Hecke algebras on homogeneous trees and relation with Hankel and - PowerPoint PPT Presentation

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on homogeneous trees Composition rules for χ n ’s X – homogeneous tree of degree deg ( X ) = q + 1 χ 0 ◦ χ n = χ n , χ 1 ◦ χ 1 = χ 2 + ( q + 1) χ 0 , χ 1 ◦ χ n = χ n +1 + q χ n − 1 Definition of the Hecke algebra H ( X ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on homogeneous trees Composition rules for χ n ’s X – homogeneous tree of degree deg ( X ) = q + 1 χ 0 ◦ χ n = χ n , χ 1 ◦ χ 1 = χ 2 + ( q + 1) χ 0 , χ 1 ◦ χ n = χ n +1 + q χ n − 1 Definition of the Hecke algebra H ( X ) The Hecke algebra H ( X ) on the homogeneous tree X is the composition algebra generated by the kernels { χ n : n = 0 , 1 , 2 , . . . } . Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on homogeneous trees Composition rules for χ n ’s X – homogeneous tree of degree deg ( X ) = q + 1 χ 0 ◦ χ n = χ n , χ 1 ◦ χ 1 = χ 2 + ( q + 1) χ 0 , χ 1 ◦ χ n = χ n +1 + q χ n − 1 Definition of the Hecke algebra H ( X ) The Hecke algebra H ( X ) on the homogeneous tree X is the composition algebra generated by the kernels { χ n : n = 0 , 1 , 2 , . . . } . Remark: H ( X ) is generated by χ 1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Theorem 1 If deg ( X ) ≥ 3 then H ( X ) ⊂ F ( X ) is a maximal abelian subalgebra. Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Theorem 1 If deg ( X ) ≥ 3 then H ( X ) ⊂ F ( X ) is a maximal abelian subalgebra. Proof: ϕ ∈ F ( X ) To show: ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇒ ϕ ∈ H ( X ). Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Theorem 1 If deg ( X ) ≥ 3 then H ( X ) ⊂ F ( X ) is a maximal abelian subalgebra. Proof: ϕ ∈ F ( X ) To show: ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇒ ϕ ∈ H ( X ). Equivalently, if, for all x , y ∈ V � � ϕ ( x , z ) = ϕ ( w , y ) z ∈ V , d ( z , y )=1 w ∈ V , d ( w , x )=1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Theorem 1 If deg ( X ) ≥ 3 then H ( X ) ⊂ F ( X ) is a maximal abelian subalgebra. Proof: ϕ ∈ F ( X ) To show: ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇒ ϕ ∈ H ( X ). Equivalently, if, for all x , y ∈ V � � ϕ ( x , z ) = ϕ ( w , y ) z ∈ V , d ( z , y )=1 w ∈ V , d ( w , x )=1 then N � ϕ = a n χ n . n =0 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Take arbitrary x , y ∈ V with d ( x , y ) = m Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Take arbitrary x , y ∈ V with d ( x , y ) = m and consider the unique path connecting them, Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Take arbitrary x , y ∈ V with d ( x , y ) = m and consider the unique path connecting them, i.e. a sequence of distinct edges ( x = x 0 , x 1 , . . . , x m − 1 , x m = y ) , where d ( x j , x j +1 ) = 1 . Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Take arbitrary x , y ∈ V with d ( x , y ) = m and consider the unique path connecting them, i.e. a sequence of distinct edges ( x = x 0 , x 1 , . . . , x m − 1 , x m = y ) , where d ( x j , x j +1 ) = 1 . Then � ( ϕ ◦ χ 1 )( x , y ) = ϕ ( x , z ) z ∈ V , d ( z , y )=1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Take arbitrary x , y ∈ V with d ( x , y ) = m and consider the unique path connecting them, i.e. a sequence of distinct edges ( x = x 0 , x 1 , . . . , x m − 1 , x m = y ) , where d ( x j , x j +1 ) = 1 . Then � ( ϕ ◦ χ 1 )( x , y ) = ϕ ( x , z ) = ϕ ( x , x m − 1 ) , z ∈ V , d ( z , y )=1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Take arbitrary x , y ∈ V with d ( x , y ) = m and consider the unique path connecting them, i.e. a sequence of distinct edges ( x = x 0 , x 1 , . . . , x m − 1 , x m = y ) , where d ( x j , x j +1 ) = 1 . Then � ( ϕ ◦ χ 1 )( x , y ) = ϕ ( x , z ) = ϕ ( x , x m − 1 ) , z ∈ V , d ( z , y )=1 � ( χ 1 ◦ ϕ )( x , y ) = ϕ ( w , y ) w ∈ V , d ( w , x )=1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation ϕ ∈ F ( X ), so there is m ∈ N such that ϕ ( x , y ) = 0 if d ( x , y ) ≥ m . Take arbitrary x , y ∈ V with d ( x , y ) = m and consider the unique path connecting them, i.e. a sequence of distinct edges ( x = x 0 , x 1 , . . . , x m − 1 , x m = y ) , where d ( x j , x j +1 ) = 1 . Then � ( ϕ ◦ χ 1 )( x , y ) = ϕ ( x , z ) = ϕ ( x , x m − 1 ) , z ∈ V , d ( z , y )=1 � ( χ 1 ◦ ϕ )( x , y ) = ϕ ( w , y ) = ϕ ( x 1 , y ) w ∈ V , d ( w , x )=1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ( x , x m − 1 ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ϕ ( x , x m − 1 ) → Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ϕ ( x , x m − 1 ) → ( x 1 , y ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ϕ ( x , x m − 1 ) → ( x 1 , y ) Induction: ( x 1 , y ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ϕ ( x , x m − 1 ) → ( x 1 , y ) Induction: ϕ ( x 1 , y ) → ( y , y m − 1 ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ϕ ( x , x m − 1 ) → ( x 1 , y ) Induction: ϕ ϕ ( x 1 , y ) → ( y , y m − 1 ) → ( z m − 1 , y ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ϕ ( x , x m − 1 ) → ( x 1 , y ) Induction: ϕ ϕ ϕ ( x 1 , y ) → ( y , y m − 1 ) → ( z m − 1 , y ) → ( y , x 1 ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ◦ χ 1 = χ 1 ◦ ϕ is equivalent to ϕ ( x , x m − 1 ) = ϕ ( x 1 , y ) for all x , y ∈ V ( ∗ ) Geometric interpretation: ϕ ( x , x m − 1 ) → ( x 1 , y ) Induction: ϕ ϕ ϕ ( x 1 , y ) → ( y , y m − 1 ) → ( z m − 1 , y ) → ( y , x 1 ) x m − 1 , z 1 – distinct neighbours of y Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ( x 1 , y ) = ϕ ( y , x 1 ) for any x 1 , y ∈ V , with d ( x 1 , y ) = m − 1, Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ( x 1 , y ) = ϕ ( y , x 1 ) for any x 1 , y ∈ V , with d ( x 1 , y ) = m − 1, so the kernel ϕ is symmetric for such pairs. Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ( x 1 , y ) = ϕ ( y , x 1 ) for any x 1 , y ∈ V , with d ( x 1 , y ) = m − 1, so the kernel ϕ is symmetric for such pairs. Moreover, ϕ ( x 1 , y ) = ϕ ( u , v ) for any u , v ∈ V with d ( u , v ) = m − 1. Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is Maximal Abelian SubAlgebra Proof - continuation Hence ϕ ( x 1 , y ) = ϕ ( y , x 1 ) for any x 1 , y ∈ V , with d ( x 1 , y ) = m − 1, so the kernel ϕ is symmetric for such pairs. Moreover, ϕ ( x 1 , y ) = ϕ ( u , v ) for any u , v ∈ V with d ( u , v ) = m − 1. Hence is constant on such pairs. ϕ Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is MASA in F 1 ( X ) Assume: deg ( X ) = q + 1 ≥ 3 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is MASA in F 1 ( X ) Assume: deg ( X ) = q + 1 ≥ 3 completion F 1 ( X ) of F ( X ) in: � � � � � ϕ � 1 = inf C > 0 : sup | ϕ ( x , y ) | ≤ C , sup | ϕ ( x , y ) | ≤ C x y y x Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is MASA in F 1 ( X ) Assume: deg ( X ) = q + 1 ≥ 3 completion F 1 ( X ) of F ( X ) in: � � � � � ϕ � 1 = inf C > 0 : sup | ϕ ( x , y ) | ≤ C , sup | ϕ ( x , y ) | ≤ C x y y x Remark F 1 ( X ) is a subalgebra in [ l 1 ( X ) → l 1 ( X )] ∩ [ l ∞ ( X ) → l ∞ ( X )] . Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is MASA in F 1 ( X ) Assume: deg ( X ) = q + 1 ≥ 3 completion F 1 ( X ) of F ( X ) in: � � � � � ϕ � 1 = inf C > 0 : sup | ϕ ( x , y ) | ≤ C , sup | ϕ ( x , y ) | ≤ C x y y x Remark F 1 ( X ) is a subalgebra in [ l 1 ( X ) → l 1 ( X )] ∩ [ l ∞ ( X ) → l ∞ ( X )] . Definition H 1 ( X ) – completion of the Hecke algebra H ( X ) in F 1 ( X ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is MASA in F 1 ( X ) ϕ ∈ F 1 ( X ) if and only if   �  − sup | ϕ ( x , y ) | − − → 0  n x y ∈ V , d ( x , y ) ≥ n Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is MASA in F 1 ( X ) ϕ ∈ F 1 ( X ) if and only if   �  − sup | ϕ ( x , y ) | − − → 0  n x y ∈ V , d ( x , y ) ≥ n and   �  − sup | ϕ ( x , y ) | − − → 0  n y x ∈ V , d ( x , y ) ≥ n Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra is MASA in F 1 ( X ) ϕ ∈ F 1 ( X ) if and only if   �  − sup | ϕ ( x , y ) | − − → 0  n x y ∈ V , d ( x , y ) ≥ n and   �  − sup | ϕ ( x , y ) | − − → 0  n y x ∈ V , d ( x , y ) ≥ n Theorem H 1 ( X ) is a maximal abelian subalgebra in F 1 ( X ). Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on integers Z Hecke algebra H ( Z ) The generator χ 1 is defined by � 1 if | j − k | = 1 χ 1 ( j , k ) = 0 otherwise Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on integers Z Hecke algebra H ( Z ) The generator χ 1 is defined by � 1 if | j − k | = 1 χ 1 ( j , k ) = 0 otherwise Commuting with χ 1 If a = ( a j , k ) j , k ∈ Z , then ( a ◦ χ 1 )( j , k ) = a j , k − 1 + a j , k +1 , Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on integers Z Hecke algebra H ( Z ) The generator χ 1 is defined by � 1 if | j − k | = 1 χ 1 ( j , k ) = 0 otherwise Commuting with χ 1 If a = ( a j , k ) j , k ∈ Z , then ( a ◦ χ 1 )( j , k ) = a j , k − 1 + a j , k +1 , ( χ 1 ◦ a )( j , k ) = a j − 1 , k + a j +1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on integers Z Hecke algebra H ( Z ) The generator χ 1 is defined by � 1 if | j − k | = 1 χ 1 ( j , k ) = 0 otherwise Commuting with χ 1 If a = ( a j , k ) j , k ∈ Z , then ( a ◦ χ 1 )( j , k ) = a j , k − 1 + a j , k +1 , ( χ 1 ◦ a )( j , k ) = a j − 1 , k + a j +1 , k Hence a ◦ χ 1 = χ 1 ◦ a Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on integers Z Hecke algebra H ( Z ) The generator χ 1 is defined by � 1 if | j − k | = 1 χ 1 ( j , k ) = 0 otherwise Commuting with χ 1 If a = ( a j , k ) j , k ∈ Z , then ( a ◦ χ 1 )( j , k ) = a j , k − 1 + a j , k +1 , ( χ 1 ◦ a )( j , k ) = a j − 1 , k + a j +1 , k Hence a ◦ χ 1 = χ 1 ◦ a ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k for all j , k ∈ Z . Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Hankel operators The Hankel operator h = ( h j , k ) j , k ∈ Z depends only on the sum of indexes: Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Hankel operators The Hankel operator h = ( h j , k ) j , k ∈ Z depends only on the sum of indexes: there exists a sequence u = ( u ( j )) j ∈ Z such that h j , k = u ( k + j ) , i. e. h j − 1 , k +1 = h j +1 , k − 1 = h j , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Hankel operators The Hankel operator h = ( h j , k ) j , k ∈ Z depends only on the sum of indexes: there exists a sequence u = ( u ( j )) j ∈ Z such that h j , k = u ( k + j ) , i. e. h j − 1 , k +1 = h j +1 , k − 1 = h j , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Hankel operators The Hankel operator h = ( h j , k ) j , k ∈ Z depends only on the sum of indexes: there exists a sequence u = ( u ( j )) j ∈ Z such that h j , k = u ( k + j ) , i. e. h j − 1 , k +1 = h j +1 , k − 1 = h j , k Then h ◦ χ 1 = χ 1 ◦ h , Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Hankel operators The Hankel operator h = ( h j , k ) j , k ∈ Z depends only on the sum of indexes: there exists a sequence u = ( u ( j )) j ∈ Z such that h j , k = u ( k + j ) , i. e. h j − 1 , k +1 = h j +1 , k − 1 = h j , k Then h ◦ χ 1 = χ 1 ◦ h , since h j , k − 1 + h j , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Hankel operators The Hankel operator h = ( h j , k ) j , k ∈ Z depends only on the sum of indexes: there exists a sequence u = ( u ( j )) j ∈ Z such that h j , k = u ( k + j ) , i. e. h j − 1 , k +1 = h j +1 , k − 1 = h j , k Then h ◦ χ 1 = χ 1 ◦ h , since h j , k − 1 + h j , k +1 = u ( k − 1 + j ) + u ( k + 1 + j ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Hankel operators The Hankel operator h = ( h j , k ) j , k ∈ Z depends only on the sum of indexes: there exists a sequence u = ( u ( j )) j ∈ Z such that h j , k = u ( k + j ) , i. e. h j − 1 , k +1 = h j +1 , k − 1 = h j , k Then h ◦ χ 1 = χ 1 ◦ h , since h j , k − 1 + h j , k +1 = u ( k − 1 + j ) + u ( k + 1 + j ) = h j − 1 , k + h j +1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Toeplitz operators The Toeplitz operator t = ( t j , k ) j , k ∈ Z depends only on the difference of indexes: Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Toeplitz operators The Toeplitz operator t = ( t j , k ) j , k ∈ Z depends only on the difference of indexes: there exists a sequence v = ( v ( j )) j ∈ Z such that t j , k = v ( k − j ) , i. e. t j − 1 , k − 1 = t j , k = t j +1 , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Toeplitz operators The Toeplitz operator t = ( t j , k ) j , k ∈ Z depends only on the difference of indexes: there exists a sequence v = ( v ( j )) j ∈ Z such that t j , k = v ( k − j ) , i. e. t j − 1 , k − 1 = t j , k = t j +1 , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Toeplitz operators The Toeplitz operator t = ( t j , k ) j , k ∈ Z depends only on the difference of indexes: there exists a sequence v = ( v ( j )) j ∈ Z such that t j , k = v ( k − j ) , i. e. t j − 1 , k − 1 = t j , k = t j +1 , k +1 Then t ◦ χ 1 = χ 1 ◦ t , Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Toeplitz operators The