A distance on the state space The self-adjoint part of a C*-algebra A is denoted by sa ( A ) The Gromov- Hausdorff while its state space is denoted by S ( A ) and the smallest Propinquity unital C*-algebra containing A is denoted by uA . Frédéric Latrémolière, PhD Definition Quantum A Lipschitz pair ( A , L ) is a C*-algebra A and a densely Compact Metric Spaces defined seminorm L on sa ( uA ) such that The Monge Kantorovich distance { a ∈ sa ( uA ) : L ( a ) = 0 } = R 1 A . Compact Quantum Metric Spaces The Gromov- Hausdorff Definition (Kantorovich (1940), Kantorovich-Rubinstein (1958), Propinquity Wasserstein (1969), Dobrushin (1970), Connes (1989), Rieffel Locally Compact (1998)) Quantum Metric Spaces The Monge-Kantorovich metric mk L on S ( A ) associated with a Lipschitz pair ( A , L ) is defined for all ϕ , ψ ∈ S ( A ) by: mk L ( ϕ , ψ ) = sup {| ϕ ( a ) − ψ ( a ) | : a ∈ sa ( A ) , L ( a ) � 1 } .
The classical Monge-Kantorovich metric The Gromov- Hausdorff Theorem Propinquity Frédéric Let ( X , m ) be a compact metric space and identify X with the pure Latrémolière, PhD state space of C ( X ) (i.e. the Gel’fand spectrum of C ( X ) ). Let L be the Lipschitz seminorm for m . Then: Quantum Compact Metric Spaces ∀ x , y ∈ X m ( x , y ) = mk L ( x , y ) . The Monge Kantorovich distance Compact Quantum Metric Spaces The Gromov- Hausdorff Propinquity Locally Compact Quantum Metric Spaces
The classical Monge-Kantorovich metric The Gromov- Hausdorff Theorem Propinquity Frédéric Let ( X , m ) be a compact metric space and identify X with the pure Latrémolière, PhD state space of C ( X ) (i.e. the Gel’fand spectrum of C ( X ) ). Let L be the Lipschitz seminorm for m . Then: Quantum Compact Metric Spaces ∀ x , y ∈ X m ( x , y ) = mk L ( x , y ) . The Monge Kantorovich distance Compact Quantum Metric Spaces The Monge-Kantorovich metric is well-behaved when The Gromov- Hausdorff working over compact metric spaces: Propinquity Locally Theorem (Wasserstein, Dobrushin (1970)) Compact Quantum Metric Spaces Let ( X , m ) be a compact metric space. The Monge-Kantorovich metric mk L associated with m is a metric which metrizes the weak* topology on the state space S ( C ( X )) of C ( X ) .
GPS The Gromov- Hausdorff Propinquity Quantum Compact Metric Spaces 1 Frédéric The Monge Kantorovich distance Latrémolière, PhD Compact Quantum Metric Spaces Quantum Compact The Gromov-Hausdorff Propinquity 2 Metric Spaces The Monge The quantum Gromov-Hausdorff distance Kantorovich distance Compact Quantum The dual propinquity Metric Spaces The Gromov- The Quantum Propinquity Hausdorff Propinquity Locally Compact Quantum Metric Spaces Locally 3 Compact Topographies Quantum Metric Spaces Convergence for locally compact quantum metric spaces
Compact Quantum Metric Spaces The Gromov- Based on this observation, Rieffel introduced: Hausdorff Propinquity Definition (Rieffel, 1998) Frédéric Latrémolière, A compact quantum metric space ( A , L ) consists of an PhD order-unit space A and a seminorm L densely defined on A , Quantum satisfying: Compact Metric Spaces { a ∈ A : L ( a ) = 0 } = R 1 A , The Monge Kantorovich distance Compact Quantum and such that the distance: Metric Spaces The Gromov- Hausdorff mk L : ϕ , ψ ∈ S ( A ) �→ sup {| ϕ ( a ) − ψ ( a ) | : a ∈ A , L ( a ) � 1 } Propinquity Locally Compact metrizes the weak* topology on the state space S ( A ) . The Quantum Metric Spaces seminorm L is then called a Lip-norm .
Compact Quantum Metric Spaces The Gromov- Based on this observation, Rieffel introduced: Hausdorff Propinquity Definition (Rieffel, 1998) Frédéric Latrémolière, A compact quantum metric space ( A , L ) consists of an PhD order-unit space A and a seminorm L densely defined on A , Quantum satisfying: Compact Metric Spaces { a ∈ A : L ( a ) = 0 } = R 1 A , The Monge Kantorovich distance Compact Quantum and such that the distance: Metric Spaces The Gromov- Hausdorff mk L : ϕ , ψ ∈ S ( A ) �→ sup {| ϕ ( a ) − ψ ( a ) | : a ∈ A , L ( a ) � 1 } Propinquity Locally Compact metrizes the weak* topology on the state space S ( A ) . The Quantum Metric Spaces seminorm L is then called a Lip-norm . We shall call a quantum compact metric space a unital Lipschitz pair ( A , L ) such that ( sa ( A ) , L ) is a compact quantum metric space.
Characterization of Compact Quantum Metric Spaces The Gromov- Hausdorff Propinquity The key observation of Rieffel is that one may characterize Frédéric compact quantum metric spaces in C*-algebraic terms: Latrémolière, PhD Theorem (Rieffel, 1998) Quantum Compact A unital Lipschitz pair ( A , L ) with A unital is a compact Metric Spaces The Monge quantum metric space if and only if: Kantorovich distance Compact Quantum 1 r = diam ( S ( A ) , mk L ) < ∞ , Metric Spaces The Gromov- 2 { a ∈ sa ( A ) : L ( a ) � 1, � a � A � r } is precompact in norm. Hausdorff Propinquity Locally Compact Quantum Metric Spaces
Characterization of Compact Quantum Metric Spaces The Gromov- Hausdorff Propinquity The key observation of Rieffel is that one may characterize Frédéric compact quantum metric spaces in C*-algebraic terms: Latrémolière, PhD Theorem (Rieffel, 1998) Quantum Compact A unital Lipschitz pair ( A , L ) with A unital is a compact Metric Spaces The Monge quantum metric space if and only if: Kantorovich distance Compact Quantum 1 r = diam ( S ( A ) , mk L ) < ∞ , Metric Spaces The Gromov- 2 { a ∈ sa ( A ) : L ( a ) � 1, � a � A � r } is precompact in norm. Hausdorff Propinquity Locally Compact Proof. Quantum Metric Spaces Use Kadison functional representation and Arzéla-Ascoli theorems.
Examples: Ergodic Actions of Compact Groups with continuous Lengths For any C*-algebra A , let sa ( A ) be its self-adjoint part and The Gromov- Hausdorff � · � A be its norm. Propinquity Frédéric Theorem (Rieffel, 1998) Latrémolière, PhD Let α be a strongly continuous action of a compact group G on a Quantum unital C*-algebra A and ℓ be a continuous length function on G. Compact Metric Spaces Let e ∈ G be the unit of G. For all a ∈ A , define: The Monge Kantorovich distance Compact Quantum Metric Spaces � � α g ( a ) − a � A � L ( a ) = sup : g ∈ G \ { e } . The Gromov- ℓ ( g ) Hausdorff Propinquity Locally If { a ∈ A : ∀ g ∈ G α g ( a ) = a } = C 1 A , then ( sa ( A ) , L ) is a Compact Quantum compact quantum metric space. Metric Spaces This result uses the fact that spectral subspaces for such actions are finite dimensional (Hoegh-Krohn, Landstad, Stormer, 1981).
