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

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

Mar 11, 2024 111 likes •1.46k views

Homogeneous trees Algebraic MASA: deg ( X ) > 2 Hecke MASA: general case. Integers: Toeplitz and Hankel Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices Janusz Wysocza nski

slide-1
SLIDE 1
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

Janusz Wysocza´ nski

Mathematical Institute, Wroc law University

Second Najman Conference on Spectral Problems for Operators and Matrices Dubrovnik, May 10–17, 2009

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-2
SLIDE 2
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Outline

  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-3
SLIDE 3
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Homogeneous trees

Tree

A tree X = (V , E) is a connected graph with no loops, where V is the set of vertices, and E is the set of edges.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-4
SLIDE 4
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Homogeneous trees

Tree

A tree X = (V , E) is a connected graph with no loops, where V is the set of vertices, and E is the set of edges.

Homogeneous tree

A tree is called homogeneous if there exists a positive integer q, such that in each vertex there is exactly q edges: |{y ∈ V : {x, y} ∈ E}| = q for every x ∈ V

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-5
SLIDE 5
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Homogeneous trees

Tree

A tree X = (V , E) is a connected graph with no loops, where V is the set of vertices, and E is the set of edges.

Homogeneous tree

A tree is called homogeneous if there exists a positive integer q, such that in each vertex there is exactly q edges: |{y ∈ V : {x, y} ∈ E}| = q for every x ∈ V The number q is called the (homogeneity) degree of the tree (notation: q = deg(X)).

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-6
SLIDE 6
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Homogeneous trees

Tree

A tree X = (V , E) is a connected graph with no loops, where V is the set of vertices, and E is the set of edges.

Homogeneous tree

A tree is called homogeneous if there exists a positive integer q, such that in each vertex there is exactly q edges: |{y ∈ V : {x, y} ∈ E}| = q for every x ∈ V The number q is called the (homogeneity) degree of the tree (notation: q = deg(X)).

Tree of integers Z

The group Z of integers can be represented as a homogeneous tree of degree q = 2: . . .

−2

  • −1
  • 1
  • 2
  • . . .

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-7
SLIDE 7
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

F-kernels F(X). Definition

ϕ ∈ F(X) if:

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-8
SLIDE 8
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

F-kernels F(X). Definition

ϕ ∈ F(X) if:

◮ ϕ : V × V → C

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-9
SLIDE 9
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

F-kernels F(X). Definition

ϕ ∈ F(X) if:

◮ ϕ : V × V → C ◮ ∃m ∈ N :

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

slide-10
SLIDE 10
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

F-kernels F(X). Definition

ϕ ∈ F(X) if:

◮ ϕ : V × V → C ◮ ∃m ∈ N :

ϕ(x, y) = 0 if d (x, y) ≥ m.

◮ multiplication ϕ, ψ ∈ F(X)

(ϕ ◦ ψ)(x, y) =

  • z∈V

ϕ(x, z) · ψ(z, y)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-11
SLIDE 11
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

F-kernels F(X). Definition

ϕ ∈ F(X) if:

◮ ϕ : V × V → C ◮ ∃m ∈ N :

ϕ(x, y) = 0 if d (x, y) ≥ m.

◮ multiplication ϕ, ψ ∈ F(X)

(ϕ ◦ ψ)(x, y) =

  • z∈V

ϕ(x, z) · ψ(z, y)

Remark

F(X) is a unital algebra.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-12
SLIDE 12
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

The operator algebra F(X)

ϕ ∈ F(X) acts as kernel of an operator: ϕ(f )(x) =

  • y∈V

ϕ(x, y)f (y) if f : V → C is finitely supported.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-13
SLIDE 13
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

The operator algebra F(X)

ϕ ∈ F(X) acts as kernel of an operator: ϕ(f )(x) =

  • y∈V

ϕ(x, y)f (y) if f : V → C is finitely supported. This can be extended to any lp(X), with p ≥ 1.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-14
SLIDE 14
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

The operator algebra F(X)

ϕ ∈ F(X) acts as kernel of an operator: ϕ(f )(x) =

  • y∈V

ϕ(x, y)f (y) if f : V → C is finitely supported. This can be extended to any lp(X), with p ≥ 1.

