Complexes of Hilbert C -modules

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Complexes of Hilbert C -modules Svatopluk Krýsl Charles University, Prague, Czechia Nafpaktos, 8th July 2018

Aim of the talk Give a generalization of the framework for Hodge theory

Aim of the talk Give a generalization of the framework for Hodge theory Recall the Hodge theory for manifolds

Aim of the talk Give a generalization of the framework for Hodge theory Recall the Hodge theory for manifolds Description of Hilbert C -modules and Hodge theory of their bundles

Aim of the talk Give a generalization of the framework for Hodge theory Recall the Hodge theory for manifolds Description of Hilbert C -modules and Hodge theory of their bundles Example

Contents 1 Complexes on Topological vector spaces 2 The original Hodge theory for manifolds 3 Hodge theory for bundles of Hilbert C -modules

1 Complexes on Topological vector spaces 2 The original Hodge theory for manifolds 3 Hodge theory for bundles of Hilbert C -modules

Complexes and Hodge theory for inner product spaces E = (E i ) i Z a sequence of inner product spaces and

Complexes and Hodge theory for inner product spaces E = (E i ) i Z a sequence of inner product spaces and (d i : E i E i+1 ) i Z a sequence of continuous linear and adjointable maps (differentials) such that d i+1 d i = 0 (complex).

Complexes and Hodge theory for inner product spaces E = (E i ) i Z a sequence of inner product spaces and (d i : E i E i+1 ) i Z a sequence of continuous linear and adjointable maps (differentials) such that d i+1 d i = 0 (complex). Important objects: i = d i d i + d i 1 d i 1 Laplacians Hodge theory is the statement: E i Im d i 1 Im di Ker i (orthogonal sum)

Complexes and Hodge theory for inner product spaces E = (E i ) i Z a sequence of inner product spaces and (d i : E i E i+1 ) i Z a sequence of continuous linear and adjointable maps (differentials) such that d i+1 d i = 0 (complex). Important objects: i = d i d i + d i 1 d i 1 Laplacians Hodge theory is the statement: E i Im d i 1 Im di Ker i (orthogonal sum) This implies: H i (E ) = Ker d i /Im d i 1 Ker i

Complexes and Hodge theory for inner product spaces E = (E i ) i Z a sequence of inner product spaces and (d i : E i E i+1 ) i Z a sequence of continuous linear and adjointable maps (differentials) such that d i+1 d i = 0 (complex). Important objects: i = d i d i + d i 1 d i 1 Laplacians Hodge theory is the statement: E i Im d i 1 Im di Ker i (orthogonal sum) This implies: H i (E ) = Ker d i /Im d i 1 Ker i Ker i harmonic elements

Dagger categories Dagger category = any category C with a dagger functor which is a contravariant and idempotent endofunctor in C which is identity on objects (i.e., it preserves objects, reverse direction of morphisms and applied twice it is neutral) Examples: Hilbert spaces: objects = Hilbert spaces, morphisms = linear continuous maps, and the adjoint of maps as

Dagger categories Dagger category = any category C with a dagger functor which is a contravariant and idempotent endofunctor in C which is identity on objects (i.e., it preserves objects, reverse direction of morphisms and applied twice it is neutral) Examples: Hilbert spaces: objects = Hilbert spaces, morphisms = linear continuous maps, and the adjoint of maps as bord-n category: objects = smooth oriented n-dimensional manifolds, morphisms between objects M, N = any oriented (n + 1)-manifold L which bords on the objects M and N, and the orientation reversion of L as TQFT is a specific functor from the cobord-n categories to the monoidal tensor category over a fixed Hilbert space.

Pseudoinverses on additive dagger categories Let C be an additive category (each set of morphism is an abelian group and compositions are bilinear with respect to the abelian structure) with dagger (no compatibility necessary)

Pseudoinverses on additive dagger categories Let C be an additive category (each set of morphism is an abelian group and compositions are bilinear with respect to the abelian structure) with dagger (no compatibility necessary) We say that a complex (E i, d i ) i in C is parametrix possessing if for i = d i d i + d i 1 d i 1, there are morphisms g i and p i in C such that Id E i = g i i + p i = i g i + p i, d i p i = 0 and d i 1 p i = 0. Theorem (Krysl, Ann. Glob. Anal. Geom.): If (E i, d i ) i is a parametrix possessing complex in an additive dagger category, then g i are morphisms of complexes, i.e., g i+1 d i = d i g i.

Cohomology of complexes in inner product spaces Theorem (Krysl, Ann. Glob. Anal. Geom.): Let E = (E i, d i ) i be a parametrix possessing complex with adjointable differentials in the category of inner product spaces. Then p i+1 d i = 0 and the cohomology groups of E are in a linear set-bijection with Ker i.

