NONCOMMUTATIVE RIESZ-MARKOV-KAKUTANI THEOREM AND WEAK TWISTED BURNSIDE THEOREM

EVGENIJ TROITSKY

Abstract. The well-known Riesz(-Markov-Kakutani) theorem identifies the space of linear functionals on algebra A = C(X) and the space of regular measures on X. The first result of the present paper is a generalization of this theorem to the case of non-commutative C^{∗}-algebra A via Dauns-Hofmann sectional representation theorem. The corresponding measures on Glimm spectrum are functional-valued.

In extreme situation this theorem is tautological, but for group C^{∗}-algebras of discrete groups in many cases one obtains some new tool for counting twisted conjugacy classes. More precisely, let G be a countable discrete group and ϕ its automorphism. Let g[a] = L_{g}aL_{ϕ(g^{−1})} be the twisted action, R(ϕ) the number of twisted (or ϕ-) conjugacy classes, R_{∗}(ϕ) the dimension of twisted invariant elements of Fourier-Stieltijes algebra, S(ϕ) the number of fixed points of the dual action on Jacobson spectrum of G, S_{∗}(ϕ) the number of generalized fixed points on Glimm spectrum.

The second result of the paper (based on the first one) is the following weak form of twisted Burnside theorem: R_{∗}(ϕ) = S_{∗}(ϕ). This result allows to obtain the strong form of twisted Burnside theorem R(ϕ) = S(ϕ) in a number of cases. These reductions will be the subject of a forthcoming paper with A. Fel’shtyn.

Contents

1. Algebras of operator fields 1 2. Functionals and measures 2 3. Twisted-invariant AOFM 6 References 8

1. Algebras of operator fields

Let us first remind the theory of operator fields following [2].

Definition 1.1. A continuity structure for T and the {A_{t}} is a linear space F of vector fields on T, with values in the {A_{t}}, satisfying

(1) If x ∈ F, the real-valued function t ↦ ∥x(t)∥ is continuous on T;

(2) for each t ∈ T, {x(t) | x ∈ F } is dense in A_{t};

(3) F is closed under pointwise multiplication and involution.

This a part of joint research program with A. Fel’shtyn at Max-Planck-Institut für Mathematik (Bonn) in Spring 2004 in partial relation to the activity on Algebraic and Topological Dynamics. The author is grateful to MPIM for support and hospitality. The author is partially supported by RFFI Grant 02-01-00572, Grant for the support of leading scientific schools HIII-619.203.1 and Grant “Universities of Russia.”

Definition 1.2. A vector field x is continuous with respect to F at t_{0}, if for each ε > 0, there is an element y ∈ F and a neighborhood U of t_{0} such that ∥x(t) − y(t)∥ < ε for all t ∈ U. The field x is continuous on S if it is continuous at all points of S

Definition 1.3. A full algebra of operator fields is a family A of operator fields on T satisfying:

(1) A is a *-algebra, i.e., it is closed under all the pointwise algebraic operations;

(2) for each x ∈ A, the function t ↦ ∥x(t)∥ is continuous on T and vanishes at infinity;

(3) for each t, {x(t) | x ∈ A} is dense in A_{t};

(4) A is complete in the norm ∥x∥ = sup_{t} ∥x(t)∥.

A full algebra of operator fields is evidently a continuity structure. If F is any continuity structure, let us define C_{0}(F) to be the family of all vector fields x which are continuous on T with respect to F, and for which t ↦ ∥x∥ vanishes at infinity. In view of the preceding lemmas, C_{0}(F) is a full algebra of operator fields — indeed, a maximal one.

Lemma 1.4. For any full algebra A of operator fields on T, the following three conditions are equivalent:

(1) A is a maximal full algebra of operator fields;

(2) A = C_{0}(F) for some continuity structure F;

(3) A = C_{0}(A).

Such a maximal full algebra A of operator fields may sometimes be called a continuous direct sum of the {A_{t}}. It is clearly separating, in the sense that , if s, t ∈ T, s ≠ t, α ∈ A_{s}, β ∈ A_{t}, there is an x ∈ A such that x(s) = α, x(t) = β. We will denote by â the section corresponding to an element a ∈ A.