Distant-dependent kernels

For each n ∈ N we define a kernel χn as: χn(x, y) = 1 if d (x, y) = n

  • therwise

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-15
SLIDE 15
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

Composition rules for χn’s

X – homogeneous tree of degree deg(X) = q + 1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-16
SLIDE 16
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

Composition rules for χn’s

X – homogeneous tree of degree deg(X) = q + 1 χ0 ◦ χn = χn,

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-17
SLIDE 17
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

Composition rules for χn’s

X – homogeneous tree of degree deg(X) = q + 1 χ0 ◦ χn = χn, χ1 ◦ χ1 = χ2 + (q + 1)χ0,

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-18
SLIDE 18
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

Composition rules for χn’s

X – homogeneous tree of degree deg(X) = q + 1 χ0 ◦ χn = χn, χ1 ◦ χ1 = χ2 + (q + 1)χ0, χ1 ◦ χn = χn+1 + qχn−1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-19
SLIDE 19
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

Composition rules for χn’s

X – homogeneous tree of degree deg(X) = q + 1 χ0 ◦ χn = χn, χ1 ◦ χ1 = χ2 + (q + 1)χ0, χ1 ◦ χn = χn+1 + qχn−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

slide-20
SLIDE 20
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

Composition rules for χn’s

X – homogeneous tree of degree deg(X) = q + 1 χ0 ◦ χn = χn, χ1 ◦ χ1 = χ2 + (q + 1)χ0, χ1 ◦ χn = χn+1 + qχn−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 {χn : n = 0, 1, 2, . . .}.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-21
SLIDE 21
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on homogeneous trees

Composition rules for χn’s

X – homogeneous tree of degree deg(X) = q + 1 χ0 ◦ χn = χn, χ1 ◦ χ1 = χ2 + (q + 1)χ0, χ1 ◦ χn = χn+1 + qχn−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 {χn : n = 0, 1, 2, . . .}.

Remark: H(X) is generated by χ1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-22
SLIDE 22
  • 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

slide-23
SLIDE 23
  • 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 ◦ χ1 = χ1 ◦ ϕ ⇒ ϕ ∈ H(X).

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-24
SLIDE 24
  • 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 ◦ χ1 = χ1 ◦ ϕ ⇒ ϕ ∈ H(X). Equivalently, if, for all x, y ∈ V

  • z∈V , d (z,y)=1

ϕ(x, z) =

  • w∈V , d (w,x)=1

ϕ(w, y)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-25
SLIDE 25
  • 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 ◦ χ1 = χ1 ◦ ϕ ⇒ ϕ ∈ H(X). Equivalently, if, for all x, y ∈ V

  • z∈V , d (z,y)=1

ϕ(x, z) =

  • w∈V , d (w,x)=1

ϕ(w, y) then ϕ =

N

  • n=0

anχn.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-26
SLIDE 26
  • 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

slide-27
SLIDE 27
  • 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

slide-28
SLIDE 28
  • 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

slide-29
SLIDE 29
  • 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

slide-30
SLIDE 30
  • 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 = x0, x1, . . . , xm−1, xm = y), where d (xj, xj+1) = 1.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-31
SLIDE 31
  • 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 = x0, x1, . . . , xm−1, xm = y), where d (xj, xj+1) = 1. Then (ϕ ◦ χ1)(x, y) =

  • z∈V , d (z,y)=1

ϕ(x, z)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-32
SLIDE 32
  • 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 = x0, x1, . . . , xm−1, xm = y), where d (xj, xj+1) = 1. Then (ϕ ◦ χ1)(x, y) =

  • z∈V , d (z,y)=1

ϕ(x, z) = ϕ(x, xm−1),

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-33
SLIDE 33
  • 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 = x0, x1, . . . , xm−1, xm = y), where d (xj, xj+1) = 1. Then (ϕ ◦ χ1)(x, y) =

  • z∈V , d (z,y)=1

ϕ(x, z) = ϕ(x, xm−1), (χ1 ◦ ϕ)(x, y) =

  • w∈V , d (w,x)=1

ϕ(w, y)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-34
SLIDE 34
  • 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 = x0, x1, . . . , xm−1, xm = y), where d (xj, xj+1) = 1. Then (ϕ ◦ χ1)(x, y) =

  • z∈V , d (z,y)=1

ϕ(x, z) = ϕ(x, xm−1), (χ1 ◦ ϕ)(x, y) =

  • w∈V , d (w,x)=1

ϕ(w, y) = ϕ(x1, y)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-35
SLIDE 35
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-36
SLIDE 36
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, 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

slide-37
SLIDE 37
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, 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

slide-38
SLIDE 38
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-39
SLIDE 39
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

ϕ

→

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-40
SLIDE 40
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

ϕ

→ (x1, y)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-41
SLIDE 41
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

