Noncommutative Potential Theory 1 Fabio Cipriani Dipartimento di - PowerPoint PPT Presentation

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP Noncommutative Potential Theory 1 Fabio Cipriani Dipartimento di Matematica Politecnico di Milano � � joint works with U. Franz, D. Guido, T. Isola, A. Kula, J.-L. Sauvageot Villa Mondragone Frascati, 15-22 June 2014

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP Themes. Review of Classical Potential Theory CPT Dirichlet forms on Standard Forms of von Neumann algebras KMS symmetric semigroups on C ∗ -algebras Approach to equilibria in Quantum Spin Systems Quantum Lévy Processes on Compact Quantum Groups Characterization of Haagerup Approximation Property by Dirichlet forms

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP References. L. Gross, Existence and uniqueness of physical ground states , J.Funct. Anal. 10 (1972). L. Gross, Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form , Duke Math. J. 42 (1975). S. Albeverio - R. Hoegh-Krohn, Dirichlet Forms and Markovian semigroups on C*-algebras , Comm. Math. Phys. 56 (1977). J.-L. Sauvageot, Semi-groupe de la chaleur transverse sur la C ∗ -algebre d’un feuilletage riemannien , C. R. Acad. Sci. Paris, Ser. I 310 (1990). F. Cipriani, Dirichlet Forms and Markovian Semigroups on Standard Forms of von Neumann Algebras , 147 (1997) - PhD Thesis S.I.S.S.A. (1992) F. Cipriani, Dirichlet forms on Noncommutative Spaces , L.N.M. 1954 (2008) E. B. Davies - J. M. Lindsay, Noncommutative symmetric Markov semigroups , Math. Z. 210 (1992). S. Goldstein and J. M. Lindsay, Beurling-Deny conditions for KMS-symmetric dynamical semigroups , C. R. Acad. Sci. Paris, Ser. I 317 (1993). D. Guido, T. Isola, and S. Scarlatti, Non-symmetric Dirichlet forms on semifinite von Neumann algebras , J. Funct. Anal. 135 (1996).

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP References. A. Majewski, B. Zegarlinski, On quantum stochastic dynamics and noncommutative Lp spaces , Lett. Math. Phys. 36 (1996) Y.M. Park, Construction of Dirichlet forms on standard forms of von Neumann algebras , IDAQP 3 (2000) F.Cipriani, U. Franz, A, Kula, Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory , J. Funct. Anal. 266 (2014). R. Okayasu, R. Tomatsu, Haagerup approximation property for arbitrary von Neumann algebras , arXiv:1312.1033 M. Caspers, A. Skalski, The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms , arXiv:1404.6214

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP Classical Potential Theory concerns properties of the Dirichlet integral � R d |∇ u | 2 dm : D : L 2 ( R d , m ) → [ 0 , + ∞ ] D [ u ] := lower semicontinuous quadratic form on the Hilbert space L 2 ( R d , m ) finite on the Sobolev space H 1 ( R d ) closed form of the Laplace operator � d √ ∂ 2 ∆ u � 2 ∆ = − D [ u ] = � k 2 k = 1 generator of the heat semigroup e − t ∆ : L 2 ( R d , m ) → L 2 ( R d , m ) whose heat kernel e − t ∆ ( x , y ) = ( 4 π t ) − d / 2 e − | x − y | 2 4 t is the fundamental solution of the heat equation ∂ t u + ∆ u = 0

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP The contraction property or Markovianity D [ u ∧ 1 ] ≤ D [ u ] is responsible for Maximum Principle for solution of the Laplace equation ∆ u = 0 Maximum Principle for solutions of the heat equation ∂ t u + ∆ u = 0 contractivity, positivity preserving and continuity properties of the heat semigroup e − t ∆ on the spaces L 2 ( R d , m ) , L ∞ ( R d , m ) , L 1 ( R d , m ) . The Brownian motion (Ω , P x , X t ) is the stochastic processes on R d associated to D ( e − t ∆ u )( x ) = P x ( u ◦ X t ) whose polar sets B (avoided by the processes) are the Cap ( B ) = 0 sets for the electrostatic capacity associated to D . The above properties are proved by the knowledge of the Green function � ∆ − 1 u ( x ) = G ( x , y ) = | x − y | 2 − d R d G ( x , y ) u ( y ) m ( dy ) d ≥ 3 . Beurling and Deny (late ’50) developed a kernel free potential theory generalizing the notion of Dirichlet integral to locally compact spaces. Fukushima (middle ’60) achieved the construction of the associated Hunt process.