2. Functionals and measures

Definition 2.1. A (bounded additive) algebra of operator field measure (BA AOFM) related to a maximal full algebra of operator fields A = Γ(A) is a set function μ : S → Γ(A)^{∗} = A^{∗}, where S ∈ Σ, some algebra of sets,

• being additive: μ(⊔S_{i})(a) = ∑_{i} μ(S_{i})(a)

• μ(S)(a) = 0 if supp â ∩ S = ∅.

• bounded: sup over partitions {S_{i}} of T of ∑_{i} ∥μ(S_{i})∥ is finite and denoted by ∥μ∥

Definition 2.2. It is *-weak regular (RBA AOFM) if for each E ∈ Σ, a ∈ A, and ε > 0 there is a set F ∈ Σ whose closure is contained in E and a set G ∈ Σ whose interior contains E such that ∥μ(C)a∥ < ε for every C ∈ Σ with C ⊂ G \ F.

We will use as Σ all sets and the algebra generated by closed sets in T.

Definition 2.3. Let AOFM λ be defined on a field Σ of sets in T for which λ(∅) = 0. A set E ∈ Σ is called λ-set if for any M ∈ Σ

λ(M) = λ(M ∩ E) + λ(M ∩ (T \ E)).

Lemma 2.4. Let λ be an AOFM defined on a field Σ of sets in T with λ(∅) = 0. The family of λ-sets is a subfield of Σ on which λ is additive. Furthemore, if E is the union of a finite sequence {E_{n}} of disjoint λ-sets and M ∈ Σ, then λ(M∩E) = ∑_{n} λ(M∩E_{n}).

Proof. It is clear, that the void set, the whole space, and the complement of any λ-set are λ-sets. Now let X and Y be λ-sets, and M ∈ Σ. Then, since X is a λ-set,

(1)  λ(M ∩ Y) = λ(M ∩ Y ∩ X) + λ(M ∩ Y ∩ (T \ X)),

and since Y is a λ-set,

(2)  λ(M) = λ(M ∩ Y) + λ(M ∩ (T \ Y)),

λ(M ∩ (T \ (X ∩ Y))) = λ(M ∩ (T \ (X ∩ Y)) ∩ Y) + λ(M ∩ (T \ (X ∩ Y)) ∩ (T \ Y)), hence,

(3)  λ(M ∩ (T \ (X ∩ Y))) = λ(M ∩ (T \ X) ∩ Y) + λ(M ∩ (T \ Y)).

From (1) and (2) it follows that

λ(M) = λ(M ∩ Y ∩ X) + λ(M ∩ Y ∩ (T \ X)) + λ(M ∩ (T \ Y)),

and from (3) that

λ(M) = λ(M ∩ Y ∩ X) + λ(M ∩ (T \ (X ∩ Y))).

Thus X ∩ Y is a λ-set. Since ∪X_{n} = T \ ∩(T \ X_{n}), we conclude that the λ-sets form a field. Now if E_{1}, and E_{2} are disjoint λ-sets, it follows, by replacing M by M ∩(E_{1} ∪E_{2}) in Definition 2.3, that

λ(M ∩ (E_{1} ∪ E_{2})) = λ(M ∩ E_{1}) + λ(M ∩ E_{2}).

The final conclusion of the lemma follows from this by induction. □

It is clear that AOFM can be decomposed in a canonical way into a linear combination of positive AOFM, i.e., such that

μ(E)[a^{∗}a] ≥ 0

for any E ∈ Σ. Such a set function is non-decreasing.

Lemma 2.5. The sets F and G in the Definition 2.2 can be chosen in such a way that ∥μ(C)(fa)∥ < ε for any continuous function f : T → [0, 1].