Toeplitz operator t = ( t j , k ) j , k ∈ Z depends only on the difference of indexes: there exists a sequence v = ( v ( j )) j ∈ Z such that t j , k = v ( k − j ) , i. e. t j − 1 , k − 1 = t j , k = t j +1 , k +1 Then t ◦ χ 1 = χ 1 ◦ t , since t j , k − 1 + t j , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Toeplitz operators The Toeplitz operator t = ( t j , k ) j , k ∈ Z depends only on the difference of indexes: there exists a sequence v = ( v ( j )) j ∈ Z such that t j , k = v ( k − j ) , i. e. t j − 1 , k − 1 = t j , k = t j +1 , k +1 Then t ◦ χ 1 = χ 1 ◦ t , since t j , k − 1 + t j , k +1 = v ( k − 1 − j ) + v ( k + 1 − j ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hankel and Toeplitz operators Toeplitz operators The Toeplitz operator t = ( t j , k ) j , k ∈ Z depends only on the difference of indexes: there exists a sequence v = ( v ( j )) j ∈ Z such that t j , k = v ( k − j ) , i. e. t j − 1 , k − 1 = t j , k = t j +1 , k +1 Then t ◦ χ 1 = χ 1 ◦ t , since t j , k − 1 + t j , k +1 = v ( k − 1 − j ) + v ( k + 1 − j ) = t j − 1 , k + t j +1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Notation ◮ F 0 ( Z ) := B ( l 1 ( Z )) ∩ B ( l ∞ ( Z )) - bounded Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Notation ◮ F 0 ( Z ) := B ( l 1 ( Z )) ∩ B ( l ∞ ( Z )) - bounded ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hankel operators Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Notation ◮ F 0 ( Z ) := B ( l 1 ( Z )) ∩ B ( l ∞ ( Z )) - bounded ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hankel operators ◮ T 0 ( Z ) ⊂ F 0 ( Z ) – Toeplitz operators Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Notation ◮ F 0 ( Z ) := B ( l 1 ( Z )) ∩ B ( l ∞ ( Z )) - bounded ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hankel operators ◮ T 0 ( Z ) ⊂ F 0 ( Z ) – Toeplitz operators ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hecke algebra (distance depending kernels) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Notation ◮ F 0 ( Z ) := B ( l 1 ( Z )) ∩ B ( l ∞ ( Z )) - bounded ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hankel operators ◮ T 0 ( Z ) ⊂ F 0 ( Z ) – Toeplitz operators ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hecke algebra (distance depending kernels) ◮ H 0 ( Z ) ′ ⊂ F 0 ( Z ) – the commutant of H 0 ( Z ) in F 0 ( Z ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Notation ◮ F 0 ( Z ) := B ( l 1 ( Z )) ∩ B ( l ∞ ( Z )) - bounded ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hankel operators ◮ T 0 ( Z ) ⊂ F 0 ( Z ) – Toeplitz operators ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hecke algebra (distance depending kernels) ◮ H 0 ( Z ) ′ ⊂ F 0 ( Z ) – the commutant of H 0 ( Z ) in F 0 ( Z ) Remark Hecke ⇒ Toeplitz Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Notation ◮ F 0 ( Z ) := B ( l 1 ( Z )) ∩ B ( l ∞ ( Z )) - bounded ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hankel operators ◮ T 0 ( Z ) ⊂ F 0 ( Z ) – Toeplitz operators ◮ H 0 ( Z ) ⊂ F 0 ( Z ) – Hecke algebra (distance depending kernels) ◮ H 0 ( Z ) ′ ⊂ F 0 ( Z ) – the commutant of H 0 ( Z ) in F 0 ( Z ) Remark Hecke ⇒ Toeplitz H 0 ( Z ) ⊂ T 0 ( Z ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Theorem The Hecke algebra H 0 ( Z ) is not maximal abelian. Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Theorem The Hecke algebra H 0 ( Z ) is not maximal abelian. The commutant H 0 ( Z ) ′ in F 0 ( Z ) is a direct sum of the Hankel and Toeplitz operators: Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z Theorem The Hecke algebra H 0 ( Z ) is not maximal abelian. The commutant H 0 ( Z ) ′ in F 0 ( Z ) is a direct sum of the Hankel and Toeplitz operators: H 0 ( Z ) ′ = H 0 ( Z ) ⊕ T 0 ( Z ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 a j , k +1   a j − 1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 + a j , k +1   a j − 1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 + a j , k +1   a j − 1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 a j , k +1   a j − 1 , k Toeplitz: t j , k := a j , k − 1 − a j − 1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 a j , k +1   a j − 1 , k Toeplitz: t j , k := a j , k − 1 − a j − 1 , k = a j +1 , k − a j , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 a j , k +1   a j − 1 , k Toeplitz: t j , k := a j , k − 1 − a j − 1 , k = a j +1 , k − a j , k +1 = t j +1 , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 a j , k +1   a j − 1 , k Toeplitz: t j , k := a j , k − 1 − a j − 1 , k = a j +1 , k − a j , k +1 = t j +1 , k +1 Hankel: h j , k := a j , k − 1 − a j +1 , k Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 a j , k +1   a j − 1 , k Toeplitz: t j , k := a j , k − 1 − a j − 1 , k = a j +1 , k − a j , k +1 = t j +1 , k +1 Hankel: h j , k := a j , k − 1 − a j +1 , k = a j − 1 , k − a j , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( ϕ = ( a j , k )) Commutation with χ 1 ϕ ◦ χ 1 = χ 1 ◦ ϕ ⇔ a j , k − 1 + a j , k +1 = a j − 1 , k + a j +1 , k   a j +1 , k a j , k − 1 a j , k +1   a j − 1 , k Toeplitz: t j , k := a j , k − 1 − a j − 1 , k = a j +1 , k − a j , k +1 = t j +1 , k +1 Hankel: h j , k := a j , k − 1 − a j +1 , k = a j − 1 , k − a j , k +1 = h j − 1 , k +1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( n ≥ 0) Hankel: a k , k − a k +1 , k +1 = a 2 k , 0 − a 2 k +1 , 1 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( n ≥ 0) Hankel: a k , k − a k +1 , k +1 = a 2 k , 0 − a 2 k +1 , 1 a 0 , 0 − a n , n = ( a 0 , 0 − a 1 , 1 ) + ( a 1 , 1 − a 2 , 2 ) + · · · + ( a n − 1 , n − 1 − a n , n ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( n ≥ 0) Hankel: a k , k − a k +1 , k +1 = a 2 k , 0 − a 2 k +1 , 1 a 0 , 0 − a n , n = ( a 0 , 0 − a 1 , 1 ) + ( a 1 , 1 − a 2 , 2 ) + · · · + ( a n − 1 , n − 1 − a n , n ) = ( a 0 , 0 − a 1 , 1 ) + ( a 2 , 0 − a 3 , 1 ) + · · · + ( a 2 n − 2 , 0 − a 2 n − 1 , 1 ) Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel • • • • Hecke algebra on Z – proof ( n ≥ 0) Hankel: a k , k − a k +1 , k +1 = a 2 k , 0 − a 2 k +1 , 1 a 0 , 0 − a n , n = ( a 0 , 0 − a 1 , 1 ) + ( a 1 , 1 − a 2 , 2 ) + · · · + ( a n − 1 , n − 1 − a n , n ) = ( a 0 , 0 − a 1 , 1 ) + ( a 2 , 0 − a 3 , 1 ) + · · · + ( a 2 n − 2 , 0 − a 2 n − 1 , 1 ) Toeplitz: a k , − k − a k +1 , − ( k +1) = a 2 k +1 , 1 − a 2 k +2 , 0 Janusz Wysocza´ nski: Mathematical Institute, Wroc� law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices