endomorphism. Thus the isomorphism of L(F (n)) and L(F (∞)) would imply the automorphic orbit of Y is not closed.

4.5 Remark. In connection with free probability, there is a third kind of orbits which occur: the orbits of the relation of equivalence in distribution. The distribution orbit of Y denoted DO(Y) is the set {Y′ ∈ M_{h}^{n} | Y′ and Y equivalent in distribution} where equivalence in distribution means τ(Y_{i_{1}} … Y_{i_{p}}) = τ(Y′_{i_{1}} … Y′_{p}) for all p ∈ ℕ and i_{j} ∈ {1, …, n}. Equivalently, Y′ ∈ DO(Y) if there is a unital trace-preserving isomorphism α : W^{∗}(I, Y_{1}, …, Y_{n}) → W^{∗}(I, Y′_{1}, …, Y′_{n}) so that α(Y_{j}) = Y′_{j}. Clearly DO(Y) ⊃ EO(Y). Then the tangent sets T DO(Y), T_{p}DO(Y), wT DO(Y), wT DO_{p}(Y) are defined similarly to the endomorphic and automorphic cases and we have the analogue of Proposition 4.3, i.e.:

wT DO(Y) ⊂ (δℂ_{⟨n⟩}^{⊥})(Y) .

5 Real and complex cyclomorphic maps

5.1 Throughout section 5, by (M, τ), (N, ν), (A, α), (B, β), (C, γ) we shall denote von Neumann algebras with normal faithful trace-states.

Definition. A differentiable map f : Ω → ℝ, where Ω ⊂ M_{h}^{n} is an open set in the norm-topology, is ℝ-cyclomorphic if for every Y ∈ Ω there is T ∈ (L^{1}(M_{h}, τ))^{n} such that for K ∈ M_{h}^{n} we have a

df[Y](K_{1}, …, K_{n}) = ∑_{1≤j≤n} τ(T_{j}K_{j})

and T is in the L^{1}-norm closure of δℂ_{⟨n⟩}(Y) ∩ M_{h}^{n}. We shall abbreviate saying f is ℝcm.

5.2 Definition. A differentiable map f : Ω → N_{h}^{p}, where Ω ⊂ M_{h}^{n} is open in the normtopology, is ℝ-cyclomorphic if for every ℝ-cyclomorphic map f : ω → ℝ, where ω ⊂ N_{h}^{p} is open in the norm-topology, we have that f ∘ F : Ω ∩ F^{−1}(ω) → ℝ is ℝ-cyclomorphic. Also in this case this will be abbreviated by F is ℝcm.

5.3 Since ℝ = ℂ_{h}, a ℝcm map f : Ω → ℝ is also a map f : Ω → ℂ_{h}. It is easily seen that definition 5.1 for f : Ω → ℝ and definition 5.2 for f : Ω → ℂ_{h} are equivalent. For this it suffices to remark that the ℝcm maps g : ω → ℝ, ω ⊂ ℂ_{h} are just the differentiable maps and that if g : ω → ℝ is differentiable, f(Y) ∈ ω, Y ∈ Ω, then d(g ∘ f)[Y] = g′(f(Y))dg[Y].

5.4 Examples. a) If P = P^{∗} ∈ ℂ_{⟨n⟩} then the map f_{P} : M_{h}^{n} → ℝ, defined by f_{P} (Y) = τ(P (Y)) is ℝcm. Indeed df_{P} [Y](K) = τ(∑_{j}(δ_{j}P)(Y)K_{j}).

b) If f_{1}, f_{2} : Ω → ℝ, where Ω ⊂ M_{h}^{n}, are ℝcm, then also f_{1} + f_{2} and f_{1}f_{2} are ℝcm. The ℝcm-maps from Ω to ℝ form an algebra over ℝ with unit.

5.5 Lemma. A differentiable map F : Ω → N_{h}^{p}, where Ω ⊂ M_{h}^{n} is ℝcm iff for every P = P^{∗} ∈ ℂ_{⟨p⟩} the map f_{P} ∘ F is ℝcm, where f_{P} : N_{h}^{p} → ℝ, f_{P} (T) = ν(P (T)).

Proof. The “only if ” is obvious by 5.4a). To prove the “if ”-part, remark that Definition 5.1 for a differentiable map f : ω → ℝ, ω ⊂ N^{p} open, is equivalent to saying that f is ℝcm iff for every T ∈ N_{h}^{p} there is a sequence of polynomials P_{s} = P_{s}^{∗} ∈ ℂ_{⟨p⟩} such that ∥df_{P_{s}}[T] − df[T]∥ → 0 as s → ∞ in L(N_{h}^{p}, ℝ). If Y ∈ Ω, F (Y) ∈ ω, then d(f ∘ F)[Y] = df[F (Y)] ∘ dF [Y], so that

∥d(f ∘ F)[Y] − d(f_{P_{s}} ∘ F)[Y]∥ ≤ ∥df[F (Y)] − df_{P_{s}}[F (Y)]∥ ∥dF [Y]∥ .

Hence d(f ∘ F)[Y] is in the closure of differentials d(f_{P_{s}} ∘ F)[Y], i.e. in the closure of linear maps given by cyclic gradients. Hence f ∘ F is ℝcm. □

5.6 Examples. a) If P_{j} = P_{j}^{∗} ∈ ℂ_{⟨n⟩}, 1 ≤ j ≤ p, then the map F_{p} : M_{h}^{n} → M_{h}^{p} given by F_{P} (Y) = (P_{j}(Y))_{1≤j≤p} is ℝcm. Indeed if Q = Q^{∗} ∈ ℂ_{⟨p⟩}, then

(f_{Q} ∘ F_{P})(Y) = τ(Q(P (Y)) = f_{Q∘P}

and the assertion follows from Lemma 5.5 and Example 5.4a).

b) Let Ω_{1} ⊂ A_{h}^{n}, Ω_{2} ⊂ B_{h}^{p}, be open sets in the norm-topology, and let F : Ω_{1} → B_{h}^{p}, G : Ω_{2} → C_{h}^{q} be ℝcm maps, so that F (Ω_{1}) ⊂ Ω_{2}. Then F ∘ G is also an ℝcm map.

c) If f : Ω → ℝ and F : Ω → N_{h}^{p}, where Ω ⊂ M_{h}^{n}, are ℝcm, then H : Ω → N_{h}^{p} defined as H(Y) = f(Y)F (Y) is ℝcm. Indeed by Lemma 5.5 it suffices to show that if P = P^{∗} ∈ ℂ_{⟨p⟩} we have f_{P} ∘ H is ℝcm. Clearly there is no loss of generality to assume P is homogeneous of degree q. Then (f_{P} ∘ H)(Y) = (f(Y))^{q}(f_{P} ∘ H)(Y) is ℝcm by 5.4b).

5.7 Proposition. Let f : Ω → ℝ, Ω ⊂ M_{h}^{n} be a ℝcm map and let (α_{t})_{t≥0} ⊂ End(M, τ) be a semigroup, i.e., α_{0} = id_{M}, α_{s+t} = α_{s} ∘ α_{t}. Let further Y ∈ ω be such that for t ∈ [0, 1], α_{t}(Y) ∈ Ω, t → α_{t}(Y) is norm-continuous and the right derivative

lim_{t↓0} t^{−1}(α_{t}(Y) − Y) = K ∈ M_{n}^{h}

exists in norm-convergence. Then

f(α_{t}(Y)) = f(Y)

for all t ∈ [0, 1].

Proof. By Proposition 4.3 we have that K ∈ δℂ_{⟨n⟩}^{⊥}(Y) which implies df[Y](K) = 0, i.e. lim_{t↓0} t^{−1}(f(α_{t}(Y)) − f(Y)) = 0. By the semigroup property we have lim_{t↓0} t^{−1}(α_{s+t}(Y) − α_{s}(Y)) = α_{s}(K) and the same argument applied to α_{s}(Y), 0 ≤ s < 1 gives

lim_{t↓0} t^{−1}(f(α_{t+s}(Y)) − f(α_{s}(Y))) = 0 .

Therefore f(α_{t}(Y)) as a function of t ∈ [0, 1] is continuous and has zero right derivatives at each point of [0,1). We infer f(α_{t}(Y)) is constant on [0,1]. □

5.8 Corollary. If f : M_{h}^{n} → ℝ is ℝcm then f is unitary-invariant, i.e., if U ∈ M is unitary and Y = (Y_{j})_{1≤j≤n} ∈ M_{h}^{n} then f((UY_{j}U^{∗})_{1≤j≤n}) = f(Y).

5.9 Proposition. Let Ω ⊂ M_{h} be open in the norm-topology. A differentiable map f : Ω → ℝ is ℝcm iff df[Y](m) = τ(ξm) for some ξ = ξ^{∗} ∈ L^{1}(W^{∗}(Y), τ) for each Y ∈ Ω.

Proof. This follows from the fact that δℂ_{⟨1⟩}(Y)∩M_{h} = M_{h}∩ℂ_{⟨1⟩}(Y) is a dense subset of L^{1}(W^{∗}(Y)_{h}, τ), which is clearly a closed subset of L(M_{h}, ℝ) via the identification ξ ⇝ τ(ξ·).

□

5.10 Corollary. Let Minv_{h} be the elements with bounded inverse in M_{h}. The map

Minv_{h} ∋ Y → Y^{−1} ∈ M_{h}

is ℝcm.

Proof. By 5.5 it suffices to check that

Minv_{h} ∋ Y → τ(Y^{−k}) ∈ ℝ

is ℝcm for all k > 0. Since the differential is

M_{h} ∋ K → −kτ(Y^{−k−1}K)

the assertion follows from Proposition 5.9.

5.11 Proposition. Let F : Ω → N_{h}^{p} be a differential map, where Ω ⊂ M_{h}^{n} is open in the norm-topology. If for each Y there are ℝcm maps F_{k} : Ω → N_{h}^{p} such that for k → ∞,

∥F_{k}(Y) − F (Y)∥ → 0
∥dF_{k}[Y] − dF [Y]∥ → 0

then F is ℝcm.

Proof. We shall use Lemma 5.5. Let P ∈ ℂ_{⟨p⟩} and f_{P} : N^{p} → ℝ, f_{P} (Y) = ν(P (Y)). We must prove that f_{P} ∘ F is ℝcm. Remark that for k → ∞ we have

∥(f_{P} ∘ F)(Y) − (f_{P} ∘ F_{k})(Y)∥ → 0
∥d(f_{P} ∘ F)[Y] − d(f_{P} ∘ F_{k})[Y]∥ → 0

where for the second equality we used the fact that

∥df_{P} [F (Y)] − df_{P} [F (Y_{k})]∥ → 0 .

Since d(f_{P} ∘ F_{k})[Y] converges to d(f_{P} ∘ F)[Y] we infer f_{P} ∘ F is ℝcm. □

5.12 Corollary. Let Ω = {Y ∈ M_{h}^{n} | ∥Y_{j}∥ < R, 1 ≤ j ≤ n} and let G_{j} = G_{j}^{∗} ∈ ℂ_{⟨n⟩,R,0} be a noncommutative power series. Then the map F_{G} defined by Ω ∋ Y → (G_{j}(Y))_{1≤j≤p} ∈ M_{h}^{p} is ℝcm.

5.13 Theorem. Let A ⊂ B be a unital inclusion of von Neumann algebras and let Z = (Z_{1}, …, Z_{m}) ⊂ B_{h}^{m}. Let Φ_{Z} : A_{h}^{n} → B_{h}^{n+m} be the map defined by Φ_{Z}(Y_{1}, …, Y_{n}) = (Y_{1}, …, Y_{n}, Z_{1}, …, Z_{m}) ∈ B_{h}^{n+m}. If {Z_{1}, …, Z_{m}} and A are independent or freely independent in (B, β), then Φ_{Z} is ℝcm.

Proof. Using Lemma 5.5 it suffices to prove under the given assumptions that f_{P} ∘Φ_{Z} is ℝcm where P = P^{∗} ∈ ℂ_{⟨n+m⟩}. In both cases τ(P (Y_{1}, …, Y_{n}, Z_{1}, …, Z_{m}) is a polynomial in the moments of (Y_{1}, …, Y_{n}) and the moments of (Z_{1}, …, Z_{m}). The moments of (Z_{1}, …, Z_{m}) are constants, so f_{P} ∘ Φ_{Z} is a polynomial in the moments of (Y_{1}, …, Y_{m}), or being a real polynomial in the real and imaginary parts of such moments it is ℝcm by 5.4b). □

5.14 Proposition. Let Y_{j} ∈ M_{h}, Z_{j} ∈ M_{h}, 1 ≤ j ≤ n be such that {Y_{1}, …, Y_{n}} and {Z_{1}, …, Z_{n}} are independent or freely independent. Let W_{Y +Z} and W_{Z} be the von Neumann subalgebras of M generated by {I, Y_{1} + Z_{1}, … Y_{n} + Z_{n}} and respectively {I, Z_{1}, …, Z_{n}} and let E be the conditional expectation of M onto W_{Y +Z}. Then, if ξ ∈ ((W_{Z}, τ))_{h}^{n} is such that τ(∑_{1≤j≤n} ξ_{j}δ_{j}P (Z)) = 0 for all P ∈ ℂ_{⟨n⟩}, then τ(∑_{1≤j≤n} ξ_{j}δ_{j}P (Y + Z)) = 0 for all P ∈ ℂ_{⟨n⟩}. In particular (Eξ_{j})_{1≤j≤n} is in the orthogonal to δℂ_{⟨n⟩}(Y + Z) in (W_{Y +Z,h})^{n}.

Proof. The map S : M_{h}^{2n} → M_{h}^{n}, S((R_{j})_{1≤j≤2n} = (R_{j} + R_{n+j})_{1≤j≤n} is ℝcm and hence S ∘ Φ_{Z} = Ψ_{Z} is ℝcm, where Φ_{Z} is the map in Theorem 5.13. We have

Ψ_{Z} : M_{h}^{n} → M_{h}^{n} , Ψ_{Z}(Y_{1}, …, Y_{n}) = (Y_{j} + Z_{j})_{1≤j≤n} .

If P = P^{∗} ∈ ℂ_{⟨n⟩} then d(f_{P} ∘Ψ_{Z})[Y] is of the form ∑_{j} τ(T_{j}·) with (T_{j})_{1≤j≤n} in the L^{1} closure of δℂ_{⟨n⟩}(Y + Z) (the L^{1} closure is actually superfluous here). Since ξ ∈ Ker d(f_{P} ∘ Ψ_{Z})[Y] we infer dΨ_{Z}[Y](ξ) ∈ Ker df_{P} [Y +Z] for all P = P^{∗} ∈ ℂ_{⟨n⟩}. Since dΨ_{Z}[Y](ξ) = ξ this yields the assertion of the proposition. □

5.15 On M_{h}^{n} we define the ℝcm-equivalence relation, by Y ∼ Y′, if f(Y) = f(Y′) for all ℝcm maps f : M_{h}^{n} → ℝ. In particular taking f = f_{P} where P = P^{∗} ∈ ℂ_{⟨n⟩} we see that Y ∼ Y′ implies that (Y_{1}, …, Y_{n}) and (Y′_{1}, …, Y′_{n}) are equivalent in distribution. This means that Y ∼ Y′ implies there is an isomorphism ρ : W^{∗}(I, Y_{1}, …, Y_{n}) → W^{∗}(I, Y′_{1}, …, Y′_{n}) such that ρ(Y_{j}) = Y′_{j} and which is trace-preserving. On the other hand, 5.7 and 5.8 give some partial converses.

5.16 A k + 1-times differentiable map F : Ω → N_{h}^{p}, where Ω ⊂ M_{h}^{n} is open in the norm-topology, is k − ℝcm (k ∈ ℕ ∪ ∞) if for all i ≤ j ≤ k the j -th order differential d^{j}F : Ω × M_{h}^{j} → N_{h}^{p} is ℝcm.

5.17 Examples. The maps F_{P} defined in 5.6a) are ∞−ℝcm. Indeed d^{j}F_{P} : M_{h}^{j+1} → M_{h}^{p} are also maps of the type defined in 5.6a).