ϕ

→ (x1, y) Induction: (x1, y)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-42
SLIDE 42
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

ϕ

→ (x1, y) Induction: (x1, y)

ϕ

→ (y, ym−1)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-43
SLIDE 43
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

ϕ

→ (x1, y) Induction: (x1, y)

ϕ

→ (y, ym−1)

ϕ

→ (zm−1, y)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-44
SLIDE 44
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

ϕ

→ (x1, y) Induction: (x1, y)

ϕ

→ (y, ym−1)

ϕ

→ (zm−1, y)

ϕ

→ (y, x1)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-45
SLIDE 45
  • 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 ◦ χ1 = χ1 ◦ ϕ is equivalent to ϕ(x, xm−1) = ϕ(x1, y) for all x, y ∈ V (∗) Geometric interpretation: (x, xm−1)

ϕ

→ (x1, y) Induction: (x1, y)

ϕ

→ (y, ym−1)

ϕ

→ (zm−1, y)

ϕ

→ (y, x1) xm−1, z1 – distinct neighbours of y

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-46
SLIDE 46
  • 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(x1, y) = ϕ(y, x1) for any x1, y ∈ V , with d (x1, y) = m − 1,

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-47
SLIDE 47
  • 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(x1, y) = ϕ(y, x1) for any x1, y ∈ V , with d (x1, 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

slide-48
SLIDE 48
  • 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(x1, y) = ϕ(y, x1) for any x1, y ∈ V , with d (x1, y) = m − 1, so the kernel ϕ is symmetric for such pairs. Moreover, ϕ(x1, 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

slide-49
SLIDE 49
  • 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(x1, y) = ϕ(y, x1) for any x1, y ∈ V , with d (x1, y) = m − 1, so the kernel ϕ is symmetric for such pairs. Moreover, ϕ(x1, y) = ϕ(u, v) for any u, v ∈ V with d (u, v) = m − 1. Hence ϕ is constant on such pairs.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-50
SLIDE 50
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra is MASA in F1(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

slide-51
SLIDE 51
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra is MASA in F1(X)

Assume: deg(X) = q + 1 ≥ 3 completion F1(X) of F(X) in: ϕ1 = inf

  • C > 0 : sup

x

  • y

|ϕ(x, y)| ≤ C, sup

y

  • x

|ϕ(x, y)| ≤ C

  • Janusz Wysocza´

nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-52
SLIDE 52
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra is MASA in F1(X)

Assume: deg(X) = q + 1 ≥ 3 completion F1(X) of F(X) in: ϕ1 = inf

  • C > 0 : sup

x

  • y

|ϕ(x, y)| ≤ C, sup

y

  • x

|ϕ(x, y)| ≤ C

  • Remark

F1(X) is a subalgebra in [l1(X) → l1(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

slide-53
SLIDE 53
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra is MASA in F1(X)

Assume: deg(X) = q + 1 ≥ 3 completion F1(X) of F(X) in: ϕ1 = inf

  • C > 0 : sup

x

  • y

|ϕ(x, y)| ≤ C, sup

y

  • x

|ϕ(x, y)| ≤ C

  • Remark

F1(X) is a subalgebra in [l1(X) → l1(X)] ∩ [l∞(X) → l∞(X)] .

Definition

H1(X) – completion of the Hecke algebra H(X) in F1(X)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-54
SLIDE 54
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra is MASA in F1(X)

ϕ ∈ F1(X) if and only if sup

x

 

  • y∈V , d (x,y)≥n

|ϕ(x, y)|   − − − →

n

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-55
SLIDE 55
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra is MASA in F1(X)

ϕ ∈ F1(X) if and only if sup

x

 

  • y∈V , d (x,y)≥n

|ϕ(x, y)|   − − − →

n

and sup

y

 

  • x∈V , d (x,y)≥n

|ϕ(x, y)|   − − − →

n

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-56
SLIDE 56
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra is MASA in F1(X)

ϕ ∈ F1(X) if and only if sup

x

 

  • y∈V , d (x,y)≥n

|ϕ(x, y)|   − − − →

n

and sup

y

 

  • x∈V , d (x,y)≥n

|ϕ(x, y)|   − − − →

n

Theorem

H1(X) is a maximal abelian subalgebra in F1(X).