GPS The Gromov- Hausdorff Propinquity Quantum Compact Metric Spaces 1 Frédéric The Monge Kantorovich distance Latrémolière, PhD Compact Quantum Metric Spaces Quantum Compact The Gromov-Hausdorff Propinquity 2 Metric Spaces The quantum Gromov-Hausdorff distance The Gromov- Hausdorff The dual propinquity Propinquity The quantum The Quantum Propinquity Gromov-Hausdorff distance The dual propinquity The Quantum Locally Compact Quantum Metric Spaces 3 Propinquity Locally Topographies Compact Quantum Convergence for locally compact quantum metric Metric Spaces spaces
Convergence of Compact Metric Spaces The Gromov- Definition Hausdorff Propinquity Let ( X , m X ) and ( Y , m Y ) be two compact metric spaces. A Frédéric distance m on X ∐ Y is admissible for ( m X , m Y ) when the Latrémolière, PhD canonical injections ( X , m X ) ֒ → ( X ∐ Y , m ) and Quantum ( Y , m Y ) ֒ → ( X ∐ Y , m ) are isometries. Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Convergence of Compact Metric Spaces The Gromov- Definition Hausdorff Propinquity Let ( X , m X ) and ( Y , m Y ) be two compact metric spaces. A Frédéric distance m on X ∐ Y is admissible for ( m X , m Y ) when the Latrémolière, PhD canonical injections ( X , m X ) ֒ → ( X ∐ Y , m ) and Quantum ( Y , m Y ) ֒ → ( X ∐ Y , m ) are isometries. Compact Metric Spaces The Gromov- Notation Hausdorff Propinquity The Hausdorff distance on the compact subsets of a metric space The quantum Gromov-Hausdorff ( X , m ) is denoted by Haus m . distance The dual propinquity The Quantum Propinquity Definition (Gromov, 1981) Locally Compact The Gromov-Hausdorff distance between two compact metric Quantum Metric Spaces spaces ( X , m X ) and ( Y , m Y ) is the infimum of the set: { Haus m ( X , Y ) : m is admissible for ( m X , m Y ) } .
McShane’s Theorem The Gromov- How to formulate “isometric embeddings” in the Hausdorff noncommutative world? Propinquity Frédéric Latrémolière, PhD Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
McShane’s Theorem The Gromov- How to formulate “isometric embeddings” in the Hausdorff noncommutative world? Propinquity Frédéric Theorem (McShane, 1934) Latrémolière, PhD Let ( Z , m ) be a metric space and X ⊆ Z. If f : X → R has Quantum Lipschitz constant l, then there exists g : Z → R with Lipschitz Compact Metric Spaces constant l and whose restriction to X is f. The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
McShane’s Theorem The Gromov- How to formulate “isometric embeddings” in the Hausdorff noncommutative world? Propinquity Frédéric Theorem (McShane, 1934) Latrémolière, PhD Let ( Z , m ) be a metric space and X ⊆ Z. If f : X → R has Quantum Lipschitz constant l, then there exists g : Z → R with Lipschitz Compact Metric Spaces constant l and whose restriction to X is f. The Gromov- Hausdorff Propinquity Thus, the Lipschitz seminorm on C ( X → R ) is the quotient The quantum Gromov-Hausdorff of the Lipschitz seminorm on C ( Z → R ) . distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
McShane’s Theorem The Gromov- How to formulate “isometric embeddings” in the Hausdorff noncommutative world? Propinquity Frédéric Theorem (McShane, 1934) Latrémolière, PhD Let ( Z , m ) be a metric space and X ⊆ Z. If f : X → R has Quantum Lipschitz constant l, then there exists g : Z → R with Lipschitz Compact Metric Spaces constant l and whose restriction to X is f. The Gromov- Hausdorff Propinquity Thus, the Lipschitz seminorm on C ( X → R ) is the quotient The quantum Gromov-Hausdorff of the Lipschitz seminorm on C ( Z → R ) . More generally, a distance The dual propinquity map ι : X → Z between two compact metric spaces is an The Quantum Propinquity isometry if and only: Locally Compact Quantum L X ( f ) = inf { L Z ( g ) : g ∈ C ( Z → R ) , g ◦ ι = f } Metric Spaces for all f ∈ C ( X → R ) .
McShane’s Theorem The Gromov- How to formulate “isometric embeddings” in the Hausdorff noncommutative world? Propinquity Frédéric Theorem (McShane, 1934) Latrémolière, PhD Let ( Z , m ) be a metric space and X ⊆ Z. If f : X → R has Quantum Lipschitz constant l, then there exists g : Z → R with Lipschitz Compact Metric Spaces constant l and whose restriction to X is f. The Gromov- Hausdorff Propinquity Thus, the Lipschitz seminorm on C ( X → R ) is the quotient The quantum Gromov-Hausdorff of the Lipschitz seminorm on C ( Z → R ) . More generally, a distance The dual propinquity map ι : X → Z between two compact metric spaces is an The Quantum Propinquity isometry if and only: Locally Compact Quantum L X ( f ) = inf { L Z ( g ) : g ∈ C ( Z → R ) , g ◦ ι = f } Metric Spaces for all f ∈ C ( X → R ) . This result requires that we work with R -valued Lipschitz functions.
The quantum Gromov-Hausdorff distance The Gromov- Hausdorff Definition (Rieffel, 2000) Propinquity Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two compact quantum metric Frédéric Latrémolière, spaces. A Lip-norm L on A 1 ⊕ A 2 is admissible for ( L 1 , L 1 ) PhD � � when, for all { j , k } = { 1, 2 } and a j ∈ sa A j : Quantum Compact Metric Spaces L j ( a ) = inf { L ( a 1 , a 2 ) : a k ∈ sa ( A k ) } . The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The quantum Gromov-Hausdorff distance The Gromov- Hausdorff Definition (Rieffel, 2000) Propinquity Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two compact quantum metric Frédéric Latrémolière, spaces. A Lip-norm L on A 1 ⊕ A 2 is admissible for ( L 1 , L 1 ) PhD � � when, for all { j , k } = { 1, 2 } and a j ∈ sa A j : Quantum Compact Metric Spaces L j ( a ) = inf { L ( a 1 , a 2 ) : a k ∈ sa ( A k ) } . The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance Proposition (Rieffel, 1999) The dual propinquity The Quantum Propinquity If L is an admissible Lip-norm for ( L A , L B ) then the canonical Locally injections ( S ( A ) , mk L A ) ֒ → ( S ( A ⊕ B ) , mk L ) is an isometry Compact Quantum (and similarly with ( B , L B ) ). Metric Spaces
The quantum Gromov-Hausdorff distance The Gromov- Hausdorff Definition (Rieffel, 2000) Propinquity Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two compact quantum metric Frédéric Latrémolière, spaces. A Lip-norm L on A 1 ⊕ A 2 is admissible for ( L 1 , L 1 ) PhD � � when, for all { j , k } = { 1, 2 } and a j ∈ sa A j : Quantum Compact Metric Spaces L j ( a ) = inf { L ( a 1 , a 2 ) : a k ∈ sa ( A k ) } . The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance Definition (Rieffel, 2000) The dual propinquity The Quantum Propinquity The quantum Gromov-Hausdorff distance Locally dist q (( A , L A ) , ( B , L B )) between two compact quantum Compact Quantum metric spaces ( A , L A ) and ( B , L B ) is the infimum of the set: Metric Spaces { Haus mk L ( S ( A ) , S ( B )) : L is admissible for ( L A , L B ) } .