5.18 We shall now take a look at some weakenings of the requirement of cyclomorphy.

Definition. Let Ω ⊂ M_{h}^{n} be open. A map F : Ω → N_{h}^{p} is weakly ℝcm (abbreviated wℝcm) at a given point Y ∈ Ω if F is differentiable at Y and

(dF [Y])(δℂ_{⟨n⟩}^{⊥}(Y) ∩ M_{h}^{n}) ⊂ δℂ_{⟨n⟩}^{⊥}(F (Y)) ∩ N_{h}^{p} .

In particular a map f : Ω → ℝ is wℝcm if f is differentiable at Y and

δℂ_{⟨n⟩}^{⊥}(Y) ∩ M_{h}^{n} ⊂ Ker df .

A map f : Ω → ℝ is pseudo ℝcm (abbreviated ψℝcm) at Y ∈ Ω if f is continuous at Y and f(Y + T) = f(Y) + o(∥T ∥) for T ∈ (Y + δℂ_{⟨n⟩}^{⊥}(Y) ∩ M_{h}^{n}) ∩ Ω.

5.19 The difference between global wℝcm and ℝcm has to do with certain ultraweak continuity requirements for the differential. We record these facts as the next proposition, the easy proof of which is left to the reader.

Proposition. Let Ω ⊂ M_{h}^{n} be open and let F : Ω → N_{h}^{p} and f : Ω → ℝ be differentiable maps. Then f is ℝcm iff f is wℝcm at Y and df[Y] is ultraweakly continuous for each Y ∈ Ω. If F is ℝcm then F is wℝcm at each point y ∈ Ω. If F is wℝcm at Y and dF [Y] is ultraweakly continuous at each point Y ∈ Ω, then F is ℝcm.

5.20 We pass now to our brief discussion of complex cyclomorphy. We shall need to consider spaces of cyclic gradients evaluated at non-selfadjoint n-tuples Z = (Z_{j})_{1≤j≤n} ∈ M^{n}.

Definition. Let Θ ⊂ M^{n} be an open set. A complex-analytic map h : Θ → ℂ is complex cyclomorphic (abbreviated ℂcm) if for each Z ∈ Ω there is S ∈ (L^{1}(M, τ))^{n} such that for W ∈ M^{n} we have dh[Z](W_{1}, …, W_{n}) = ∑_{1≤j≤n} τ(S_{j}W_{j}) and S is in the L^{1}-norm-closure of δℂ_{⟨n⟩}(Z) ∩ M^{n}. A complex-analytic map H : Θ → N^{p} is ℂcm if for every h : θ → ℂ, where θ ⊂ N^{p} is open, we have that h ∘ H : Θ ∩ h^{−1}(θ) → ℂ is ℂcm.

5.21 Many of the results for ℝ-cyclomorphic maps have correspondents for ℂ-cyclomorphic maps and the proofs are usually analogous. We shall briefly state, without proofs, a few of these.

Proposition. A complex analytic map H : Θ → N^{p}, where Θ ⊂ M^{n} is open, is ℂcm iff for every P ∈ ℂ_{⟨p⟩} the map h_{p} ∘ F is ℂcm, where h_{P} : N^{p} → ℂ, h_{P} (T) = ν(P (T)).

5.22 Examples. a) If P ∈ ℂ_{⟨n⟩}, then h_{P} : M^{n} → ℂ defined by h_{P} (Z = τ(P (Z)) is ℂcm.

b) If h ∈ Ω → ℂ is ℂcm, H : Ω → N^{p} is ℂcm, where Ω is open, then also Hh : Ω → N^{p} is ℂcm. In particular the ℂcm maps from Ω to ℂ form an algebra over ℂ with unit.

c) If P_{j} ∈ ℂ_{⟨n⟩}, 1 ≤ j ≤ p, then the map H_{P} : M^{n} → M^{p}, given by H_{P} (Z) = (P_{j}(Z))_{1≤j≤p} is ℂcm.

d) Let Θ_{1} ⊂ A^{n}, Θ ⊂ B^{p} be open sets, and let H : Θ_{1} → B^{p}, K : Θ_{2} → C^{q} be ℂcm maps such that H(Θ_{1}) ⊂ Θ_{2}. Then K ∘ H is ℂcm.

5.23 Proposition. Let Θ ⊂ M be open. A complex analytic map h : Θ → ℂ is ℂcm iff dh[Z](T) = τ(ξT) for some ξ ∈ L^{1}-closure of ℂ_{⟨1⟩}(Z).

5.24 Proposition. Let M^{inv} be the elements with bounded inverse in M. The map M^{inv} ∋ Z → Z^{−1} ∈ M is ℂcm.

5.25 Proposition. Let H : Θ → N^{p} be a complex-analytic map, where Θ ⊂ M^{n} is open. If for each Z ∈ M^{n} there are ℂcm maps H_{k} : Θ → N^{p} such that for k → ∞

∥H_{k}(Z) − H(Z)∥ → 0
∥dH_{k}[Z] − dH[Z]∥ → 0

then H is ℂcm.

5.26 Proposition. Let Θ = {Z ∈ M^{n} | ∥Z_{j}∥ < R, 1 ≤ j ≤ n} and let G_{j} ∈ ℂ_{⟨n⟩,R,0} be noncommutative power series. Then the map H_{G} defined by

Θ ∋ Z → (G_{j})(Z))_{1≤j≤p} ∈ M^{p}

is ℂcm.

5.27 Theorem. Let A ⊂ B be a unital inclusion of von Neumann algebras and let W = (W_{1}, …, W_{m}) ∈ B^{m}. Let Ξ_{W} : A^{n} → B^{n+m} be the map defined by Ξ_{W} (Z_{1}, …, Z_{n}) = (Z_{1}, …, Z_{n}, W_{1}, …, W_{m}) ∈ B^{n+m}. If {W_{1}, …, W_{m}} and A are independent or freely independent in (B, β) then Ξ_{W} is ℂcm.

5.28 Definition. A complex-analytic map H : Θ → N^{p}, where Θ ⊂ M^{n} is open, is k−ℂcm (k ∈ ℕ ∪ ∞) if for all 1 ≤ j ≤ k, the j -th order complex differential d^{j}H : Ω × M^{j} → N^{p} is ℂcm.

5.29 Example. If P_{i} ∈ ℂ_{⟨n⟩}, 1 ≤ i ≤ p, then the map H_{P} : M^{n} → M^{p} given by H_{P} (Z) = (P_{i}(Z))_{1≤i≤p} is ∞ − ℂcm.

5.30 To define weak ℂcm maps we first must adapt some notation. For non-selfadjoint Z ∈ M^{n} we shall denote by δℂ_{⟨n⟩}^{⊥}(Z) the set

{T ∈ (L^{1}(M, τ))^{n} | ∑_{1≤j≤n} τ(T_{j}(δ_{j}P)(Z)) = 0 for all P ∈ ℂ_{⟨n⟩}} .

In case there may be some confusion with the notation in the selfadjoint case, we may emphasize the non-selfadjointness by writing (δℂ_{⟨n⟩}^{⊥})_{nh}(Z).

Definition. Let Θ ⊂ M^{n} be open. A map H : Θ → N^{p} is weakly ℂcm (abbreviated wℂcm) at a point Z ∈ Θ if H is ℂ-differentiable at Z and its ℂ-differential satisfies

(dH[Z])(δℂ_{⟨n⟩}^{⊥}(Z) ∩ M^{n}) ⊂ δℂ_{⟨p⟩}^{⊥}(H(Z)) ∩ N^{p} .

In particular a map h : Θ → ℂ is wℂcm at Z if h is ℂ-differentiable at Z and δℂ_{⟨n⟩}^{⊥}(Z) ∩ M^{n} ⊂ Ker df.

5.31 Proposition. Let Θ ⊂ M^{n} be open and let H : Θ → N^{p} and h : Θ → ℂ be maps. Then h is ℂcm iff h is wℂcm at Z and dh[Z] is ultraweakly continuous for each Z ∈ Θ.

If H is ℂcm then H is wℂcm at each point Z ∈ Θ. If H is wℂcm at Z and dH[Z] is ultraweakly continuous at each point Z ∈ Θ, then H is ℂcm.

5.32 ℝcm and ℂcm manifolds. If Ω_{1} and Ω_{2} are open in M_{h}^{n} (resp. in M^{n}) then a ℝcm-isomorphism (resp. ℂcm isomorphism) is a ℝcm (resp. ℂcm) map F : Ω_{1} → Ω_{2} which is a bijection and the inverse of which F^{−1} : Ω_{2} → Ω_{1} is also ℝcm (resp. ℂcm).

