Generalized Lefschetz numbers of unitary endomorphisms of W*-elliptic complexes
by
Alexandre Pavlov
Moscow State University

Suppose A is a von Neumann algebra, M_r(A) is the set of r×r matrices with entries in A, M_\infty(A) is an inductive limit of the sequence {M_r(A)}_r=1^\infty, and M_\infty(A)_n is the set of normal elements for M_\infty(A). Let p, q be projections in M_\infty(A); then by p =~ q we denote the stable equivalence relation. Suppose a is a normal element and E is a Borel subset of the spectrum sp(a); then by P_a(E) we denote the spectral projection of a corresponding to the set E. A Borel set E subset C is called admissible if 0 not in [`E].

Definition. Call elements a, b in M_\infty(A) equivalent, denoted a =~ b, if and only if P_a(E) =~ P_b(E) for any admissible Borel subset E subset C.

Note that this equivalence relation coinsides with the usual stable equivalence relation whenever a, b are projections. Let N(A)=M_\infty(A)_n / =~ . By [a] denote the equivalence class of a in M_\infty(A)_n. We have P_a\oplusb(E)=P_a(E)\oplusP_b(E) for any a, b in M_\infty(A)_n and Borel E subset C. Therefore the set N(A) is an abelian semigroup with respect to the direct sum operation.

Definition. The symmetrization of N(A) is called the N-group of A and is denoted by N₀(A).

Proposition 0. K₀(A) is a subgroup of the group N₀(A).

Our next task is to establish a functorial property for N₀. Let A, B be von Neumann algebras and j: A --> B be an ultra-strong continuous unital *-homomorphism. By definition, put j_* ([a])=[j(a)] for a in M_\infty(A)_n.

Proposition 0. The map j_*:N₀(A) --> N₀(B) is a well defined homomorphism.

Suppose M_\infty(A)_fin subset M_\infty(A)_n is the subset of elements a in M_\infty (A) such that sp(a) is finite; then by N₀(A)_fin we denote the subgroup of N₀(A) generated by the set M_\infty(A)_fin. Consider the map h:N₀(A)_fin --> K₀(A)\otimes_ZC defined as follows

\labeleq1 h([a]-[b])= n å i=1  [Pa(\lambdai)]\otimesZ\lambdai- m å j=1  [Pb(\muj)]\otimesZ\muj,

(\theequation)

where a, b in M_\infty(A)_fin and sp(a)={\lambda₁, ..., \lambda_n}, sp(b)={\mu₁, ..., \mu_m}.

Proposition 0. The map h is a well defined surjective homomorphism. The kernel of h coinsides with a free abelian subgroup in N₀(A)_fin with the following generators: { [p(\lambda+\mu)]-[p\lambda\oplusp\mu] :\lambda, \mu in C, where p is a projection in M_\infty(A)}.

Let us consider an A-elliptic complex (E, d) and its unitary endomorphism U. The following result was proved in .

Proposition 0. For an A-Fredholm operator  F=d+d^*:\Gamma(E_ev) --> \Gamma(E_od)  there exists a decomposition  F: M₀\oplusN₀ --> M₁\oplusN₁, F:M₀ =~ M₁  such that N₀=\oplus_i N_2i, N₁=\oplus_i N_2i+1, N_m subset \Gamma(E_m), where N_m are projective U-invariant Hilbert A-modules.

Definition. We define the generalized Lefschetz number L₁ as

L1(E, U)= å i  (-1)i[U|Ni] in N0(A).

Now suppose U=U_g for some representation U_g of a compact group G and let L₁(g, E) in K₀(A)\otimesC is the Lefschetz number (of the first type) defined in .

One can establish the following result directly (cf. 5.2.17).

Theorem 0. Suppose U=U_g is an unitary endomorphism of an A-elliptic complex (E, d). Then  L₁(E, U_g)=h(L₁(E, U_g)),   where h is defined by ().

The author thanks V.M. Manuilov, A.S. Mishchenko, and E.V. Troitsky for many fruitful discussions.

The work is partially supported by the RFBR grant N 99-01-01202 and INTAS grant N 96-1099.

References

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Bratteli O., Robinson D. Operator algebras and quantum statistical mechanics. - New York - Heidelberg - Berlin, 1979.

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Murphy G.J. C^*-algebras and operator theory. - Academic Press, 1990.

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Frank M., Troitsky E. Lefschetz numbers and geometry of operators in W*-modules. \\ Func. Anal. i Pril. - 1996, V.30, N 4, 45-57 (in Russian). (English translation: Funct. Anal. Appl.- 1996, V.30, 257-266).

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Solovyov Yu.P., Troitsky E.V. C*-algebras and elliptic operators in differential topology. - M.: Factorial, 1996 (in Russian). (Revised English translation - Amer. Math. Soc., 1999, to appear).

Date received: June 7, 1999