Basic Properties of dist q The Gromov- Theorem (Rieffel, 2000) Hausdorff Propinquity For any three quantum compact metric spaces ( A , L A ) , ( B , L B ) Frédéric and ( D , L D ) , we have: Latrémolière, PhD 1 diam ( S ( A ) , mk L A ) + diam ( S ( B ) , mk L B ) � Quantum dist q (( A , L A ) , ( B , L B )) = dist q (( B , L B ) , ( A , L A )) � 0 , Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Basic Properties of dist q The Gromov- Theorem (Rieffel, 2000) Hausdorff Propinquity For any three quantum compact metric spaces ( A , L A ) , ( B , L B ) Frédéric and ( D , L D ) , we have: Latrémolière, PhD 1 diam ( S ( A ) , mk L A ) + diam ( S ( B ) , mk L B ) � Quantum dist q (( A , L A ) , ( B , L B )) = dist q (( B , L B ) , ( A , L A )) � 0 , Compact Metric Spaces dist q (( A , L A ) , ( D , L D )) � 2 The Gromov- Hausdorff dist q (( A , L A ) , ( B , L B )) + dist q (( B , L B ) , ( D , L D )) , Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Basic Properties of dist q The Gromov- Theorem (Rieffel, 2000) Hausdorff Propinquity For any three quantum compact metric spaces ( A , L A ) , ( B , L B ) Frédéric and ( D , L D ) , we have: Latrémolière, PhD 1 diam ( S ( A ) , mk L A ) + diam ( S ( B ) , mk L B ) � Quantum dist q (( A , L A ) , ( B , L B )) = dist q (( B , L B ) , ( A , L A )) � 0 , Compact Metric Spaces dist q (( A , L A ) , ( D , L D )) � 2 The Gromov- Hausdorff dist q (( A , L A ) , ( B , L B )) + dist q (( B , L B ) , ( D , L D )) , Propinquity The quantum dist q is complete, 3 Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Basic Properties of dist q The Gromov- Theorem (Rieffel, 2000) Hausdorff Propinquity For any three quantum compact metric spaces ( A , L A ) , ( B , L B ) Frédéric and ( D , L D ) , we have: Latrémolière, PhD 1 diam ( S ( A ) , mk L A ) + diam ( S ( B ) , mk L B ) � Quantum dist q (( A , L A ) , ( B , L B )) = dist q (( B , L B ) , ( A , L A )) � 0 , Compact Metric Spaces dist q (( A , L A ) , ( D , L D )) � 2 The Gromov- Hausdorff dist q (( A , L A ) , ( B , L B )) + dist q (( B , L B ) , ( D , L D )) , Propinquity The quantum dist q is complete, 3 Gromov-Hausdorff distance The dual propinquity dist q is dominated by the Gromov-Hausdorff distance in the 4 The Quantum Propinquity classical case, Locally Compact Quantum Metric Spaces
Basic Properties of dist q The Gromov- Theorem (Rieffel, 2000) Hausdorff Propinquity For any three quantum compact metric spaces ( A , L A ) , ( B , L B ) Frédéric and ( D , L D ) , we have: Latrémolière, PhD 1 diam ( S ( A ) , mk L A ) + diam ( S ( B ) , mk L B ) � Quantum dist q (( A , L A ) , ( B , L B )) = dist q (( B , L B ) , ( A , L A )) � 0 , Compact Metric Spaces dist q (( A , L A ) , ( D , L D )) � 2 The Gromov- Hausdorff dist q (( A , L A ) , ( B , L B )) + dist q (( B , L B ) , ( D , L D )) , Propinquity The quantum dist q is complete, 3 Gromov-Hausdorff distance The dual propinquity dist q is dominated by the Gromov-Hausdorff distance in the 4 The Quantum Propinquity classical case, Locally Compact dist q (( A , L A ) , ( B , L B )) = 0 iff there exists a 5 Quantum Metric Spaces order-unit-space isomorphism from sa ( A ) to sa ( B ) whose dual map is an isometry from ( S ( B ) , mk L B ) to ( S ( A ) , mk L A ) .
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum 2 Replace the state space by the graph of the Compact Metric Spaces multiplication restricted to the unit Lip-ball: Li’s The Gromov- Hausdorff C*-algebraic distance Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum 2 Replace the state space by the graph of the Compact Metric Spaces multiplication restricted to the unit Lip-ball: Li’s The Gromov- Hausdorff C*-algebraic distance Propinquity 3 Work entirely within the C*-algebra category. The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum 2 Replace the state space by the graph of the Compact Metric Spaces multiplication restricted to the unit Lip-ball: Li’s The Gromov- Hausdorff C*-algebraic distance Propinquity 3 Work entirely within the C*-algebra category. The quantum Gromov-Hausdorff distance Li’s nuclear distance based on Lip-balls, The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum 2 Replace the state space by the graph of the Compact Metric Spaces multiplication restricted to the unit Lip-ball: Li’s The Gromov- Hausdorff C*-algebraic distance Propinquity 3 Work entirely within the C*-algebra category. The quantum Gromov-Hausdorff distance Li’s nuclear distance based on Lip-balls, The dual propinquity The Quantum FL approach based on Leibniz Lip-norms : Propinquity Locally Compact Quantum Metric Spaces
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum 2 Replace the state space by the graph of the Compact Metric Spaces multiplication restricted to the unit Lip-ball: Li’s The Gromov- Hausdorff C*-algebraic distance Propinquity 3 Work entirely within the C*-algebra category. The quantum Gromov-Hausdorff distance Li’s nuclear distance based on Lip-balls, The dual propinquity The Quantum FL approach based on Leibniz Lip-norms : Propinquity FL’s quantum propinquity based on Lip-balls. Locally 1 Compact Quantum Metric Spaces
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum 2 Replace the state space by the graph of the Compact Metric Spaces multiplication restricted to the unit Lip-ball: Li’s The Gromov- Hausdorff C*-algebraic distance Propinquity 3 Work entirely within the C*-algebra category. The quantum Gromov-Hausdorff distance Li’s nuclear distance based on Lip-balls, The dual propinquity The Quantum FL approach based on Leibniz Lip-norms : Propinquity FL’s quantum propinquity based on Lip-balls. Locally 1 Compact FL’s dual propinquity based on state space. 2 Quantum Metric Spaces
The Distance Zero Problem The Gromov- How to get *-isomorphism as necessary for distance zero? Hausdorff Propinquity 1 Replace the state space by 2 × 2-matrix-valued Frédéric Latrémolière, completely positive unital maps: Kerr’s matricial PhD Gromov-Hausdorff distance Quantum 2 Replace the state space by the graph of the Compact Metric Spaces multiplication restricted to the unit Lip-ball: Li’s The Gromov- Hausdorff C*-algebraic distance Propinquity 3 Work entirely within the C*-algebra category. The quantum Gromov-Hausdorff distance Li’s nuclear distance based on Lip-balls, The dual propinquity The Quantum FL approach based on Leibniz Lip-norms : Propinquity FL’s quantum propinquity based on Lip-balls. Locally 1 Compact FL’s dual propinquity based on state space. 2 Quantum Metric Spaces Thus, our new approach focuses on keeping the noncommutative Monge-Kantorovich metric and shift the focus to the relationship between Lip-norms and multiplicative structure.