A ℝcm (resp. ℂcm) n-manifold over M_{h} (resp. M) is then a manifold modeled on M_{h}^{n} (resp. M^{n}) with an equivalence class of atlases (U_{1}, φ_{1}) such that the maps φ_{j}φ_{i}^{−1} : φ_{i}(U_{i}∩U_{j}) → φ_{j}(U_{i}∩U_{j}) are ℝcm-isomorphisms (resp. ℂcm isomorphisms) (See for instance [7] for the definitions in the differentiable case.)

5.33 Remarks. a) It is easy to produce examples of ℝcm or ℂcm manifolds by gluing open sets in M_{h}^{n}, or respectively M^{n} by ℝcm, or respectively ℂcm isomorphisms.

b) It seems that some natural candidates for ℝcm manifolds don’t satisfy the definitions. For instance the unitary group U(M) = {u ∈ M | u^{∗}u = uu^{∗} = I} is such a candidate, but there is a problem with the charts which one would expect to be connected with the Cayley transform u → i(u − e^{iθ}I)(u − e^{−iθ}I)^{−1} or to logarithms. Unfortunately the natural domain for the Cayley transform are the unitaries such that Ker(u − e^{−iθ}I) = {0}, which is not an open set while the range would consist of unbounded selfadjoint operators affiliated with M. Since these charts are the natural ones, the solution could be to find relaxations of the definitions so that also these charts be admissible.

5.34 Cyclicgrad submanifolds. To conclude the discussion around cyclomorphy here is another interesting class of geometric objects that arise in this context: the real (respectively, complex) cyclicgrad submanifolds in M_{h}^{n} (respectively, in M^{n}). A differentiable (respectively, complex-analytic) submanifold V ⊂ M_{h}^{n} (respectively, W ⊂ M^{n}) is real-cyclicgrad (respectively, complex-cyclicgrad) if for every Y ∈ V (respectively, Z ∈ W), T_{Y} V ⊂ L^{2}(δℂ_{⟨n⟩}[Y]) (respectively, for the complex tangent space T_{Z}W ⊂ L^{2}(δℂ_{⟨n⟩}[Z]). Note that these manifolds are perpendicular to automorphic orbits since the latter have tangent spaces contained in the orthogonal to cyclic gradients.

6 The Lie algebras

6.1 Vect ℂ_{⟨n⟩} is a Lie algebra under the bracket

[P, Q] = (D_{P} Q_{j} − D_{Q}P_{j})_{1≤j≤n}

where P = (P_{j})_{1≤j≤n}, Q = (Q_{j})_{1≤j≤n}. Possible confusion with the commutator on ℂ_{⟨n⟩} which is denoted the same way can be avoided by checking the context.

Since the map

Vect ℂ_{⟨n⟩} ∋ P → D_{P} ∈ Der ℂ_{⟨n⟩}

into the derivations of ℂ_{⟨n⟩} is injective, the equality D_{[P,Q]} = [D_{P} , D_{Q}] implies that [P, Q] is indeed a Lie algebra bracket. The last equality in turn is a straightforward computation

D_{P} D_{Q}X_{i_{1}} … X_{i_{p}} = D_{P} ∑_{1≤j≤p} X_{i_{1}} … X_{i_{j−1}}Q_{i_{j}}X_{i_{j+1}} … X_{i_{p}}
= ∑_{{1≤j,k≤p \atop j≠k}} X_{i_{1}} … X_{i_{j}}P_{ij}X_{i_{j+1}} … X_{i_{k−1}}Q_{i_{k}}X_{i_{k}} … X_{i_{p}}

+ ∑_{1≤j≤p} X_{i_{1}} … X_{i_{j−1}}(D_{P} Q_{i_{j}})X_{i_{j+1}} … X_{i_{p}} .

Combining with the analogues formula for D_{Q}D_{P} we get

[D_{P} D_{Q}]X_{i_{1}} … X_{i_{p}} = ∑_{1≤j≤p} X_{i_{1}} … X_{i_{j−1}}(D_{P} Q_{i_{j}} − D_{Q}P_{i_{j}})X_{i_{j+1}} … X_{i_{p}}
= D_{[P,Q]}X_{i_{1}} … X_{i_{p}} .

6.2 We have |[P, Q]|_{R,0} ≤ |P |_{R,0}|Q|_{R,1} + |P |_{R,1}|Q|_{R,0}. In particular if R′ > R the definition of the bracket extends to P, Q ∈ Vect ℂ_{⟨n⟩,R′,0} with [P, Q] ∈ Vect ℂ_{⟨n⟩,R,0}. This turns

Vect ℂ_{⟨n⟩,>R} = ⋃_{R′>R} Vect ℂ_{⟨n⟩,R,0}

into a Lie algebra.

6.3 Lemma. Let H, K ∈ Vect ℂ_{⟨n⟩} and assume

Ψ_{n}(H) ≤ C_{1}(1 − αX_{1})^{−e}
Ψ_{n}(K) ≤ C_{2}(1 − αX_{1})^{−f}

Then

Ψ_{n}([H, K]) ≤ C_{1}C_{2}(e + f)(1 − αX_{1})^{−e−f−1} .

Proof. This is a straightforward computation based on

Ψ_{n}([H, K]) ≤ Ψ_{n}((D_{|H|}|K_{1}|), …, (D_{|H|}|K_{n}|)) + Ψ_{n}((D_{|K|}|H_{1}|), …, (D_{|K|}|H_{n}|))
≤ D_{Ψ_{n}(H)}Ψ_{n}(K) + D_{Ψ_{n}(K)}Ψ_{n}(H)
≤ C_{1}C_{2}(D_{(1−αX_{1})^{−e}}(1 − αX_{1})^{−f} + D_{(1−αX_{1})^{−f}} (1 − αX_{1})^{−e})
□

6.4 Proposition. Let K^{(j)} ∈ Vect ℂ_{⟨n⟩}, 0 ≤ j ≤ m, and let M ≥ |K^{(j)}|_{R′,0}, 0 ≤ j ≤ m. If R′ > R > 0, then

|ad K^{(n)}ad K^{(n−1)}ad K^{(1)}K^{(0)}|_{R,0} ≤ M^{m+1}2^{m}m!(1 − R/R′)^{−2−1}

Proof. The assumption M ≥ |K^{(j)}|_{R′,0} gives Ψ_{n}(K^{(j)}) ≤ M(1 − X_{1}/R′)^{−1}. We then prove by induction that

Ψ_{n}(ad K^{(n)}ad K^{(n−1)} … ad K^{(1)}ad K^{(0)}) ≤ M^{m+1}2^{m}m!(1 − X_{1}/R′)^{−2m−1}

using Lemma 2.9. Setting X_{1} = R then yields the desired result. □

6.5 It is easily seen that the Lie bracket we considered extends to Vect ℂ⟨⟨X_{1}, …, X_{n}⟩⟩ (see for instance the discussion fo the grading in 6.9). It is immediate from Lemma 6.3 that the following proposition holds.

Proposition. The set {H ∈ Vect ℂ⟨⟨X_{1}, …, X_{n}⟩⟩ | ψ_{n}(H) ≤ C(1 − αX_{1})^{−p} for some C > 0 and p ∈ ℕ} is a Lie subalgebra.

6.6 Remark that the Lie algebra in Proposition 6.5 is contained in Vect ℂ_{⟨n⟩,R′,0}, where αR′ < 1, and contain Vect ℂ_{⟨n⟩,R′′,0}, when αR′′ > 1. Combining this with Proposition 6.5 gives the following result.

Corollary. Vect ℂ_{⟨n⟩,>R} is a Lie subalgebra of Vect C⟨⟨X_{1}, …, X_{n}⟩⟩.

6.7 Remark. Unless in some unexpected way the estimates in Proposition 6.4 can be vastly improved, it seems unlikely that convergence of the Campbell-Hausdorff series (see [2]) can be obtained in the operator algebra context.

6.8 We set aside the completions now to point out some simple features of Vect ℂ_{⟨n⟩} and we leave the reader to carry over the obvious considerations for the completions.