Proof. Let us take the decompositions μ = ∑_{i=1}^{4} x_{i}μ_{i}, a = ∑_{j=1}^{4} y_{j}a_{j}, where μ_{i} and a_{j} are positive, x_{i}, y_{j} are complex numbers of norm ≤ 1. Let us choose the sets F and G, as in Definition 2.2, for ε/16 and for all pairs μ_{i}, a_{j} simultaneously. Then

0 ≤ μ_{i}(C)(f · a_{j}) = μ_{i}(C)((a_{j})^{1/2}f(a_{j})^{1/2}) ≤ μ_{i}(C)(a_{j}) ≤ \frac{ε}{16},

and

∥μ_{i}(C)(f · a)∥ ≤ ∑_{i,j=1}^{4} |x_{i}y_{j}| · |μ_{i}(C)(a_{j})| ≤ 16 · \frac{ε}{16} = ε.

□

Theorem 2.6. The functionals on Γ(A) can be identified with RBA AOFM.

Proof. Obviously RBA AOFM form a linear normed space with respect to ∥.∥.

First, we want to prove that the natural linear map μ ↦ μ(T) is an isometry of RBA AOFM into A^{∗}. Since ∥μ(T)∥ ≤ ∥μ∥, it is of norm ≤ 1. Let now take an arbitrary small ε > 0. Let E_{1}, …, E_{n} be a partition of T such that

∑_{i=1}^{n} ∥μ(E_{i})∥ ≥ ∥μ∥ − ε.

Let a_{i} ∈ A be such elements of norm 1, that μ(E_{i})(a_{i}) ≥ ∥μ(E_{i})∥ − ε/n.

By *-weak regularity of μ and normality of T one can take closed sets C_{i}, disjoint open sets G_{i}, and continuous functions f_{i} : T → [0, 1] such that C_{i} ⊂ E_{i}, ∥μ(E_{i} \ C_{i})(a_{j})∥ ≤ ε/n^{2}, C_{i} ⊂ G_{i}, ∥μ(G_{i} \ C_{i})(a_{j})∥ ≤ ε/n^{2}, (and estimations hold for multiplication by positive functions as well, as in Lemma 2.5) f_{i}(s) = 0 if s ∉ G_{i}, f_{i}(s) = 1 if s ∈ C_{i}, i, j = 1, …, n.

Consider the element a := ∑_{i} f_{i}a_{i} ∈ Γ(A) = A. Then ∥a∥ ≤ 1 and

| μ(S)(a) − ∥μ∥ | ≤ ∑_{i=1}^{n} |μ(E_{i})(a) − μ(E_{i})| + ε

≤ ∑_{i=1}^{n} |μ(E_{i} \ C_{i})(a) + μ(C_{i})(a) − μ(E_{i})(a_{i})| + 2ε

= ∑_{i=1}^{n} |∑_{j=1}^{n} μ(E_{i} \ C_{i})(f_{j}a_{j}) + μ(C_{i})(a_{i}) − μ(E_{i})(a_{i})| + 2ε

≤ ∑_{i,j=1}^{n} |μ(E_{i} \ C_{i})(f_{j}a_{j})| + ∑_{i=1}^{n} |μ(E_{i} \ C_{i})(a_{i})| + 2ε ≤ n^{2} \frac{ε}{n^{2}} + n \frac{ε}{n^{2}} + 2ε ≤ 4ε.

Since ε is arbitrary small, ∥μ∥ = ∥μ(S)∥.

It remains to represent the general functional φ by RBA AOFM. This functional on Γ(A) can be extended by Hahn-Banach theorem to a continuous functional ψ on B(A) = ∏_{t∈T} A_{t} (the algebra of possibly discontinuous cross-sections of A with supnorm). This functional can be decomposed ψ = ∑_{i=1}^{4} α_{i}ψ_{i}, where ψ_{i} are positive functionals, α_{i} ∈ ℂ, |α_{i}| = 1, ∥ψ_{i}∥ ≤ ∥ψ∥. Let us define

λ(E)(a) := ψ(χ_{E}a), λ_{i}(a) := ψ_{i}(χ_{E}a), i = 1, …, 4.

where a ∈ Γ(A), and χ_{E} is the characteristic function of E. Evidently, λ(T)(a) = ψ(a) and λ is a BA AOFM. Indeed, the first two properties of Definition 2.1 are evident. The third one can be verified for each λ_{i}, i = 1, …, 4:

∑_{j=1}^{N} |λ_{i}(E_{j})| = ∑_{j=1}^{N} λ_{i}(E_{j})(1) = ∑_{j=1}^{N} ψ_{i}(χ_{E_{j}}1) = ψ_{i}(1) ≤ ∥ψ_{i}∥,

hence,

∑_{j=1}^{N} |λ(E_{j})| ≤ ∑_{i=1}^{4} ∑_{j=1}^{N} |λ_{i}(E_{j})| ≤ ∑_{i=1}^{4} ∥ψ_{i}∥ ≤ 4 · ∥ψ∥.