The Leibniz inequality The Gromov- The main problem of dist q is that it does not involve the Hausdorff multiplication at all, and in fact, neither does the definition Propinquity of compact quantum metric spaces. Frédéric Latrémolière, PhD Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The Leibniz inequality The Gromov- The main problem of dist q is that it does not involve the Hausdorff multiplication at all, and in fact, neither does the definition Propinquity of compact quantum metric spaces. Yet, most important Frédéric Latrémolière, examples of quantum locally compact metric space have a PhD very important additional property: Quantum Compact Metric Spaces Definition The Gromov- A seminorm L on a C*-algebra A has the Leibniz property Hausdorff Propinquity when: The quantum Gromov-Hausdorff distance The dual propinquity ∀ a , b ∈ A L ( ab ) � � a � A L ( b ) + L ( a ) � b � A . The Quantum Propinquity Locally Compact Quantum Metric Spaces
The Leibniz inequality The Gromov- The main problem of dist q is that it does not involve the Hausdorff multiplication at all, and in fact, neither does the definition Propinquity of compact quantum metric spaces. Yet, most important Frédéric Latrémolière, examples of quantum locally compact metric space have a PhD very important additional property: Quantum Compact Metric Spaces Definition The Gromov- A seminorm L on a C*-algebra A has the Leibniz property Hausdorff Propinquity when: The quantum Gromov-Hausdorff distance The dual propinquity ∀ a , b ∈ A L ( ab ) � � a � A L ( b ) + L ( a ) � b � A . The Quantum Propinquity Locally In most cases, the Lip-norms of quantum locally compact Compact Quantum metric space comes from derivations, spectral triples or Metric Spaces similar constructions which gives the Leibniz property. This is a natural connection between metric and multiplicative structures of quantum locally compact metric space .
The role of the Leibniz inequality The Gromov- The Leibniz inequality plays a central role in Rieffel’s Hausdorff Propinquity recent work on convergence of vector bundles. It Frédéric appears that one should work within the framework of Latrémolière, PhD C*-metric spaces, where Lip-norms are defined on C*-algebras and satisfy a strong form of the Leibniz Quantum Compact property (cf Rieffel’s work on convergence of matrix Metric Spaces The Gromov- algebras to spheres, for instance). Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The role of the Leibniz inequality The Gromov- The Leibniz inequality plays a central role in Rieffel’s Hausdorff Propinquity recent work on convergence of vector bundles. It Frédéric appears that one should work within the framework of Latrémolière, PhD C*-metric spaces, where Lip-norms are defined on C*-algebras and satisfy a strong form of the Leibniz Quantum Compact property (cf Rieffel’s work on convergence of matrix Metric Spaces The Gromov- algebras to spheres, for instance). Hausdorff Propinquity Yet, the quotient of a Leibniz seminorm is not Leibniz in The quantum Gromov-Hausdorff general. This means that if one asks for admissible distance The dual propinquity Lip-norms to be Leibniz in the definition of dist q , one The Quantum Propinquity only gets a pseudo-semi-metric (Rieffel’s proximity). Locally Compact Quantum Metric Spaces
The role of the Leibniz inequality The Gromov- The Leibniz inequality plays a central role in Rieffel’s Hausdorff Propinquity recent work on convergence of vector bundles. It Frédéric appears that one should work within the framework of Latrémolière, PhD C*-metric spaces, where Lip-norms are defined on C*-algebras and satisfy a strong form of the Leibniz Quantum Compact property (cf Rieffel’s work on convergence of matrix Metric Spaces The Gromov- algebras to spheres, for instance). Hausdorff Propinquity Yet, the quotient of a Leibniz seminorm is not Leibniz in The quantum Gromov-Hausdorff general. This means that if one asks for admissible distance The dual propinquity Lip-norms to be Leibniz in the definition of dist q , one The Quantum Propinquity only gets a pseudo-semi-metric (Rieffel’s proximity). Locally Compact Quantum Hard Problem Metric Spaces How does one define a non-trivial metric on *-isomorphic, quantum isometric classes of C*-metric spaces?
GPS The Gromov- Hausdorff Propinquity Quantum Compact Metric Spaces 1 Frédéric The Monge Kantorovich distance Latrémolière, PhD Compact Quantum Metric Spaces Quantum Compact The Gromov-Hausdorff Propinquity 2 Metric Spaces The quantum Gromov-Hausdorff distance The Gromov- Hausdorff The dual propinquity Propinquity The quantum The Quantum Propinquity Gromov-Hausdorff distance The dual propinquity The Quantum Locally Compact Quantum Metric Spaces 3 Propinquity Locally Topographies Compact Quantum Convergence for locally compact quantum metric Metric Spaces spaces
Leibniz quantum compact metric spaces The Gromov- We first choose a category of quantum compact metric Hausdorff Propinquity spaces. Frédéric Latrémolière, PhD Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Leibniz quantum compact metric spaces The Gromov- We first choose a category of quantum compact metric Hausdorff Propinquity spaces. Frédéric For a , b elements of a C*-algebra A , let a ◦ b = ab + ba be the Latrémolière, 2 PhD Jordan product of a , b and { a , b } = ab − ba be the Lie product 2 i Quantum of a , b . Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Leibniz quantum compact metric spaces The Gromov- We first choose a category of quantum compact metric Hausdorff Propinquity spaces. Frédéric For a , b elements of a C*-algebra A , let a ◦ b = ab + ba be the Latrémolière, 2 PhD Jordan product of a , b and { a , b } = ab − ba be the Lie product 2 i Quantum of a , b . Compact Metric Spaces Definition (Latrémolière, 2013) The Gromov- Hausdorff A quantum compact metric space ( A , L ) is a Leibniz quantum Propinquity The quantum compact metric space when, for all a , b ∈ sa ( A ) we have: Gromov-Hausdorff distance The dual propinquity The Quantum L ( a ◦ b ) � � a � A L ( b ) + L ( a ) � b � A Propinquity Locally Compact and Quantum Metric Spaces L ( { a , b } ) � � a � A L ( b ) + L ( a ) � b � A , while L is lower semi-continuous.
Bridges and Tunnels The Gromov- We propose the following notion of a pair of isometric Hausdorff Propinquity embeddings of Leibniz quantum compact metric spaces: Frédéric Latrémolière, Definition (Latrémolière, 2013) PhD Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two Leibniz quantum compact Quantum Compact metric spaces. A tunnel ( D , L D , π 1 , π 2 ) is a Leibniz quantum Metric Spaces compact metric space ( D , L D ) together with two surjective The Gromov- Hausdorff *-morphisms π 1 and π 2 such that: Propinquity The quantum Gromov-Hausdorff distance � � � L j ( a ) = inf L D ( d ) � π j ( d ) = a The dual propinquity The Quantum Propinquity � � for all j ∈ { 1, 2 } and a ∈ sa . A j Locally Compact Quantum Metric Spaces
Bridges and Tunnels The Gromov- We propose the following notion of a pair of isometric Hausdorff Propinquity embeddings of Leibniz quantum compact metric spaces: Frédéric Latrémolière, Definition (Latrémolière, 2013) PhD Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two Leibniz quantum compact Quantum Compact metric spaces. A tunnel ( D , L D , π 1 , π 2 ) is a Leibniz quantum Metric Spaces compact metric space ( D , L D ) together with two surjective The Gromov- Hausdorff *-morphisms π 1 and π 2 such that: Propinquity The quantum Gromov-Hausdorff distance � � � L j ( a ) = inf L D ( d ) � π j ( d ) = a The dual propinquity The Quantum Propinquity � � for all j ∈ { 1, 2 } and a ∈ sa . A j Locally Compact Quantum Metric Spaces We do not require the tunnel to be of the form ( A ⊕ B , L , π A , π B ) with π A , π B canonical surjections.