6.9 Vect ℂ_{⟨n⟩} has a grading ⨁_{k≥−1} V_{k} where V_{k} consists of n-tuples P = (P_{1}, …, P_{n}) which are homogeneous of degree k + 1. It is easily seen that [V_{k}, V_{ℓ}] ⊂ V_{k+ℓ}.

6.10 It is easily seen that V_{−1} is isomorphic to the commutative Lie algebra ℂ^{n}. Also V_{≥0} = ⨁_{k≥0} V_{k} is a Lie subalgebra and V_{≥p} = ⨁_{k≥p} V_{k} where p ∈ ℕ are ideals in V_{≥0}. V_{0} is also a Lie algebra. Let E_{ik} = (δ_{ij}X_{k})_{1≤j≤n} (here δ_{ij} is the Kronecker symbol). Then [E_{ab}, E_{cd}] = δ_{bc}E_{ad} − δ_{da}E_{cb}, i.e., V_{0} is isomorphic to 𝔤l(n, ℂ) with E_{ab} corresponding to the matrix with (i, j)-entry equal to δ_{ai}δ_{bj}. In particular the center of V_{0} is ℂ(X_{j})_{1≤j≤n}.

It is also easily seen that V_{≥0} is the semidirect product of V_{≥1} and V_{0} (V_{0} acting on V_{≥1}).

6.11 The involution on Vect ℂ_{⟨n⟩} being defined component-wise, we have D_{P^{∗}}Q_{j}^{∗} = (D_{P} Q_{j})^{∗} so that P ⇝ P^{∗} is a conjugate-linear automorphism of Vect ℂ_{⟨n⟩}. Then the selfadjoint part Vect ℂ_{⟨n⟩}^{sa} is a real Lie-algebra. Remark also that V_{0}^{sa} is isomorphic to 𝔤l(n, ℝ).

6.12 If τ : ℂ_{⟨n⟩} → ℂ is a trace, we define

Vect ℂ_{⟨n|τ⟩} = {P ∈ Vect ℂ_{⟨n⟩} | ∑_{1≤j≤n} τ(P_{j}(δ_{j}R)) = 0 for all R ∈ ℂ_{⟨n⟩}} .

Then Vect ℂ_{⟨n|τ⟩} is a Lie subalgebra of Vect ℂ_{⟨n⟩}. It can be viewed as a noncommutative analogue of the Lie algebras of volume-preserving vector fields. The defining relations can be also written τ(D_{P} R) = 0. If P, Q ∈ ℂ_{⟨n|τ⟩} then τ(D_{[P,Q]}R) = 0 since

τ(D_{[P,Q]}R) = τ(D_{P} D_{Q}R − D_{Q}D_{P} R) = 0 .

6.13 Let P ∈ ℂ_{⟨n⟩} and consider ([P, X_{j}])_{1≤j≤n} ∈ Vect ℂ_{⟨n⟩} (these brackets are commutators). Then D_{([P,X_{j}])_{1≤j≤n}}Q = [P, Q], i.e. the derivation defined by this element of Vect ℂ_{⟨n⟩} is just the commutator with P. We have

[([P, X_{1}], …, [P, X_{n}]), (P_{1}, …, P_{n})] = (D_{([P,X_{j}])_{1≤j≤n}}P_{k})_{1≤k≤n}
−([D_{(P_{k})_{1≤k≤n}}P, X_{j}])_{1≤j≤n} − ([P, P_{j}])_{1≤j≤n}
= −([D_{(P_{k})_{1≤k≤n}}P, X_{j}])_{1≤j≤n} .

So {([P, X_{j}])_{1≤j≤n} ∈ Vect ℂ_{⟨n⟩} | P ∈ ℂ_{⟨n⟩}} is an ideal in Vect ℂ_{⟨n⟩}. This ideal corresponds to the inner derivations. In particular since a trace applied to a commutator gives zero, we infer that

{([P, X_{j}])])_{1≤j≤n} ∈ Vect ℂ_{⟨n⟩} | P ∈ ℂ_{⟨n⟩}} ⊂ Vect ℂ_{⟨n|τ⟩} .

6.14 Clearly

Vect ℂ_{⟨n|τ⟩}^{sa} = Vect ℂ_{⟨n⟩}^{sa} ∩ Vect ℂ_{⟨n|τ⟩}

is a real Lie subalgebra of Vect ℂ_{⟨n⟩}. Note also that

{(i[P, X_{j}])_{1≤j≤n} ∈ Vect ℂ_{⟨n⟩} | P ∈ ℂ_{⟨n⟩}^{sa}} ⊂ Vect ℂ_{⟨n|τ⟩}^{sa} .

7 The Lie algebraVect ℂ_{⟨n|τ⟩} in the semicircular case

7.1 We return to the context of 1.4 and we will compute Vect ℂ_{⟨n|τ⟩} in the semicircular case. An important consequence will be that the assumptions for the exponentiation result, Theorem 3.8, are satisfied in the semicircular case.

T (ℂ^{n}) identifies on one hand with the L^{2}-space of W^{∗}(s_{1}, …, s_{n}), but also with the L^{2}-space of the non-selfadjoint algebra generated by ℓ_{1}, …, ℓ_{n}, w.r.t. the scalar product which corresponds to the vacuum on the C^{∗}-algebra generated by ℓ_{1}, …, ℓ_{n}. Thus we will be able to look at cyclic gradients w.r.t. s_{1}, …, s_{n} and ℓ_{1}, …, ℓ_{n} within the same Hilbert space. We shall denote by

δ^{ℓ} = (δ_{1}^{ℓ}, …, δ_{n}^{ℓ})ℂ⟨ℓ_{1}, …, ℓ_{n}⟩ → Vect ℂ⟨ℓ_{1}, …, ℓ_{n}⟩ and δ^{s} = (δ_{1}^{s}, …, δ_{n}^{s}) : ℂ⟨s_{1}, …, s_{n}⟩ → Vect ℂ⟨s_{1}, …, s_{n}⟩

the corresponding cyclic gradients. Similarly ∂^{ℓ} and ∂^{s} will denote the two free difference quotient gradients.

7.2 Proposition. Assume i_{j} ≠ i_{j+1} (1 ≤ j < p) and assume k_{1} > 0, …, k_{p} > 0. Then:

(∂_{j}^{s}P_{k_{1}}(s_{i_{1}}) … P_{k_{p}}(s_{i_{p}}))(1 ⊗ 1) = (∂_{j}^{ℓ}ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{p}}^{k_{p}})(1 ⊗ 1)

If moreover i_{1} ≠ i_{p} if p > 1, then

(δ_{j}^{s}P_{k_{1}}(s_{i_{1}}) … P_{k_{p}}(s_{i_{p}}))1 = (δ_{s}^{ℓ}ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{p}}^{k_{p}})1 .

Proof. By the recurrence relation for Gegenbauer polynomials which we did recall in 1.4 we have

(1 − rt + r^{2})^{−1} = ∑_{n≥0} P_{n}(t)r^{n} .

In particular computing the difference quotient for the left- and right-hand sides, we get

∑_{n≥0} \frac{P_{n}(t_{1}) − P_{n}(t_{2})}{t_{1} − t_{2}} r^{n} = r(1 − rt_{1} + r^{2})^{−1}(1 − rt_{2} + r^{2})^{−1}

= ∑_{n≥1} ( ∑_{0≤k≤n−1} P_{k}(t_{1})P_{n−1−k}(t_{2})) r^{n} .

It follows that

\frac{P_{n}(t_{1}) − P_{n}(t_{2})}{t_{1} − t_{2}} = ∑_{0≤k≤n−1} P_{k}(t_{1})P_{n−1−k}(t_{2})

i.e.,

∂P_{n} = ∑_{0≤k≤n−1} P_{k} ⊗ P_{n−1−k} .