Now we want to find an RBA AOFM μ such that μ(T)(a) = λ(T)(a). Without loss of generality it is sufficient to do this for a positive λ = λ_{i}.

Let F represent the general closed subset, G the general open subset, E the general subset of S. Define μ_{1} and μ_{2} by putting

μ_{1}(F)(a^{∗}a) = inf_{G⊃F} λ(G)(a^{∗}a), μ_{2}(E)(a^{∗}a) = sup_{F ⊂E} μ_{1}(F)(a^{∗}a),

and then by taking linear extension. More precisely, due to separability one can take a confinal sequence {G_{i}}. The unit ball in the dual space is weakly compact and one can take a weakly convergent sequence λ(G_{i_{k}}). Its limit ψ is a positive functional on A enjoing inf-property on positive elements. In particular, it is independent of the choice of {G_{i}} and {G_{i_{k}}}. In a similar way for sup.

These set functions are non-negative and non-decreasing. Let G_{1} be open and F_{1} be closed. Then if G ⊃ F_{1} \ G_{1}, it follows that G_{1} ∩ G ⊃ F_{1} and λ(G_{1}) ≤ λ(G_{1}) + λ(G) so that μ_{1}(F_{1}) ≤ λ(G_{1}) + λ(G). Since G is an arbitrary open set containing F_{1} \ G_{1} we have

μ_{1}(F_{1}) ≤ λ(G_{1}) + μ_{1}(F_{1} \ G_{1}).

If F is a closed set it follows from this inequality, by allowing G_{1} to range over all open sets containing F ∩ F_{1}, that

μ_{1}(F_{1}) ≤ μ_{1}(F ∩ F_{1}) + μ_{2}(F_{1} \ F).

If E is an arbitrary subset of T and F_{1} ranges over the closed subsets of E it follows from the preceding inequality that

(4)  μ_{2}(E) ≤ μ_{2}(E ∩ F) + μ_{2}(E \ F).

It will next be shown that for an arbitrary set E in T and arbitrary closed set F in T we have

(5)  μ_{2}(E) ≥ μ_{2}(E ∩ F) + μ_{2}(E \ F).

To see this let F_{1} and F_{2} be disjoint closed sets. Since T is normal there are disjoint neighborhoods G_{1} and G_{2} of F_{1} and F_{2} respectively. If G is an arbitrary neighborhood of F_{1} ∪ F_{2} then λ(G) ≥ λ(G ∩ G_{1}) + λ(G ∩ G_{2}) so that

μ_{1}(F_{1} ∩ F_{2}) ≥ μ_{1}(F_{1}) + μ_{2}(F_{2}).

We now let E and F be arbitrary sets in T with F closed and let F_{1} range over closed subsets of E ∩ F while F_{2} ranges over the closed subsets of E \ F. The preceding inequality then proves (5). From (4) and (5) we have

(6)  μ_{2}(E) = μ_{2}(E ∩ F) + μ_{2}(E ∩ (T \ F))

for any E in T and F closed. The function μ_{2} is defined on the field of all subsets of T and it follows from (6) that every closed set F is a μ_{2}-set. If μ is the restriction of μ_{2} on the field determined by the closed sets it follows from Lemma 2.4 that μ is additive on this field. It is clear from the definition of μ_{1} and μ_{2} that μ_{1}(F) = μ_{2}(F) = μ(F) if F is closed and thus μ(E) = sup_{F ⊂E} μ(F). This shows that μ is *-weak regular and since ∥μ(T)∥ < ∞, we have μ is RBA AOFM.