Bridges and Tunnels The Gromov- We propose the following notion of a pair of isometric Hausdorff Propinquity embeddings of Leibniz quantum compact metric spaces: Frédéric Latrémolière, Definition (Latrémolière, 2013) PhD Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two Leibniz quantum compact Quantum Compact metric spaces. A tunnel ( D , L D , π 1 , π 2 ) is a Leibniz quantum Metric Spaces compact metric space ( D , L D ) together with two surjective The Gromov- Hausdorff *-morphisms π 1 and π 2 such that: Propinquity The quantum Gromov-Hausdorff distance � � � L j ( a ) = inf L D ( d ) � π j ( d ) = a The dual propinquity The Quantum Propinquity � � for all j ∈ { 1, 2 } and a ∈ sa . A j Locally Compact Quantum Metric Spaces We can add various conditions on the Leibniz quantum com- pact metric space of a tunnel: strong Leibniz Lip-norm, com- pact C*-metric space, etc...
Bimodules and Bridges The Gromov- A particular, common type of tunnels is given by the Hausdorff following construction for two Leibniz quantum compact Propinquity metric spaces ( A , L A ) and ( B , L B ) : Frédéric Latrémolière, 1 Let Ω be a A - B -bimodule, with a norm � · � Ω such that: PhD Quantum � a ω b � Ω � � a � A � ω � Ω � b � B Compact Metric Spaces for all a ∈ A , b ∈ B and ω ∈ Ω . The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Bimodules and Bridges The Gromov- A particular, common type of tunnels is given by the Hausdorff following construction for two Leibniz quantum compact Propinquity metric spaces ( A , L A ) and ( B , L B ) : Frédéric Latrémolière, 1 Let Ω be a A - B -bimodule, with a norm � · � Ω such that: PhD Quantum � a ω b � Ω � � a � A � ω � Ω � b � B Compact Metric Spaces for all a ∈ A , b ∈ B and ω ∈ Ω . The Gromov- Hausdorff 2 Choose ω 0 ∈ Ω and γ > 0 such that, if we set: Propinquity The quantum Gromov-Hausdorff bn ω 0 , γ ( a , b ) = � a ω 0 − ω 0 b � Ω distance The dual propinquity The Quantum Propinquity and then: Locally � � Compact L A ( a ) , L B ( b ) , 1 L ( a , b ) = max Quantum γ bn ω 0 , γ ( a , b ) Metric Spaces for all a ∈ A , b ∈ B , then ( A ⊕ B , L , π A , π B ) is a tunnel (where π A , π B are canonical surjections).
Bridges The Gromov- The bimodule approach to the construction of Lip-norm is Hausdorff particularly interesting when the bimodules are Propinquity C*-algebras. We thus propose: Frédéric Latrémolière, PhD Definition (Latrémolière, 2013) Quantum Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two Leibniz quantum compact Compact Metric Spaces metric spaces. A bridge ( D , ω , ρ 1 , ρ 2 ) is a unital C*-algebra D The Gromov- → D ( j = 1, 2) and Hausdorff and two unital *-monomorphisms ρ j : A j ֒ Propinquity ω ∈ D such that there exists ϕ ∈ S ( D ) with The quantum Gromov-Hausdorff ϕ (( 1 − ω ) ∗ ( 1 − ω )) = 0 and ϕ (( 1 − ω )( 1 − ω ) ∗ ) = 0. distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Bridges The Gromov- The bimodule approach to the construction of Lip-norm is Hausdorff particularly interesting when the bimodules are Propinquity C*-algebras. We thus propose: Frédéric Latrémolière, PhD Definition (Latrémolière, 2013) Quantum Let ( A 1 , L 1 ) and ( A 2 , L 2 ) be two Leibniz quantum compact Compact Metric Spaces metric spaces. A bridge ( D , ω , ρ 1 , ρ 2 ) is a unital C*-algebra D The Gromov- → D ( j = 1, 2) and Hausdorff and two unital *-monomorphisms ρ j : A j ֒ Propinquity ω ∈ D such that there exists ϕ ∈ S ( D ) with The quantum Gromov-Hausdorff ϕ (( 1 − ω ) ∗ ( 1 − ω )) = 0 and ϕ (( 1 − ω )( 1 − ω ) ∗ ) = 0. distance The dual propinquity The Quantum Propinquity To every bridge, we can associate a tunnel. The question is Locally Compact to choose the constant γ such that: Quantum Metric Spaces � L 1 ( a ) , L 2 ( b ) , 1 � L : ( a , b ) ∈ sa ( A ⊕ B ) �→ max γ � a ω − ω b � Ω is admissible (difficulties arise: Rieffel, 0910.1968 )
Defining a Distance from Tunnels: reach The Gromov- Hausdorff Propinquity How do we define a distance from tunnels? Frédéric Latrémolière, PhD Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Defining a Distance from Tunnels: reach The Gromov- Hausdorff Propinquity How do we define a distance from tunnels? We associate Frédéric Latrémolière, numerical quantities to a tunnel. The first is: PhD Definition (Latrémolière, 2013) Quantum Compact Metric Spaces Let ( A , L A ) , ( B , L B ) be two Leibniz quantum compact The Gromov- metric spaces and τ = ( D , L D , π A , π B ) be a tunnel from Hausdorff Propinquity ( A , L A ) to ( B , L B ) . The reach ρ ( τ ) of τ is: The quantum Gromov-Hausdorff distance Haus mk L D ( π ∗ A ( S ( A )) , π ∗ The dual propinquity B ( S ( B ))) , The Quantum Propinquity Locally where Haus m is the Hausdorff distance on compact subsets Compact Quantum of a metric space ( E , m ) . Metric Spaces
Defining a Distance from Tunnels: depth The Gromov- We must also account for the greater level of generality Hausdorff from Rieffel’s admissibility. Propinquity Frédéric Latrémolière, PhD Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Defining a Distance from Tunnels: depth The Gromov- We must also account for the greater level of generality Hausdorff from Rieffel’s admissibility. The key is the quantity: Propinquity Frédéric Definition (Latrémolière, 2013) Latrémolière, PhD Let ( A , L A ) , ( B , L B ) be two Leibniz quantum compact Quantum metric spaces and τ = ( D , L D , π A , π B ) be a tunnel from Compact Metric Spaces ( A , L A ) to ( B , L B ) . The depth δ ( τ ) of τ is: The Gromov- Hausdorff Propinquity Haus mk L D ( S ( D ) , co ( π ∗ A ( S ( A )) ∪ π ∗ B ( S ( B )))) , The quantum Gromov-Hausdorff distance The dual propinquity where co ( A ) is the weak* closure of the convex hull of any The Quantum Propinquity subset A of S ( D ) . Locally Compact Quantum Metric Spaces
Defining a Distance from Tunnels: depth The Gromov- We must also account for the greater level of generality Hausdorff from Rieffel’s admissibility. The key is the quantity: Propinquity Frédéric Definition (Latrémolière, 2013) Latrémolière, PhD Let ( A , L A ) , ( B , L B ) be two Leibniz quantum compact Quantum metric spaces and τ = ( D , L D , π A , π B ) be a tunnel from Compact Metric Spaces ( A , L A ) to ( B , L B ) . The depth δ ( τ ) of τ is: The Gromov- Hausdorff Propinquity Haus mk L D ( S ( D ) , co ( π ∗ A ( S ( A )) ∪ π ∗ B ( S ( B )))) , The quantum Gromov-Hausdorff distance The dual propinquity where co ( A ) is the weak* closure of the convex hull of any The Quantum Propinquity subset A of S ( D ) . Locally Compact Quantum This quantity will prove useful in dealing with the triangle Metric Spaces inequality property of our new metric. No other approach has ever involved our more general tunnels and only look at A ⊕ B , for which the depth is always 0.