This in turn gives

(∂_{j}^{s}P_{k_{1}}(s_{i_{1}}) … P_{k_{p}}(s_{i_{p}}))(1 ⊗ 1)

∑_{{m|i_{m}=j}} ∑_{0≤h<k_{m}}(P_{k_{1}}(s_{i_{1}}) … P_{h}(s_{i_{m}})1) ⊗ (P_{k_{m}−1−h}(s_{i_{m}})P_{k_{m+1}}(s_{i_{m+1}}) … P_{k_{p}}(s_{i_{p}})1)

= ∑_{{m|i_{m}=j}} ∑_{0≤h<k_{m}} (e_{i_{1}}^{⊗k_{1}} ⊗ … ⊗ e_{i_{m}}^{⊗h}) ⊗ (e_{i_{m}}^{⊗k_{m}−1−h} ⊗ … ⊗ e_{i_{p}}^{⊗k_{p}})

= ∑_{{m|i_{m}=j}} ∑_{0≤h<k_{m}} (ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{m}}^{h}1) ⊗ (ℓ_{i_{m}}^{k_{m}−1−h} … ℓ_{i_{p}}^{k_{p}}1)

= (∂_{j}^{ℓ} ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{p}}^{k_{p}})1 ⊗ 1 .

Similarly under the additional assumption that i_{p} ≠ i_{1} in case p > 1, we have

δ_{j}^{s}P_{k_{1}}(s_{i_{1}}) … P_{k_{p}}(s_{i_{p}})1

= ∑_{{m|i_{m}=j}} ∑_{0≤h<k_{m}} P_{k_{m}−1−h}(s_{i_{m}})P_{k_{m}+1}(s_{i_{m+1}}) … P_{k_{p}}(s_{i_{p}})P_{k_{1}}(s_{i_{1}}) … P_{k_{m−1}}(s_{i_{m−1}})P_{h}(s_{i_{m}})1

= ∑_{{m|i_{m}=j}} ∑_{0≤h<k_{m}} e_{i_{m}}^{⊗k_{m}−1−h} ⊗ e_{i_{m+1}}^{⊗k_{m+1}} ⊗ … ⊗ e_{i_{p}}^{⊗k_{p}} ⊗ e_{i_{1}}^{⊗k_{1}} ⊗ … ⊗ e_{i_{m−1}}^{⊗k_{m−1}} ⊗ e_{i_{m}}^{⊗h}

= ∑_{{m|i_{m}=j}} ℓ_{i_{m}}^{k_{m}−1−h}ℓ_{i_{m+1}}^{k_{m+1}} … ℓ_{i_{p}}^{k_{p}}ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{m−1}}^{k_{m−1}}ℓ_{i_{m}}^{h}1

= (δ_{j}^{ℓ} ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{p}}^{k_{p}})1 .

□

7.3 Lemma. Let F_{k} ⊂ ℂ⟨s_{1}, …, s_{n}⟩ be the linear span of P_{k_{1}}(s_{i_{1}}) … P_{k_{p}}(s_{i_{p}}) with k_{j} > 0 (1 ≤ j ≤ p), k_{1} + ⋯ + k_{p} = k, i_{j} ≠ i_{j+1} (1 ≤ j < p) and i_{1} ≠ i_{p} if p > 1, and let F = ∑_{k≥0} F_{k}. Then F+ Ker δ^{s} = ℂ⟨s_{1}, …, s_{n}⟩.

Proof. Recall from 1.3 that Ker δ^{s} = Ker C^{s} where C^{s} is the cyclic symmetrization. Moreover Ker δ^{s} = ∑_{k≥0} Ker δ^{s,k} is a direct sum decomposition where δ^{s,k} is the restriction of δ^{s} to the homogeneous noncommutative polynomials of degree k and similarly

Ker C^{s} = ∑_{k≥0} Ker C^{s,k}

so that Ker δ^{s,k} = Ker C^{s,k}.

Let G_{k} be the linear span of s_{i_{1}}^{k_{1}} … s_{i_{p}}^{k_{p}} with k_{j} > 0 (1 ≤ j ≤ p), k_{1} + ⋯ + k_{p} = k, i_{j} ≠ i_{j+1} (1 ≤ j < p) and i_{1} ≠ i_{p} if p ≥ 2, and put G = ∑_{k≥0} G_{k}. It is easily seen that

G_{k} + Ker C^{s,k} = (ℂ⟨s_{1}, …, s_{n}⟩)_{k} .

It will suffice to prove that

∑_{0≤k≤m}(F_{k} + Ker C^{s,k}) = ∑_{0≤k≤m}(G_{k} + Ker C^{s,k}) .

This is shown by induction on m, using

F_{m} + ∑_{0≤k<m}(ℂ⟨s_{1}, …, s_{n}⟩)_{k} = G_{m} + ∑_{0≤k<m}(ℂ⟨s_{1}, …, s_{n}⟩)_{k}

which follows from P_{k}(s) = cs^{k} (mod ∑_{0≤h<k}(ℂ⟨s⟩)_{h}) for some c ≠ 0. □

7.4 Theorem. (δ^{s}ℂ⟨s_{1}, …, s_{n}⟩)(1 ⊕ ⋯ ⊕ 1) = (δ^{ℓ}ℂ⟨ℓ_{1}, …, ℓ_{m}⟩)(1 ⊕ ⋯ ⊕ 1).

Proof. Let L_{k} be the linear span of ℓ_{i_{1}}^{k_{1}} … ℓ_{i_{p}}^{k_{p}}, k = k_{1} + ⋯ + k_{p}, i_{j} ≠ i_{j+1}, (1 ≤ j ≤ p), i_{1} ≠ i_{p} if p ≥ 2, and L = ∑_{k≥0} L_{k}. Then L+Ker C^{ℓ} = ℂ⟨ℓ_{1}, …, ℓ_{n}⟩ and hence L+Ker δ^{ℓ} = ℂ⟨ℓ_{1}, …, ℓ_{n}⟩.

It follows that

(δ^{ℓ}ℂ⟨ℓ_{1}, …, ℓ_{n}⟩)(1 ⊕ ⋯ ⊕ 1)
= (δ^{ℓ}L)(1 ⊕ ⋯ ⊕ 1)
= (δ^{s}F)(1 ⊕ ⋯ ⊕ 1)
= (δ^{s}ℂ⟨s_{1}, …, s_{n}⟩)(1 ⊕ ⋯ ⊕ 1)

where we used 7.2 and 7.3 in the last two steps. □

7.5 Theorem. Identifying T (ℂ^{n}) and L^{2}(W (s_{1}, …, s_{n}), τ) we have:

(T (ℂ^{n}))^{n} ⊖ \overline{δ^{s}ℂ⟨s_{1}, …, s_{n}⟩} = \{((ℓ_{j}^{∗} − r_{j}^{∗})ξ)_{1≤j≤n} | ξ ∈ T (ℂ^{n})\} .

In particular

Vect ℂ⟨s_{1}, …, s_{n} | τ⟩ = ∑_{k≥0} X_{k}

where X_{k} ⊂ (T_{k}(ℂ^{n}))^{n}, where X_{0} = 0 and X_{k} for k ≥ 1 is spanned by

(δ_{j,i_{0}}e_{i_{1}} ⊗ … ⊗ e_{i_{k}} − δ_{j,i_{k}}e_{i_{0}} ⊗ … ⊗ e_{i_{k−1}})_{1≤j≤n}

where (i_{0}, …, i_{k}) ∈ {1, …, n}^{k+1}.

Proof. δ^{s}ℂ⟨s_{1}, …, s_{n}⟩ identifies with

δ^{s}ℂ⟨s_{1}, …, s_{n}⟩)(1 ⊕ ⋯ ⊕ 1) = (δ^{ℓ}ℂ⟨ℓ_{1}, …, ℓ_{n}⟩)(1 ⊕ ⋯ ⊕ 1) .

To compute (T (ℂ^{n}))^{n} ⊖ \overline{δ^{ℓ}ℂ⟨ℓ_{1}, …, ℓ_{n}⟩(1 ⊕ ⋯ ⊕ 1)} we use the exact sequence 1.3. The maps in the exact sequence are homogeneous, so we get exact sequences

(ℂ_{⟨n⟩})_{k} ⟶^{δ} V_{k−2} ⟶^{θ} (ℂ_{⟨n⟩})_{k}

(recall that V_{k} are the noncommutative vector fields with components homogeneous of degree k + 1). Endowing (ℂ_{⟨n⟩})_{k} with the scalar product in which the monomials X_{i_{1}} … X_{i_{k}} form an orthonormal basis we have isometries

(ℂ_{⟨n⟩})_{k} ⟶ T_{k}(ℂ^{n})
V_{k−2} ⟶ (T_{k−1}(ℂ^{n}))^{n}

which map X_{i_{1}} … X_{i_{k}} to e_{i_{1}} ⊗ … ⊗ e_{i_{k}}. In this correspondence δ identifies with δ^{ℓ} acting in the L^{2}-space and θ with the map

θ^{ℓ}((ξ_{j})_{1≤j≤n}) = ∑_{1≤j≤n}(ℓ_{j} − r_{j})ξ_{j} .