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP Let ( M , L 2 ( M ) , L 2 + ( M ) , J ) be a standard form of a von Neumann algebra M . Let ξ 0 ∈ L 2 + ( M ) be a fixed cyclic and separating vector and ξ ∧ ξ 0 ∈ L 2 + ( M ) be the projection of a real vector ξ = J ξ ∈ L 2 ( M ) onto the positive cone L 2 + ( M ) . Definition. (Dirichlet form) A Dirichlet form E : L 2 ( M ) → ( −∞ , + ∞ ] is a l.s.c., quadratic form such that the domain F := { ξ ∈ L 2 ( M ) : E [ ξ ] < + ∞} is dense in L 2 ( M ) E [ J ξ ] = E [ ξ ] real E [ ξ ∧ ξ 0 ] ≤ E [ ξ ] Markovian ( E , F ) is a complete Dirichlet form if its matrix expansions for n ≥ 1 � E n [( ξ ij ) ij ] := E [ ξ ij ] ij are Dirichlet forms on M ⊗ M n ( C ) (tacitly assumed since now on) The domain F is called Dirichlet space when endowed with the graph norm � E [ ξ ] + � ξ � 2 � ξ � F := L 2 ( M ) .

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP Definition. (Markovian semigroup) A self-adjoint C 0 -semigroup { T t : t ≥ 0 } on L 2 ( M ) is Markovian if T t J = JT t t ≥ 0 ξ ≤ ξ 0 ⇒ T t ξ ≤ ξ 0 t ≥ 0 { T t : t ≥ 0 } on L 2 ( M ) is completely Markovian if its matrix expansions T n t ([ ξ ij ] ij ) := [ T t ξ ij ] ij are Markovian semigroups on L 2 ( M ⊗ M n ( C )) (tacitly assumed since now on) i 0 ( x ) := ∆ 1 / 4 i 0 : M → L 2 ( M ) Consider the symmetric embedding ξ 0 x ξ 0 and the faithful, normal state ω 0 : M → C ω 0 ( x ) := ( ξ 0 | x ξ 0 ) 2 . Theorem. (Modular ω 0 -symmetry) Markovian semigroups are in 1:1 correspondence with C ∗ 0 -continuous, positively preserving, contractive semigroups { S t : t ≥ 0 } on M which are ω 0 -symmetric ω 0 ( S t ( x ) σ ω 0 − i / 2 ( y )) = ω 0 ( σ ω 0 − i / 2 ( x ) S t ( y )) x , y ∈ M σ ω 0 , t > 0 i 0 ( S t ( x )) = T t ( i 0 ( x )) x ∈ M . through

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP Theorem. (Generalized Beurling-Deny correspondence) Dirichlet forms are in 1:1 correspondence with Markovian semigroups by 1 E [ ξ ] = lim t ( ξ | a − T t ξ ) a ∈ F t → 0 or through the self-adjoint generator ( L , dom ( L )) √ √ T t = e − tL La � 2 E [ a ] = � a ∈ F = dom ( L ) . L 2 ( A ,τ ) In particular, Dirichlet forms are nonnegative E ≥ 0 and Markovian semigroups are positivity preserving and contractive. Extending Markovian semigroups from M to L 2 ( M ) via non symmetric embeddings i α ( x ) := ∆ α ξ 0 x ξ 0 α ∈ [ 0 , 1 / 2 ] α � = 1 / 4 , produces semigroups on L 2 ( M ) which automatically commute with ∆ ξ 0 . By duality and interpolation, Markovian semigroups extend to C 0 -semigroups on noncommutative L p ( M ) spaces, p ∈ [ 1 , + ∞ ) .

Overview Classical Potential Theory Dirichlet forms on von Neumann algebras KMS-symmetry Quantum Spins Compact Quantum Groups Lévy Process HAP Theorem. (Ergodic Markovian semigroups) The following properties are equivalent: the Markovian semigroup { T t : t ≥ 0 } on L 2 ( M , ω ) is ergodic: for ξ, η ∈ L 2 + ( M , ω ) there exists t > 0 such that ( ξ | T t η ) 2 > 0 the Markovian semigroup { T t : t ≥ 0 } on L 2 ( M , ω ) is indecomposable: for some t > 0 , T t leaves invariant no proper face of the cone L 2 + ( M , ω ) λ := inf {E [ ξ ] : � ξ � 2 = 1 } is a Perron-Frobenius eigenvalue: it is a simple eigenvalue with cyclic eigenvector ξ λ ∈ L 2 + ( M , ω ) . Faces F of the self-polar cone L 2 + ( M , ω ) are in 1:1 correspondence with Peirce projections P e = eJeJ associated to projections e ∈ Proj ( M ) F = P e ( L 2 + ( M , ω )) . In the trace case, the above equivalences were established by L. Gross in his paper Existence and uniqueness of physical ground states , J. Funct. Anal. 10 (1972).