Finally, by the definition, μ(S)(a) = λ(S)(a) = ψ(a) = φ(a) for a ∈ Γ(A). □

3. Twisted-invariant AOFM

The most part of the argument is valid for various representations of algebras by operator fields, but now we will pass to the case of group C^{∗}-algebra of a discrete group and concentrate ourselves on the following important case due to Dauns and Hofmann [1, Corollary 8.13]:

Theorem 3.1. Consider a C^{∗}-algebra A (with or without an identity) and the set B of all its primitive ideals in the hull-kernel topology Q.

(1) The complete regularization φ : (B, Q) → (M, T) can be taken to consist of closed ideals m of A satisfying m = ∩φ^{−1}(m). The topology T of M contains the weak-star topology T^{∗} induced on M by A.

Denote by K the family of all T -compact subsets of M of the form {m ∈ M |∥a+m∥ ≥ ε}, a ∈ A, 0 < ε. Let (π′′ : E′ → (M, T) be the uniform field of C^{∗}-algebras obtained by first forming the canonical field from A, M and then enlarging the weak-star topology T^{∗} of M up to T. For each a ∈ A, â is the map â : M → E′, â(m) = a + m ∈ A/m. The C^{∗}-algebra of all sections vanishing at infinity with respect to the class K is denoted by Γ_{0}(π′′).

(2) Then the map A → Γ_{0}(π′′), a ↦ â is an isometric star-isomorphism. In particular, if M is T -compact, then M ∈ K and A ≅ Γ(π′′).

Definition 3.2. Ideals m from the previous theorem, i.e., points of M, are called Glimm ideals.

We will use the notation T instead of M for Glimm spectrum to agree with the first section.

Now we consider a countable discrete group G and its automorphism ϕ. Also, A = C^{∗}(G). One has the twisted action of G on A:

g[a] = L_{g}aL_{ϕ(g^{−1})},

where L_{g} is the left shift, and respectively on functionals.

Definition 3.3. The dimension R_{∗}(ϕ) of the space of twisted invariant functionals on C^{∗}(G) is called generalized Reidemeister number of ϕ.

Definition 3.4. A (Glimm) ideal I is a generalized fixed point of ϕ̂ if the linear span of elements b − g[b] is not dense in A_{I} = A/I, where g[.] is the twisted action, i.e. its closure K_{I} does not coincide with A_{I}.

If we have a finite number of such fixed points, then the twisted invariant RBA AOFM’s are concentrated in these points. Indeed, let us describe the action of G on RBA AOFM’s in more detail. The action of G on measures is defined by the identification of the measures and functionals on A.

Lemma 3.5. For any Borel E ⊂ T the functional μ(E) is twisted invariant.

Proof. This is an immediate consequence of *-weak regularity. Let a ∈ A, g ∈ G and ε > 0 be an arbitrary small number, and U and F as in Lemma 2.5, for a and g[a] simultaneously. Let us choose a continuous function f : T → [0, 1], with f|_{F} = 1 and f|_{T \U} = 0. Then

|μ(E)(a − g[a])| = |μ(E \ F)(a) + μ(F)(a) − μ(E \ F)(g[a]) − μ(F)(g[a])| ≤
≤ |μ(F)(a) − μ(F)(g[a])| + 2ε = |μ(F)(fa) − μ(F)(g[fa])| + 2ε =
= |μ(U)(fa) − μ(U \ F)(fa) − μ(U)(g[fa]) + μ(U \ F)(g[fa])| + 2ε ≤
≤ |μ(U)(fa) − μ(U)(g[fa])| + 4ε = |μ(T)(fa) − μ(T)(g[fa])| + 4ε = 4ε.
□

Lemma 3.6. For any twisted-invariant functional φ on C^{∗}(G) the corresponding measure μ is concentrated in the set GF P of generalized fixed points.

Proof. Let ∥μ∥ = 1. Suppose opposite: there exists an element a ∈ A, ∥a∥ = 1, vanishing on generalized fixed points, such that φ(a) ≠ 0. Let ε := |φ(a)| > 0. In each point t ∉ GF P we can find elements b_{t}^{i} ∈ A, g_{t}^{i} ∈ G, i = 1, … k(t), such that

∥a(t) − ∑_{i=1}^{k(t)}(g_{t}^{i}[b_{t}^{i}](t) − b_{t}^{i}(t))∥ < ε/4.

Then there exists a neighborhood U_{t} such that for s ∈ U_{t} one has

∥a(s) − ∑_{i=1}^{k(t)}(g_{t}^{i}[b_{t}^{i}](s) − b_{t}^{i}(s))∥ < ε/2.

Let us choose a finite subcovering {U_{t_{j}}}, j = 1, …, n, of {U_{t}} and a Borel partition E_{1}, …, E_{n} subordinated to this subcovering. Then

φ(a) = ∑_{j=1}^{n} μ(E_{j})(a) = ∑_{j=1}^{n} μ(E_{j}) (a − ∑_{i=1}^{k(t_{j})}(g_{t_{j}}^{i}[b_{t_{j}}^{i}] − b_{t_{j}}^{i})) +

+ ∑_{j=1}^{n} ∑_{i=1}^{k(t_{j})} μ(E_{j})(g_{t_{j}}^{i}[b_{t_{j}}^{i}] − b_{t_{j}}^{i}).

By Lemma 3.5 each summand in the second term iz zero. The absolute value of the first term is less then ∑_{j} ∥μ(E_{j})∥ε/2 ≤ ∥μ∥ · ε/2 = ε/2. A contradiction with |φ(a)| = ε. □

Since a functional φ on A_{I} is twisted-invariant if and only if Ker φ ⊃ K_{I}, the dimension of the space of these functionals equals the dimension of the space of functionals on A_{I}/K_{I} and is finite dimensional if and only if the space A_{I}/K_{I} is finite dimensional. In this case the dimension of the space of twisted-invariant functionals on A_{I} equals dim(A_{I}/K_{I}).

Definition 3.7. Generalized number S_{∗}(ϕ) of fixed points of ϕ̂ on Glimm spectrum is

S_{∗}(ϕ) := ∑_{I∈GF P} dim(A_{I}/K_{I}).

The argument above gives the following statement.

Theorem 3.8 (Weak generalized Burnside).

R_{∗}(ϕ) = S_{∗}(ϕ)

if one of these numbers is finite.

In [4] (see also [3, 5]) it was proved for groups of type I that the Reidemeister number R(ϕ), i.e., the number of twisted (or ϕ-)conjugacy classes, coincides with the number of fixed points of the corresponding action ϕ̂ on the dual space Ĝ. This was a generalization of Burnside theorem. The proof used identification of R(ϕ) and the dimension of the space of (L^{∞}-) twisted invariant functions on G, i.e. twisted invariant functionals on L^{1}(G). Since only part of L^{∞}-functions defines functionals on C^{∗}(G) (namely, Fourier-Stieltijes functions), so, a priori one has R_{∗}(ϕ) ≤ R(ϕ). Nevertheless, the functions with some symmetry conditions very often are in Fourier-Stieltijes algebra, so one can conjecture that R(ϕ) = R_{∗}(ϕ) if R(ϕ) < ∞. This is the case for all known examples.

References

[1] J. Dauns and K. H. Hofmann, Representations of rings by continuous sections, Mem. Amer. Math. Soc., vol. 83, Amer. Math. Soc., Providence, RI, 1968.

[2] J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280. MR 29 #1547

[3] Alexander Fel’shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion, Mem. Amer. Math. Soc. 147 (2000), no. 699, xii+146. MR 2001a:37031

[4] Alexander Fel’shtyn and Evgenij Troitsky, A twisted Burnside theorem for countable groups and Reidemeister numbers, Noncommutative Geometry and Number Theory (Bonn, 2003), Vieweg, Braunschweig, 2004 (MPIM (Bonn) Preprint, MPIM2004-65), pp. ??–??

[5] Alexander Fel’shtyn, Evgenij Troitsky, and Anatoly Vershik, Twisted Burnside theorem for type II_{1} groups: an example, Preprint MPIM2004-85, 2004.

Dept. of Mech. and Math., Moscow State University, 119992 GSP-2 Moscow, Russia

E-mail address: troitsky@mech.math.msu.su
URL: http://mech.math.msu.su/~troitsky