Clearly, then

(T_{k}(ℂ^{n}))^{n} ⊖ δ^{ℓ}(ℂ⟨ℓ_{1}, …, ℓ_{n}⟩)_{k+1} = (θ^{ℓ})^{∗}ξ = ((ℓ_{j}^{∗} − r_{j}^{∗})ξ)_{1≤j≤n} .

Taking ξ = e_{i_{0}} ⊗ e_{i_{1}} ⊗ … ⊗ e_{i_{k}} we get the last part of the statement. □

7.6. Transcribing the last part of the preceding theorem in terms of the s_{j} ’s instead of the e_{j} ’s, we get the following result.

Corollary The elements

F_{I} = (δ_{i_{0},j}P_{k_{0}−1}(s_{i_{0}})P_{k_{1}}(s_{i_{1}}) … P_{k_{p}}(s_{i_{p}}) − δ_{i_{p},j}P_{k_{0}}(s_{i_{0}})P_{k_{1}}(s_{i_{1}}) … P_{k_{p}−1}(s_{i_{p}}))_{1≤j≤n}

where I = (\underbrace{i_{0}, …, i_{0}}_{k_{0}-times}, …, \underbrace{i_{p}, …, i_{p}}_{k_{p}-times}), k_{r} > 0 (0 ≤ r ≤ p), span Vect ℂ⟨s_{1}, …, s_{n} | τ⟩.

7.7. We will now clarify the relations among the elements F_{I}. On T_{k}(ℂ^{n}) let R_{k} be the cyclic permutation R_{k}e_{i_{1}} ⊗…⊗e_{i_{k}} = e_{i_{k}} ⊗e_{i_{1}} ⊗…⊗e_{i_{k−1}} (in particular R_{0}1 = 0, R_{1}e_{j} = e_{j} and R_{2}e_{i} ⊗ e_{j} = e_{j} ⊗ e_{i}). Let R = ⨁_{k≥1} R_{k} be the operator on T (ℂ^{n}). Let also P be the projection onto ℂ1.

Lemma Let T_{j} = ℓ_{j} − r_{j}. Then ∑_{1≤j≤n}T_{j}T_{j}^{∗} = 2I − 2P − (R + R^{∗}). In particular

Ker (θ^{ℓ})^{∗} = Ker ∑_{1≤j≤n} T_{j}T_{j}^{∗} = Ker(I − P − R) .

Proof.

∑_{1≤j≤n} T_{j}T_{j}^{∗} = ∑_{1≤j≤n} (ℓ_{j}ℓ_{j}^{∗} + r_{j}r_{j}^{∗} − ℓ_{j}r_{j}^{∗} − r_{j}ℓ_{j}^{∗})
= (I − P) + (I − P) − R − R^{∗} .

Since R |_{T_{+}} is a unitary operator, we have Ker((2I − R − R^{∗})|_{T_{+}} = Ker(I − R)|_{T_{+}} . □

7.8. Remark. Let N be the number operator, i.e. N1 = 0, Ne_{i_{1}} ⊗ … ⊗ e_{i_{k}} = ke_{i_{1}} ⊗ … ⊗ e_{i_{k}}. Then

Ker((I − R)|_{T_{+}} = Ker(N − C)|_{T_{+}}

7.9. Remark. To get a basis of Vect ℂ⟨s_{1}, …, s_{n}|τ⟩ we shall use a basis of (I − R)(T_{+}(ℂ^{n})). Consider on {1, …, n}^{k+1} the lexicographic order ≺ and let

Ω(k + 1)={(i_{0}, …, i_{k})∈{i, …, n}^{k+1} |(i_{0}, …, i_{k})≺(i_{k}, i_{0}, …, i_{k−1}) and (i_{0}, …, i_{k}) ≠(i_{k}, i_{0}, …, i_{k−1})} Then

Ψ = ⋃_{k≥1}{F_{i_{0}...i_{k}} − F_{i_{k}i_{0}...i_{k−1}} | (i_{0} … i_{k}) ∈ Ω(k + 1)}

is a basis of Vect ℂ⟨s_{1}, …, s_{n}|τ⟩ over ℂ. Indeed the set

Φ = ⋃_{k≥1}{e_{i_{0}} ⊗ … ⊗ e_{i_{k}} − e_{i_{k}} ⊗ e_{i_{0}} … e_{i_{k−1}} | (i_{0} … i_{k}) ∈ Ω(k + 1)}

is a homogeneous basis of (⨁_{k≥2} T_{k}(ℂ^{n}) ⊖ Ker(I − P − R) which in view of Lemma 7.7 is mapped by (θ^{ℓ})^{∗} to Vect ℂ⟨s_{1}, …, s_{n}|τ⟩. We have used here that (θ^{ℓ})^{∗}ξ = (T_{j}^{∗}ξ)_{1≤j≤n}, so that Ker(θ^{ℓ})^{∗} = Ker ∑_{1≤j≤n} T_{j}T_{j}^{∗}. Also F_{i_{0}...i_{k}} = (θ^{ℓ})^{∗}e_{i_{0}} ⊗ … ⊗ e_{i_{k}}.

7.10. Remark. The same kind of considerations as in 7.9 yield also another natural basis. If I =(i_{0}, …, i_{k}) ∈ {1, …, n}^{k+1}, let per(I) be the least non-zero cyclic period of I, i.e., the least m ∈ {1, …, k} such that i_{s} = i_{t} whenever s ≡ t (mod m). If ζ^{m} = 1, ζ ≠ 1 and m = per(I) let F (I, ζ) = ∑_{0≤j<m} ζ^{j}F_{(i_{j},i_{j+1},...,i_{k},i_{0},...,i_{j−1})}. Let also ω(k + 1) = {(i_{0} … i_{k}) ∈ {1, …, h}^{k+1} | (i_{0}, … i_{k}) ≺ (i_{j}, i_{j+1}, …, i_{k}, i_{0}, …, i_{j−1}), j = 1, …, k} and ρ(m) = {ζ ∈ ℂ | ζ ≠ 1, ζ^{m} = 1}. Then

⋃_{k≥1}{F (I, ζ) | I ∈ ω(k + 1), ζ ∈ ρ(per(I))}

is a basis of Vect ℂ⟨s_{1}, …, s_{n}|τ⟩.

7.11. To construct a basis over ℝ of the real Lie algebra Vect ℂ^{sa}⟨s_{1}, …, s_{n}|τ⟩ we can take the hermitian and antihermitian parts of the elements of the previous bases. In doing so one must pay attention to the fact that if per(I) = 2 then F_{i_{0}...i_{k}} −F_{i_{k}i_{0}...i_{k−1}} is antihermitian. For instance the basis in 7.9 then gives the basis:

⋃_{k≥1} {F_{i_{0}...i_{k}} + F_{i_{k}...i_{0}} − F_{i_{k}i_{0}...i_{k−1}} − F_{i_{k−1}...i_{0}i_{k−1}} | (i_{0}, …, i_{k}) ∈ Ω(k + 1), per(i_{0}, …, i_{k}) > 2}

∪ ⋃_{k≥1} \{\sqrt{−1}(F_{i_{0}...i_{k}} − F_{i_{k}...i_{0}} − F_{i_{k}i_{0}...i_{k−1}} − F_{i_{k−1}...i_{0}i_{k−1}} | (i_{0}, …, i_{k}) ∈ Ω(k + 1)\}

7.12. Putting together the results of this section and the results about exponentiation and automorphic orbits we see that we have proved the following result.

Theorem If s_{1}, …, s_{n} is a semicircular system and M = W^{∗}(s_{1}, …, s_{n}) then TAO_{∞}(s_{1}, …, s_{n})∩Vect ℂ^{sa}⟨s_{1}, …, s_{n}⟩ = Vect ℂ^{sa}⟨s_{1}, …, s_{n} | τ⟩ is dense in (δℂ_{⟨n⟩}^{⊥})(s_{1}, …, s_{n}).