Cohomology of complexes in inner product spaces Theorem (Krysl, Ann. Glob. Anal. Geom.): Let E = (E i, d i ) i be a parametrix possessing complex with adjointable differentials in the category of inner product spaces. Then p i+1 d i = 0 and the cohomology groups of E are in a linear set-bijection with Ker i. We say nothing about the continuity of this bijection.

1 Complexes on Topological vector spaces 2 The original Hodge theory for manifolds 3 Hodge theory for bundles of Hilbert C -modules

Suppose 1) M a compact manifold, q i : E i M, i Z, sequence of finite rank vector bundles

Suppose 1) M a compact manifold, q i : E i M, i Z, sequence of finite rank vector bundles 2) Smooth functions with values in E i (sections: p i s = Id E i ) have an inner product space topology: Equip M with a Riemannian metric g and each E i with a hermitian inner product h. Then (s, t) = x M h(s(x), t(x))dµ(g) x

Suppose 1) M a compact manifold, q i : E i M, i Z, sequence of finite rank vector bundles 2) Smooth functions with values in E i (sections: p i s = Id E i ) have an inner product space topology: Equip M with a Riemannian metric g and each E i with a hermitian inner product h. Then (s, t) = x M h(s(x), t(x))dµ(g) x 3) complex of differential operators d i : C (E i ) C (E i+1 ) complex is elliptic = symbol of i is isomorphism out of the zero section

Suppose 1) M a compact manifold, q i : E i M, i Z, sequence of finite rank vector bundles 2) Smooth functions with values in E i (sections: p i s = Id E i ) have an inner product space topology: Equip M with a Riemannian metric g and each E i with a hermitian inner product h. Then (s, t) = x M h(s(x), t(x))dµ(g) x 3) complex of differential operators d i : C (E i ) C (E i+1 ) complex is elliptic = symbol of i is isomorphism out of the zero section Theorem (Hodge, Fredholm, Weyl): Let (C (E i ), d i ) i be an elliptic complex on finite rank vector bundles (E i ) i over a compact manifold M. Then (C (E i ), d i ) i satisfies the Hodge theory and the cohomology groups H i (E )( Ker i C (E i )) are finite dimensional.

H i (E ) = Ker d i /Im d i 1

H i (E ) = Ker d i /Im d i 1 Im d i 1 C (E i ) are closed

H i (E ) = Ker d i /Im d i 1 Im d i 1 C (E i ) are closed Cohomology groups are Hausdorff

H i (E ) = Ker d i /Im d i 1 Im d i 1 C (E i ) are closed Cohomology groups are Hausdorff The bijection between cohomology and harmonic element is homeomorphism

1 Complexes on Topological vector spaces 2 The original Hodge theory for manifolds 3 Hodge theory for bundles of Hilbert C -modules

Hilbert C -modules 1) (A,, ) a C -algebra, i.e., A is associative algebra, is a norm and is an linear idempotent map such that aa = a 2 and (A, ) is Cauchy complete

Hilbert C -modules 1) (A,, ) a C -algebra, i.e., A is associative algebra, is a norm and is an linear idempotent map such that aa = a 2 and (A, ) is Cauchy complete 2) A + = {a a = a and sp(a) [0, )} where sp(a) = {λ a λ1 does not have inverse}

Hilbert C -modules 1) (A,, ) a C -algebra, i.e., A is associative algebra, is a norm and is an linear idempotent map such that aa = a 2 and (A, ) is Cauchy complete 2) A + = {a a = a and sp(a) [0, )} where sp(a) = {λ a λ1 does not have inverse} 3) Hilbert A-module = (R, (, )) is complex vector space which is a right A-module and (, ) : R R A is A-sesquilinear, hermitian ((u, v) = (v, u) ), positive definite (v, v) A +, (v, v) = 0 implies v = 0 and (R, ) is complete where v = (v, v)

Closed images of Laplacians implies Hausdorff cohomologies Hilbert C -bundles = Banach bundles with fibers a fixed Hilbert C -module R and transition maps to the group of C - automorph. of fibers - homeomorphisms T such that T (ra) = [T (r)]a

Closed images of Laplacians implies Hausdorff cohomologies Hilbert C -bundles = Banach bundles with fibers a fixed Hilbert C -module R and transition maps to the group of C - automorph. of fibers - homeomorphisms T such that T (ra) = [T (r)]a Finitely generated = fiber is a finite linear combination m i=1 r ia i, r i R, a i A

Closed images of Laplacians implies Hausdorff cohomologies Hilbert C -bundles = Banach bundles with fibers a fixed Hilbert C -module R and transition maps to the group of C - automorph. of fibers - homeomorphisms T such that T (ra) = [T (r)]a Finitely generated = fiber is a finite linear combination m i=1 r ia i, r i R, a i A Projective = whenever R is embedded into S, S = R R OG splitting