Putting it together The Gromov- Originally, we define the length of a tunnel by: Hausdorff Propinquity Definition (Latrémolière, 2013) Frédéric Latrémolière, The length of a tunnel τ is the maximum of its reach and its PhD depth. Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Putting it together The Gromov- Originally, we define the length of a tunnel by: Hausdorff Propinquity Definition (Latrémolière, 2013) Frédéric Latrémolière, The length of a tunnel τ is the maximum of its reach and its PhD depth. Quantum Compact Metric Spaces A better, equivalent, synthetic quantity, however, is: The Gromov- Hausdorff Definition (Latrémolière, 2014) Propinquity The quantum Let τ = ( D , L D , π A , π B ) be a tunnel between two Leibniz Gromov-Hausdorff distance quantum compact metric spaces ( A , L A ) and ( B , L B ) . The The dual propinquity The Quantum Propinquity extent χ ( τ ) of τ is: Locally Compact Quantum � Haus mk L D ( S ( D ) , π ∗ Metric Spaces max A ( S ( A )) , ) � Haus mk L D ( S ( D ) , π ∗ B ( S ( B ))) .
The Dual Propinquity The Gromov- Hausdorff Propinquity We can define a new distance between Leibniz quantum Frédéric compact metric spaces: Latrémolière, PhD Definition (Latrémolière, 2013, 2014) Quantum Compact The dual propinquity Λ ∗ (( A , L A ) , ( B , L B )) between two Metric Spaces The Gromov- Leibniz quantum compact metric spaces ( A , L A ) and Hausdorff Propinquity ( B , L B ) is: The quantum Gromov-Hausdorff distance inf { χ ( τ ) | τ is a tunnel from ( A , L A ) and ( B , L B ) } . The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
The Dual Propinquity The Gromov- Hausdorff Propinquity We can define a new distance between Leibniz quantum Frédéric compact metric spaces: Latrémolière, PhD Definition (Latrémolière, 2013, 2014) Quantum Compact The dual propinquity Λ ∗ (( A , L A ) , ( B , L B )) between two Metric Spaces The Gromov- Leibniz quantum compact metric spaces ( A , L A ) and Hausdorff Propinquity ( B , L B ) is: The quantum Gromov-Hausdorff distance inf { χ ( τ ) | τ is a tunnel from ( A , L A ) and ( B , L B ) } . The dual propinquity The Quantum Propinquity Locally We originally defined the dual propinquity in terms of Compact lengths of tunnels, though this requires more care; the re- Quantum Metric Spaces sulting metrics are equivalent.
The Dual Propinquity The Gromov- Hausdorff Propinquity We can define a new distance between Leibniz quantum Frédéric compact metric spaces: Latrémolière, PhD Definition (Latrémolière, 2013, 2014) Quantum Compact The dual propinquity Λ ∗ (( A , L A ) , ( B , L B )) between two Metric Spaces The Gromov- Leibniz quantum compact metric spaces ( A , L A ) and Hausdorff Propinquity ( B , L B ) is: The quantum Gromov-Hausdorff distance inf { χ ( τ ) | τ is a tunnel from ( A , L A ) and ( B , L B ) } . The dual propinquity The Quantum Propinquity Locally We may restrict our attention to some specific classes of tun- Compact nels, and define specialized versions of the dual propinquity, Quantum Metric Spaces e.g. to compact C*-metric spaces.
Triangle Inequality The Gromov- Theorem (Latrémolière, 2014) Hausdorff Propinquity For all Leibniz quantum compact metric spaces ( A 1 , L 1 ) , ( A 2 , L 2 ) Frédéric Latrémolière, and ( A 3 , L 3 ) , we have: PhD Λ ∗ (( A 1 , L 1 ) , ( A 3 , L 3 )) � Λ ∗ (( A 1 , L 1 ) , ( A 2 , L 2 )) Quantum Compact Metric Spaces + Λ ∗ (( A 2 , L 2 ) , ( A 3 , L 3 )) . The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Triangle Inequality The Gromov- Theorem (Latrémolière, 2014) Hausdorff Propinquity For all Leibniz quantum compact metric spaces ( A 1 , L 1 ) , ( A 2 , L 2 ) Frédéric Latrémolière, and ( A 3 , L 3 ) , we have: PhD Λ ∗ (( A 1 , L 1 ) , ( A 3 , L 3 )) � Λ ∗ (( A 1 , L 1 ) , ( A 2 , L 2 )) Quantum Compact Metric Spaces + Λ ∗ (( A 2 , L 2 ) , ( A 3 , L 3 )) . The Gromov- Hausdorff Propinquity Proof. The quantum Gromov-Hausdorff distance Let τ 12 = ( D 12 , L 12 , π 1 , π 2 ) be a tunnel from ( A 1 , L 1 ) to The dual propinquity The Quantum Propinquity ( A 2 , L 2 ) and τ 23 = ( D 23 , L 23 , ρ 2 , ρ 3 ) be a tunnel from ( A 2 , L 2 ) Locally to ( A 3 , L 3 ) . Compact Quantum Metric Spaces
Triangle Inequality The Gromov- Theorem (Latrémolière, 2014) Hausdorff Propinquity For all Leibniz quantum compact metric spaces ( A 1 , L 1 ) , ( A 2 , L 2 ) Frédéric Latrémolière, and ( A 3 , L 3 ) , we have: PhD Λ ∗ (( A 1 , L 1 ) , ( A 3 , L 3 )) � Λ ∗ (( A 1 , L 1 ) , ( A 2 , L 2 )) Quantum Compact Metric Spaces + Λ ∗ (( A 2 , L 2 ) , ( A 3 , L 3 )) . The Gromov- Hausdorff Propinquity Proof. The quantum Gromov-Hausdorff distance Let D = D 12 ⊕ D 23 . For all ε > 0, set L ε ( d 12 , d 23 ) as: The dual propinquity The Quantum Propinquity Locally � � L 12 ( d 12 ) , L 23 ( d 23 ) , 1 Compact ε � π 2 ( d 12 ) − ρ 2 ( d 23 ) � A 3 max Quantum Metric Spaces for all d 12 ∈ sa ( D 12 ) , d 23 ∈ sa ( D 23 ) .