In particular for K in this dense subset of (δℂ_{⟨n⟩}^{⊥})(s_{1}, …, s_{n}), D_{K}^{ε} exponentiates to a oneparameter automorphism group of W^{∗}(s_{1}, …, s_{n}). Moreover C^{∗}(s_{1}, …, s_{n}) is invariant under these exp(tD_{K}^{ε}).

7.13. Remark It is easily seen that V_{0} ∩ Vect ℂ^{sa}⟨s_{1}, …, s_{n} | τ⟩ = 0 and V_{1} ∩Vect ℂ^{sa}⟨s_{1}, …, s_{n} | τ⟩ is isomorphic to the Lie algebra of the orthogonal group 𝔬(n, ℝ).

8 B-morphic maps

8.1 Cyclomorphic maps have a natural generalization to the context of B -von Neumann algebras, i.e. the complex field is replaced by B. More precisely we deal with (M, τ) von Neumann algebras with specified normal faithful trace-state and there is a specified unital inclusion B ↪ M so that B is a von Neumann subalgebra. For lack of better names the analogues of cyclomorphic maps will be called B -morphic. If (Y_{j})_{1≤j≤n} ∈ M_{h}^{n} we shall denote by Δ_{h}(Y_{1}, …, Y_{m} : B_{h}) the norm-closed linear span in L(M_{h}^{n}, B_{h}) of the maps

M_{h}^{n} ∋ m = (m_{j})_{1≤j≤n} ⇝ T (m) + (T (m))^{∗}

and

M_{h}^{n} ∋ m = (m_{j})_{1≤j≤n} ⇝ \sqrt{−1} (T (m) − (T (m))^{∗}

where T (m) is of the form

T (m) = ∑_{1≤j≤p} E_{B}(b_{0}Y_{i_{1}}b_{1} … Y_{i_{j−1}}b_{j−1}m_{i_{j}}b_{j}Y_{i_{j+1}}b_{j+1} … Y_{i_{p}}b_{p}) .

Similarly when (Y_{j})_{1≤j≤n} ∈ M^{n} (no selfadjointness requirement) then Δ(Y_{1}, …, Y_{n} : B) denotes the norm-closed linear span in L(M^{n}, B) of the maps m ⇝ T (m). Note that T (·) is the differential at (Y_{1}, …, Y_{n}) of the map M^{n} ∋ (X_{j})_{1≤j≤n} ⇝ E_{B}(b_{0}X_{i_{1}}b_{1} … X_{i_{n}}b_{n}). Remark that Δ(Y_{1}, …, Y_{n} : B) is a B −B -bimodule, while Δ_{h}(Y_{1}, …, Y_{n} : B_{h}) is stable under L ⇝ bL + Lb^{∗}, L ⇝ (bL − Lb^{∗})\sqrt{−1}. This last feature is why we will prefer to work with Δ_{h}(Y_{1}, …, Y_{n} : B) in the selfadjoint case, which is the norm closure of the linear span of the T (·) in L(M^{n}, B).

By Δ_{h}^{⊥}(Y_{1}, …, Y_{n} : B) we denote the set of K = (K_{1}, …, K_{n}) ∈ M_{h}^{n} so that T (K) = 0 for all T ∈ Δ_{h}(Y_{1}, …, Y_{n} : B). Similarly, Δ^{⊥}(Y_{1}, …, Y_{n} : B) is the set of K ∈ M^{n} so that T (K) = 0 for all T ∈ Δ(Y_{1}, …, Y_{n} : B).

Our discussion of B -morphic maps will be rather sketchy in this paper, we just want to explain the main idea of the generalization.

8.2 Definition. Let Ω ⊂ M_{h}^{n} be an open set. A differentiable map f : Ω → B is hermitian B -morphic (abbreviated hBm) if for every Y ∈ Ω, df[Y] ∈ Δ_{h}(Y_{1}, …, Y_{n} : B). A differentiable map f : Ω → B is weakly hermitian B -morphic (abbreviated whBm) if (df[Y])(Δ_{h}^{⊥}(Y_{1}, …, Y_{n} : B)) = 0.

8.3 Definition. Let Ω ⊂ M_{h}^{n} be an open set. A differentiable map F : Ω → N_{h}^{p} where N is also a B -von Neumann algebra is hBm if for every hBm map f : ω → B, ω ⊂ N_{h}^{p} open, the map f ∘ (F |_{F^{−1}(ω)∩Ω}) is hBm. The differentiable map F : Ω → N_{h}^{p} is whBm if

(dF [Y])(Δ_{h}^{⊥}(Y : B)) ⊂ Δ_{h}^{⊥}(F (Y) : B) .

8.4 Definition. Let Ω ⊂ M^{n} be an open set. A complex analytic map f : Ω → B is B - morphic (abbreviated Bm) if for each Y ∈ Ω the complex differential is such that dF [Y] ∈ Δ(Y_{1}, …, Y_{n} : B). A complex analytic map f : Ω → B is weakly B -morphic (abbreviated wBm) if for each Y ∈ Ω the complex differential is such that dF [Y])(Δ_{h}^{⊥}(Y_{1}, …, Y_{n} : B)) = 0.

8.5 Definition. Let Ω ⊂ M^{n} be open and let F : Ω → N^{p} be a complex analytic map. F is Bm if for every Bm map f : ω → B, ω ⊂ N^{p} open, the map f ∘ F |_{F^{−1}(ω)∩Ω} is Bm. The map F is wBm if

dF [Y](Δ^{⊥}(Y : B)) ⊂ Δ^{⊥}(F (Y) : B) .

8.6 If X_{1}, …, X_{n} are indeterminates the noncommutative polynomials with coefficients in B, i.e. B ∗_{ℂ} ℂ⟨X_{1}, …, X_{n}⟩ will be denoted by B⟨X_{1}, …, X_{n}⟩ or B_{⟨n⟩}. B_{⟨n⟩} is spanned by noncommutative monomials b_{0}X_{i_{1}}b_{1} … X_{i_{p}}b_{p}. The involution is defined so that (b_{0}X_{i_{1}}b_{1} … X_{i_{p}}b_{p})^{∗} = b_{p}^{∗}X_{i_{p}} … b_{1}^{∗}X_{i_{1}}b_{0}^{∗}.

8.7 Examples. a) If P ∈ B_{⟨n⟩} then f_{P} : M_{h}^{n} → B defined by f_{P} (Y) = E_{B}(P (Y)) is hBm.

b) The hBm maps of Ω to B form an algebra.

c) If P ∈ B_{⟨n⟩} then g_{P} : M^{n} → B defined by g_{P} (Z) = E_{B}(P (Z)) is a Bm map.

d) The Bm maps Ω → B form an algebra.

e) If P =(P_{j})_{1≤j≤m} ∈ (B_{⟨n⟩})^{m}, then G_{P} : M^{n} → M^{m} defined by G_{P} (Z) = (P_{j}(Z))_{1≤j≤m} is a Bm map.

f) If P = (P_{j})_{1≤j≤m} ∈ (B_{⟨n⟩})^{n} and P_{j} = P_{j}^{∗}, 1 ≤ j ≤ m, then F_{P} : M_{h}^{n} → M_{h}^{m} defined by F_{P} (Y) = (P_{j}(Y))_{1≤j≤m} is a hBm map.

8.8 Theorem. Assume we have I ∈ B ⊂ N ⊂ M and let Y_{1}, …, Y_{m} ∈ M_{h} be such that {Y_{1}, …, Y_{m}} and N are freely independent over B in (M, E_{B}). Then the map Φ : N_{h}^{n} → M_{h}^{n+m} defined by Φ(T_{1}, …, T_{n}) = (T_{1}, …, T_{n}, Y_{1}, …, Y_{m}) is a hBm map.

8.9 Theorem. Assume we have I ∈ B ⊂ N ⊂ M and let Z_{1}, …, Z_{m} ∈ M be such that {Z_{1}, …, Z_{m}} and N are freely independent over B in (M, E_{B}). Then the map Ψ : N^{n} → M^{n+m} defined by Ψ(T_{1}, …, T_{n}) = (T_{1}, …, T_{n}, Z_{1}, …, Z_{m}) is a Bm map.

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