Closed images of Laplacians implies Hausdorff cohomologies Hilbert C -bundles = Banach bundles with fibers a fixed Hilbert C -module R and transition maps to the group of C - automorph. of fibers - homeomorphisms T such that T (ra) = [T (r)]a Finitely generated = fiber is a finite linear combination m i=1 r ia i, r i R, a i A Projective = whenever R is embedded into S, S = R R OG splitting Theorem (S. Krysl, Ann. Glob. Anal. Geom.): Let (E i M) i be a sequence of finitely generated projective C -bundles over compact manifold M and (C (E i ), d i ) i be an elliptic complex of C -operators such that the Laplacians have closed images. Then the Hodge theory holds for the complex and the cohomology groups are finitely generated projective Hilbert C -modules.

Compactness trick not possible. The proof is based on regularity (G. Kasparov, Y. Solovyov), results of A. S. Mishchenko and A. T. Fomenko from 70

Compactness trick not possible. The proof is based on regularity (G. Kasparov, Y. Solovyov), results of A. S. Mishchenko and A. T. Fomenko from 70 The assumption on the closed images seems to be hard to verify in specific cases.

Compact operators Let A = CH be the algebra of compact operators on a Hilbert space, then extension of the image of i to Sobolev completions is closed, i.e., Theorem (S. Krysl, Jour. Geom. Phys.): Let (E i M) i be a sequence of finitely generated projective CH-bundles over compact manifold M and (C (E i ), d i ) i be an elliptic complex of C -operators. Then the Hodge theory holds for them and the cohomology groups of the complex are finitely generated projective Hilbert C -modules. Especially, 1) C (E i ) Im d i 1 Im di Ker i 2) H i (E ) Ker i is finitely generated over CH 3) Im i Im di Im d i 1 4) Ker d i Ker i Im d i 1 and Ker di Underlying result [Bakic, Guljas]. Ker i Im d i+1

An example from bundles induced by representations (M 2n, ω) real symplectic manifold, Mp(2n, R) double cover of the symplectic group, S the complexified Shale Weil representation on the Hilbert space H = L 2 (C n ), S the Shale Weil bundle induced by S over manifold M

An example from bundles induced by representations (M 2n, ω) real symplectic manifold, Mp(2n, R) double cover of the symplectic group, S the complexified Shale Weil representation on the Hilbert space H = L 2 (C n ), S the Shale Weil bundle induced by S over manifold M Global trivialization of S, product connection on S, S i = i T M S, complex d of differential operators (twisted derham ops). Bundles can be viewed as a Hilbert CH-bundle.

An example from bundles induced by representations (M 2n, ω) real symplectic manifold, Mp(2n, R) double cover of the symplectic group, S the complexified Shale Weil representation on the Hilbert space H = L 2 (C n ), S the Shale Weil bundle induced by S over manifold M Global trivialization of S, product connection on S, S i = i T M S, complex d of differential operators (twisted derham ops). Bundles can be viewed as a Hilbert CH-bundle. Theorem (S. Krysl): If M is compact than d satisfies the Hodge theory. In particular, C (S i ) Im d i 1 Im di Ker i and H i (d ) Ker i Further the cohomological groups are CH-isomorphic to HdeRham k (M) H. In particular their rank equals to the Betti numbers.

[BakGulOpThy] Bakić, D., Guljaš, B., Operators on Hilbert H -modules, J. Oper. Theory 46 (2001), no. 1, 123 137. [FomenkoMishchenkoIzvNauk] Fomenko, A., Mishchenko, A., The index of elliptic operators over C -algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 831 859, 967. [SKAGAG] Krýsl, S., Hodge theory for self-adjoint parametrix possessing complexes over C -algebras, Annals Glob. Anal. Geom. Vol. 47 (2015), no. 4, 359 372. [SKGeom.Phys2016] Krýsl, S., Elliptic complexes over C -algebras of compact operators, Journal of Geometry and Physics, Vol. 101 (2016), 27 37. [MishchenkoRMS] Mishchenko, A., Banach algebras, pseudodifferential operators, and their application to K-theory. Russ. Math. Surv. 34 (1979), no. 6, 77 91. [SchickNYMath] Schick, T., L 2 -index theorems, KK-theory, and connections, New York J. Math. 11, 387-443 (2005).

[ShaleTrans.AMS62] Shale, D., Linear symmetries of free boson fields, Trans. Amer. Math. Soc. 103 (1962), 149 167. [AGAG] Troitsky, E., The index of equivariant elliptic operators over C -algebras. Ann. Global Anal. Geom. 5 (1987), no. 1, 3 22. [WeilActa.Math.64] Weil, A., Sur certains groupes d opérateurs unitaires, Acta Math. 111 (1964), 143 211.

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