Triangle Inequality The Gromov- Theorem (Latrémolière, 2014) Hausdorff Propinquity For all Leibniz quantum compact metric spaces ( A 1 , L 1 ) , ( A 2 , L 2 ) Frédéric Latrémolière, and ( A 3 , L 3 ) , we have: PhD Λ ∗ (( A 1 , L 1 ) , ( A 3 , L 3 )) � Λ ∗ (( A 1 , L 1 ) , ( A 2 , L 2 )) Quantum Compact Metric Spaces + Λ ∗ (( A 2 , L 2 ) , ( A 3 , L 3 )) . The Gromov- Hausdorff Propinquity Proof. The quantum Gromov-Hausdorff distance For all ε > 0, we check that τ ε = ( D 12 ⊕ D 23 , L ε , π 1 , ρ 3 ) is a The dual propinquity The Quantum Propinquity tunnel from ( A 1 , L 1 ) to ( A 3 , L 3 ) with: Locally Compact Quantum χ ( τ ε ) � χ ( τ 12 ) + χ ( τ 23 ) + ε . Metric Spaces
Triangle Inequality The Gromov- Theorem (Latrémolière, 2014) Hausdorff Propinquity For all Leibniz quantum compact metric spaces ( A 1 , L 1 ) , ( A 2 , L 2 ) Frédéric Latrémolière, and ( A 3 , L 3 ) , we have: PhD Λ ∗ (( A 1 , L 1 ) , ( A 3 , L 3 )) � Λ ∗ (( A 1 , L 1 ) , ( A 2 , L 2 )) Quantum Compact Metric Spaces + Λ ∗ (( A 2 , L 2 ) , ( A 3 , L 3 )) . The Gromov- Hausdorff Propinquity Proof. The quantum Gromov-Hausdorff distance We conclude by choosing τ 12 and τ 23 such that The dual propinquity The Quantum Propinquity χ ( τ 12 ) � Λ ∗ (( A 1 , L 1 ) , ( A 2 , L 2 )) + ε Locally Compact Quantum Metric Spaces and χ ( τ 23 ) � Λ ∗ (( A 2 , L 2 ) , ( A 3 , L 3 )) + ε , then take the infi- mum over ε .
Triangle Inequality The Gromov- Theorem (Latrémolière, 2014) Hausdorff Propinquity For all Leibniz quantum compact metric spaces ( A 1 , L 1 ) , ( A 2 , L 2 ) Frédéric Latrémolière, and ( A 3 , L 3 ) , we have: PhD Λ ∗ (( A 1 , L 1 ) , ( A 3 , L 3 )) � Λ ∗ (( A 1 , L 1 ) , ( A 2 , L 2 )) Quantum Compact Metric Spaces + Λ ∗ (( A 2 , L 2 ) , ( A 3 , L 3 )) . The Gromov- Hausdorff Propinquity Proof. The quantum Gromov-Hausdorff distance Comment: the tunnels D ε are not in general of the form The dual propinquity The Quantum Propinquity ( A 1 ⊕ A 3 , . . . ) . To form such a tunnel would require tak- Locally ing a quotient, and this is why triangle inequality fails, for Compact Quantum instance, with Rieffel’s proximity, or the quantum Gromov- Metric Spaces Hausdorff distance involves non-Leibniz seminorms.
Distance Zero The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity For any two Leibniz quantum compact metric spaces ( A , L A ) and Frédéric ( B , L B ) : Latrémolière, PhD Λ ∗ (( A , L A ) , ( B , L B )) = 0 Quantum Compact if and only if there exists a *-isomorphism π : A → B such that Metric Spaces L B ◦ π = L A . The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Distance Zero The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity For any two Leibniz quantum compact metric spaces ( A , L A ) and Frédéric ( B , L B ) : Latrémolière, PhD Λ ∗ (( A , L A ) , ( B , L B )) = 0 Quantum Compact if and only if there exists a *-isomorphism π : A → B such that Metric Spaces L B ◦ π = L A . The Gromov- Hausdorff Propinquity The quantum Proof. Gromov-Hausdorff distance Fix ε > 0 and let τ ε = ( D ε , L ε , π ε A , π ε B ) be a tunnel from The dual propinquity The Quantum Propinquity ( A , L A ) to ( B , L B ) of extent ε or less. Locally Compact Quantum Metric Spaces
Distance Zero The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity For any two Leibniz quantum compact metric spaces ( A , L A ) and Frédéric ( B , L B ) : Latrémolière, PhD Λ ∗ (( A , L A ) , ( B , L B )) = 0 Quantum Compact if and only if there exists a *-isomorphism π : A → B such that Metric Spaces L B ◦ π = L A . The Gromov- Hausdorff Propinquity The quantum Proof. Gromov-Hausdorff distance For any a ∈ sa ( A ) and l � L A ( a ) , introduce the sets: The dual propinquity The Quantum Propinquity Locally l τ ε ( a | l ) = { d ∈ sa ( D ε ) : π ε A ( d ) = a , L ε ( d ) � l } , Compact Quantum Metric Spaces and t τ ε ( a | l ) = π ε B ( l τ ε ( a | l )) .
Distance Zero The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity For any two Leibniz quantum compact metric spaces ( A , L A ) and Frédéric ( B , L B ) : Latrémolière, PhD Λ ∗ (( A , L A ) , ( B , L B )) = 0 Quantum Compact if and only if there exists a *-isomorphism π : A → B such that Metric Spaces L B ◦ π = L A . The Gromov- Hausdorff Propinquity The quantum Proof. Gromov-Hausdorff distance The target sets t τ ε ( a | l ) are sort of an image of a for τ ε . If ϕ ∈ The dual propinquity The Quantum Propinquity S ( D ε ) and d ∈ l τ ε ( a | l ) then there exists ψ ∈ S ( A ) such that Locally mk L D ( ϕ , ψ ◦ π A ) � χ ( τ ) . Then: Compact Quantum Metric Spaces | ϕ ( d ) | � | ϕ ( d ) + ψ ◦ π A ( d ) | + | ψ ( a ) | � l χ ( τ ε ) + � a � A .
Distance Zero The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity For any two Leibniz quantum compact metric spaces ( A , L A ) and Frédéric ( B , L B ) : Latrémolière, PhD Λ ∗ (( A , L A ) , ( B , L B )) = 0 Quantum Compact if and only if there exists a *-isomorphism π : A → B such that Metric Spaces L B ◦ π = L A . The Gromov- Hausdorff Propinquity The quantum Proof. Gromov-Hausdorff distance One then deduces that: The dual propinquity The Quantum Propinquity Locally diam ( t τ ε ( a | l ) , � · � B ) � l χ ( τ ε ) � l ε . Compact Quantum Metric Spaces and t τ ( a | l ) is a compact subset of the norm compact set { b ∈ sa ( B ) : L ( b ) � 1, � b � � � a � + 1 } .
Distance Zero The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity For any two Leibniz quantum compact metric spaces ( A , L A ) and Frédéric ( B , L B ) : Latrémolière, PhD Λ ∗ (( A , L A ) , ( B , L B )) = 0 Quantum Compact if and only if there exists a *-isomorphism π : A → B such that Metric Spaces L B ◦ π = L A . The Gromov- Hausdorff Propinquity The quantum Proof. Gromov-Hausdorff distance Thus ( t τ ε ( a | l )) ε > 0 admits a converging subnet for the Haus- The dual propinquity The Quantum Propinquity dorff distance induced by � · � B , whose limit is a singleton. Locally We can use a diagonal argument and our norm estimates to Compact Quantum remove the dependence of the subnet on a and l . This defines Metric Spaces a map π from A to B .
Distance Zero The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity For any two Leibniz quantum compact metric spaces ( A , L A ) and Frédéric ( B , L B ) : Latrémolière, PhD Λ ∗ (( A , L A ) , ( B , L B )) = 0 Quantum Compact if and only if there exists a *-isomorphism π : A → B such that Metric Spaces L B ◦ π = L A . The Gromov- Hausdorff Propinquity The quantum Proof. Gromov-Hausdorff distance The multiplicative property of π requires the norm estimate The dual propinquity The Quantum Propinquity for l a ( l | r ) , while the linearity does not. Locally Compact Quantum Metric Spaces
Comparison with the quantum Gromov-Hausdorff distance The Gromov- Hausdorff We established: Propinquity Theorem (Latrémolière, 2013) Frédéric Latrémolière, PhD For any two Leibniz quantum compact metric spaces ( A , L A ) and ( B , L B ) : Quantum Compact Metric Spaces dist q (( A , L A ) , ( B , L B )) � Λ ∗ (( A , L A ) , ( B , L B )) . The Gromov- Hausdorff Propinquity Moreover, if ( A , L A ) = ( C ( X ) , L X ) and ( B , L B ) = ( C ( Y ) , L Y ) The quantum Gromov-Hausdorff distance where X , Y are compact metric spaces and L X and L Y are Lipschitz The dual propinquity The Quantum seminorms, then: Propinquity Locally Compact Λ ∗ (( A , L A ) , ( B , L B )) � GH ( X , Y ) . Quantum Metric Spaces Thus the dual propinquity is an analogue of the Gromov-Hausdorff distance.
Completeness The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity The dual propinquity is complete. Frédéric Latrémolière, PhD Quantum Compact Metric Spaces The Gromov- Hausdorff Propinquity The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
Completeness The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity The dual propinquity is complete. Frédéric Latrémolière, PhD Proof. Quantum It is sufficient to work with a sequence ( A n , L n ) n ∈ N of Leibniz Compact Metric Spaces quantum compact metric spaces such that for all n ∈ N there The Gromov- exists τ n = ( D n , L n , π n , ρ n ) with: Hausdorff Propinquity The quantum ∞ Gromov-Hausdorff distance ∑ λ ( τ n ) < ∞ . The dual propinquity The Quantum n = 0 Propinquity Locally Compact For any d = ( d n ) n ∈ N ∈ ∏ n ∈ N sa ( D n ) , we set: Quantum Metric Spaces S ( d ) = sup { L n ( d n ) : n ∈ N } .
Completeness The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity The dual propinquity is complete. Frédéric Latrémolière, PhD Proof. Quantum Let Compact Metric Spaces ∀ n ∈ N The Gromov- Hausdorff ( d n ) n ∈ N ∈ ∏ L = sa ( D n ) : π n + 1 ( d n ) = ρ n ( d n + 1 ) . Propinquity The quantum n ∈ N S (( d n ) n ∈ N ) < ∞ Gromov-Hausdorff distance The dual propinquity The Quantum Let F be the C*-algebra spanned by L in ∏ n ∈ N D n and: Propinquity Locally Compact I = { ( d n ) n ∈ N ∈ F : lim n → ∞ � d n � D n = 0 } . Quantum Metric Spaces Our candidate for a limit to ( A n , L n ) n ∈ N is F / I .
Completeness The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity The dual propinquity is complete. Frédéric Latrémolière, PhD Proof. Quantum If ε > 0 and d n ∈ sa ( D n ) for some n ∈ N with L n ( d n ) < ∞ Compact Metric Spaces then we can find d = ( d m ) , ∈ N with L n ( d n ) � S ( d ) � L n ( d n ) + The Gromov- 1 2 ε and Hausdorff Propinquity The quantum Gromov-Hausdorff ∞ distance � d � F � � d n � D n + 2 ( L n ( d n ) + ε ) ∑ λ ( τ n ) . The dual propinquity The Quantum n = 0 Propinquity Locally Compact = ω n ( d n ) , If a n + 1 then there exists d n + 1 in D n + 1 Quantum with L n + 1 ( d n + 1 ) L n + 1 ( a n + 1 ) + 1 Metric Spaces 2 ε and � d n + 1 � D n + 1 � � � a n + 1 � A n + 1 + 2 ( L n + 1 ( a n + 1 ) + ε ) . Now L n + 1 ( a n + 1 ) � L n ( d n ) .
Completeness The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity The dual propinquity is complete. Frédéric Latrémolière, PhD Proof. Quantum We may use our lifting lemma to show for m ∈ N : Compact Metric Spaces the map ( d n ) n ∈ N ∈ F �→ d m ∈ D m is a *-epimorphism, The Gromov- Hausdorff the Lip-norms L m are quotient of S. Propinquity The quantum We then get two estimates: Gromov-Hausdorff distance The dual propinquity The Quantum Haus mk L n ( S ( A n + 1 ) , S ( D n )) � 2 λ ( τ n ) Propinquity Locally Compact and Quantum Metric Spaces Haus mk L n ( S ( D n ) , S ( D n + 1 )) � 2 max { λ ( τ n ) , λ ( τ n + 1 ) } .
Completeness The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity The dual propinquity is complete. Frédéric Latrémolière, PhD Proof. Quantum We need a few technical lemmas to show that: Compact Metric Spaces The Gromov- diam ( S ( F ) , mk S ) < ∞ . Hausdorff Propinquity The quantum From this, we then can prove that ( F , S ) is a Leibniz quantum Gromov-Hausdorff distance compact metric space. The dual propinquity The Quantum Propinquity Using Blaschke selection theorem and our estimates, the se- Locally quences ( S ( A n )) n ∈ N and ( S ( D n )) n ∈ N converge to some Compact Quantum weak* compact convex Z in ( S ( F ) , mk S ) . Metric Spaces
Completeness The Gromov- Theorem (Latrémolière, 2013) Hausdorff Propinquity The dual propinquity is complete. Frédéric Latrémolière, PhD Proof. Quantum We now identify Z with the state space of F / I . Last, we en- Compact Metric Spaces dow F / I with the quotient of S, which is a Lip-norm. How- The Gromov- Hausdorff ever, why is it a Leibniz Lip-norm? Propinquity This is shown by truncating sequences in F which all map to The quantum Gromov-Hausdorff the same element in F / I . distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
GPS The Gromov- Hausdorff Propinquity Quantum Compact Metric Spaces 1 Frédéric The Monge Kantorovich distance Latrémolière, PhD Compact Quantum Metric Spaces Quantum Compact The Gromov-Hausdorff Propinquity 2 Metric Spaces The quantum Gromov-Hausdorff distance The Gromov- Hausdorff The dual propinquity Propinquity The quantum The Quantum Propinquity Gromov-Hausdorff distance The dual propinquity The Quantum Locally Compact Quantum Metric Spaces 3 Propinquity Locally Topographies Compact Quantum Convergence for locally compact quantum metric Metric Spaces spaces
Bridges and a new distance The Gromov- Hausdorff Propinquity For any two Leibniz quantum compact metric spaces, a Frédéric bridge γ = ( D , ω , ρ A , ρ B ) provides the ingredients for a Latrémolière, PhD tunnel, if we can find λ > 0 such that: Quantum Compact � L A ( a ) , L B ( b ) , 1 � Metric Spaces a , b �→ max λ � ρ 1 ( a ) ω − ωρ 2 ( b ) � D The Gromov- Hausdorff Propinquity is admissible, and in particular, defines a tunnel. The quantum Gromov-Hausdorff distance The dual propinquity The Quantum Propinquity Locally Compact Quantum Metric Spaces
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