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-57
SLIDE 57
  • 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 χ1 is defined by χ1(j, k) = 1 if |j − k| = 1

  • therwise

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-58
SLIDE 58
  • 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 χ1 is defined by χ1(j, k) = 1 if |j − k| = 1

  • therwise

Commuting with χ1

If a = (aj,k)j,k∈Z, then (a ◦ χ1)(j, k) = aj,k−1 + aj,k+1,

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-59
SLIDE 59
  • 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 χ1 is defined by χ1(j, k) = 1 if |j − k| = 1

  • therwise

Commuting with χ1

If a = (aj,k)j,k∈Z, then (a ◦ χ1)(j, k) = aj,k−1 + aj,k+1, (χ1 ◦ a)(j, k) = aj−1,k + aj+1,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-60
SLIDE 60
  • 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 χ1 is defined by χ1(j, k) = 1 if |j − k| = 1

  • therwise

Commuting with χ1

If a = (aj,k)j,k∈Z, then (a ◦ χ1)(j, k) = aj,k−1 + aj,k+1, (χ1 ◦ a)(j, k) = aj−1,k + aj+1,k Hence a ◦ χ1 = χ1 ◦ a

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-61
SLIDE 61
  • 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 χ1 is defined by χ1(j, k) = 1 if |j − k| = 1

  • therwise

Commuting with χ1

If a = (aj,k)j,k∈Z, then (a ◦ χ1)(j, k) = aj,k−1 + aj,k+1, (χ1 ◦ a)(j, k) = aj−1,k + aj+1,k Hence a ◦ χ1 = χ1 ◦ a ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+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

slide-62
SLIDE 62
  • 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 = (hj,k)j,k∈Z depends only on the sum

  • f indexes:

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-63
SLIDE 63
  • 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 = (hj,k)j,k∈Z depends only on the sum

  • f indexes: there exists a sequence u = (u(j))j∈Z such that

hj,k = u(k + j),

  • i. e.

hj−1,k+1 = hj+1,k−1 = hj,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-64
SLIDE 64
  • 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 = (hj,k)j,k∈Z depends only on the sum

  • f indexes: there exists a sequence u = (u(j))j∈Z such that

hj,k = u(k + j),

  • i. e.

hj−1,k+1 = hj+1,k−1 = hj,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-65
SLIDE 65
  • 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 = (hj,k)j,k∈Z depends only on the sum

  • f indexes: there exists a sequence u = (u(j))j∈Z such that

hj,k = u(k + j),

  • i. e.

hj−1,k+1 = hj+1,k−1 = hj,k Then h ◦ χ1 = χ1 ◦ h,

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-66
SLIDE 66
  • 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 = (hj,k)j,k∈Z depends only on the sum

  • f indexes: there exists a sequence u = (u(j))j∈Z such that

hj,k = u(k + j),

  • i. e.

hj−1,k+1 = hj+1,k−1 = hj,k Then h ◦ χ1 = χ1 ◦ h, since hj,k−1 + hj,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-67
SLIDE 67
  • 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 = (hj,k)j,k∈Z depends only on the sum

  • f indexes: there exists a sequence u = (u(j))j∈Z such that

hj,k = u(k + j),

  • i. e.

hj−1,k+1 = hj+1,k−1 = hj,k Then h ◦ χ1 = χ1 ◦ h, since hj,k−1 + hj,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

slide-68
SLIDE 68
  • 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 = (hj,k)j,k∈Z depends only on the sum

  • f indexes: there exists a sequence u = (u(j))j∈Z such that

hj,k = u(k + j),

  • i. e.

hj−1,k+1 = hj+1,k−1 = hj,k Then h ◦ χ1 = χ1 ◦ h, since hj,k−1 + hj,k+1 = u(k − 1 + j) + u(k + 1 + j) = hj−1,k + hj+1,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-69
SLIDE 69
  • 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 = (tj,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

slide-70
SLIDE 70
  • 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 = (tj,k)j,k∈Z depends only on the difference of indexes: there exists a sequence v = (v(j))j∈Z such that tj,k = v(k − j),

  • i. e.

tj−1,k−1 = tj,k = tj+1,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-71
SLIDE 71
  • 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 = (tj,k)j,k∈Z depends only on the difference of indexes: there exists a sequence v = (v(j))j∈Z such that tj,k = v(k − j),

  • i. e.

tj−1,k−1 = tj,k = tj+1,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-72
SLIDE 72
  • 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 = (tj,k)j,k∈Z depends only on the difference of indexes: there exists a sequence v = (v(j))j∈Z such that tj,k = v(k − j),

  • i. e.

tj−1,k−1 = tj,k = tj+1,k+1 Then t ◦ χ1 = χ1 ◦ t,

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-73
SLIDE 73
  • 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 = (tj,k)j,k∈Z depends only on the difference of indexes: there exists a sequence v = (v(j))j∈Z such that tj,k = v(k − j),

  • i. e.

tj−1,k−1 = tj,k = tj+1,k+1 Then t ◦ χ1 = χ1 ◦ t, since tj,k−1 + tj,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-74
SLIDE 74
  • 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 = (tj,k)j,k∈Z depends only on the difference of indexes: there exists a sequence v = (v(j))j∈Z such that tj,k = v(k − j),

  • i. e.

tj−1,k−1 = tj,k = tj+1,k+1 Then t ◦ χ1 = χ1 ◦ t, since tj,k−1 + tj,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

slide-75
SLIDE 75
  • 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 = (tj,k)j,k∈Z depends only on the difference of indexes: there exists a sequence v = (v(j))j∈Z such that tj,k = v(k − j),

  • i. e.

tj−1,k−1 = tj,k = tj+1,k+1 Then t ◦ χ1 = χ1 ◦ t, since tj,k−1 + tj,k+1 = v(k − 1 − j) + v(k + 1 − j) = tj−1,k + tj+1,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-76
SLIDE 76
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Notation

◮ F0(Z) := B(l1(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

slide-77
SLIDE 77
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Notation

◮ F0(Z) := B(l1(Z)) ∩ B(l∞(Z)) - bounded ◮ H0(Z) ⊂ F0(Z) – Hankel operators

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-78
SLIDE 78
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Notation

◮ F0(Z) := B(l1(Z)) ∩ B(l∞(Z)) - bounded ◮ H0(Z) ⊂ F0(Z) – Hankel operators ◮ T0(Z) ⊂ F0(Z) – Toeplitz operators

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-79
SLIDE 79
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Notation

◮ F0(Z) := B(l1(Z)) ∩ B(l∞(Z)) - bounded ◮ H0(Z) ⊂ F0(Z) – Hankel operators ◮ T0(Z) ⊂ F0(Z) – Toeplitz operators ◮ H0(Z) ⊂ F0(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

slide-80
SLIDE 80
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Notation

◮ F0(Z) := B(l1(Z)) ∩ B(l∞(Z)) - bounded ◮ H0(Z) ⊂ F0(Z) – Hankel operators ◮ T0(Z) ⊂ F0(Z) – Toeplitz operators ◮ H0(Z) ⊂ F0(Z) – Hecke algebra (distance depending

kernels)

◮ H0(Z)′ ⊂ F0(Z) – the commutant of H0(Z) in F0(Z)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-81
SLIDE 81
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Notation

◮ F0(Z) := B(l1(Z)) ∩ B(l∞(Z)) - bounded ◮ H0(Z) ⊂ F0(Z) – Hankel operators ◮ T0(Z) ⊂ F0(Z) – Toeplitz operators ◮ H0(Z) ⊂ F0(Z) – Hecke algebra (distance depending

kernels)

◮ H0(Z)′ ⊂ F0(Z) – the commutant of H0(Z) in F0(Z)

Remark

Hecke ⇒ Toeplitz

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 82
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Notation

◮ F0(Z) := B(l1(Z)) ∩ B(l∞(Z)) - bounded ◮ H0(Z) ⊂ F0(Z) – Hankel operators ◮ T0(Z) ⊂ F0(Z) – Toeplitz operators ◮ H0(Z) ⊂ F0(Z) – Hecke algebra (distance depending

kernels)

◮ H0(Z)′ ⊂ F0(Z) – the commutant of H0(Z) in F0(Z)

Remark

Hecke ⇒ Toeplitz H0(Z) ⊂ T0(Z)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 83
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Theorem

The Hecke algebra H0(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

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SLIDE 84
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Theorem

The Hecke algebra H0(Z) is not maximal abelian. The commutant H0(Z)′ in F0(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

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SLIDE 85
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z

Theorem

The Hecke algebra H0(Z) is not maximal abelian. The commutant H0(Z)′ in F0(Z) is a direct sum of the Hankel and Toeplitz operators: H0(Z)′ = H0(Z) ⊕ T0(Z)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 86
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 87
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 88
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 aj,k+1 aj−1,k  

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 89
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 + aj,k+1 aj−1,k  

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 90
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 + aj,k+1 aj−1,k  

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 91
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 aj,k+1 aj−1,k  

Toeplitz:

tj,k := aj,k−1 − aj−1,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 92
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 aj,k+1 aj−1,k  

Toeplitz:

tj,k := aj,k−1 − aj−1,k = aj+1,k − aj,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-93
SLIDE 93
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 aj,k+1 aj−1,k  

Toeplitz:

tj,k := aj,k−1 − aj−1,k = aj+1,k − aj,k+1 = tj+1,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 94
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 aj,k+1 aj−1,k  

Toeplitz:

tj,k := aj,k−1 − aj−1,k = aj+1,k − aj,k+1 = tj+1,k+1

Hankel:

hj,k := aj,k−1 − aj+1,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 95
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 aj,k+1 aj−1,k  

Toeplitz:

tj,k := aj,k−1 − aj−1,k = aj+1,k − aj,k+1 = tj+1,k+1

Hankel:

hj,k := aj,k−1 − aj+1,k = aj−1,k − aj,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 96
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (ϕ = (aj,k))

Commutation with χ1

ϕ ◦ χ1 = χ1 ◦ ϕ ⇔ aj,k−1 + aj,k+1 = aj−1,k + aj+1,k   aj+1,k aj,k−1 aj,k+1 aj−1,k  

Toeplitz:

tj,k := aj,k−1 − aj−1,k = aj+1,k − aj,k+1 = tj+1,k+1

Hankel:

hj,k := aj,k−1 − aj+1,k = aj−1,k − aj,k+1 = hj−1,k+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-97
SLIDE 97
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 98
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-99
SLIDE 99
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n) = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-100
SLIDE 100
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n) = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: ak,−k − ak+1,−(k+1) = a2k+1,1 − a2k+2,0

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 101
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n) = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: ak,−k − ak+1,−(k+1) = a2k+1,1 − a2k+2,0

a0,0 − an,−n = a0,0 − a1,−1 + a1,−1 − a2,−2 + · · · + an−1,−n+1 − an,−n

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-102
SLIDE 102
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n) = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: ak,−k − ak+1,−(k+1) = a2k+1,1 − a2k+2,0

a0,0 − an,−n = a0,0 − a1,−1 + a1,−1 − a2,−2 + · · · + an−1,−n+1 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

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SLIDE 103
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n) = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: ak,−k − ak+1,−(k+1) = a2k+1,1 − a2k+2,0

a0,0 − an,−n = a0,0 − a1,−1 + a1,−1 − a2,−2 + · · · + an−1,−n+1 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Combining: (a0,0 − an,n) + (a0,0 − an,−n)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-104
SLIDE 104
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n) = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: ak,−k − ak+1,−(k+1) = a2k+1,1 − a2k+2,0

a0,0 − an,−n = a0,0 − a1,−1 + a1,−1 − a2,−2 + · · · + an−1,−n+1 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Combining: (a0,0 − an,n) + (a0,0 − an,−n)

2a0,0 − an,n − an,−n = a0,0 − a2n,0

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-105
SLIDE 105
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an,n = (a0,0 − a1,1) + (a1,1 − a2,2) + · · · + (an−1,n−1 − an,n) = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: ak,−k − ak+1,−(k+1) = a2k+1,1 − a2k+2,0

a0,0 − an,−n = a0,0 − a1,−1 + a1,−1 − a2,−2 + · · · + an−1,−n+1 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Combining: (a0,0 − an,n) + (a0,0 − an,−n)

2a0,0 − an,n − an,−n = a0,0 − a2n,0 a0,0 = an,n + an,−n − a2n,0

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-106
SLIDE 106
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-107
SLIDE 107
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-108
SLIDE 108
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-109
SLIDE 109
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: lim

n→+∞ an,−n = h(0) exists

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-110
SLIDE 110
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: lim

n→+∞ an,−n = h(0) exists

a0,0 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-111
SLIDE 111
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: lim

n→+∞ an,−n = h(0) exists

a0,0 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Combining with lim

n→+∞ a2n,0 = 0

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-112
SLIDE 112
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: lim

n→+∞ an,−n = h(0) exists

a0,0 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Combining with lim

n→+∞ a2n,0 = 0

a0,0 = an,n + an,−n − a2n,0

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-113
SLIDE 113
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: lim

n→+∞ an,−n = h(0) exists

a0,0 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Combining with lim

n→+∞ a2n,0 = 0

a0,0 = an,n + an,−n − a2n,0 − − − − →

n→+∞ t(0) + h(0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-114
SLIDE 114
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≥ 0)

Hankel: lim

n→+∞ an,n = t(0) exists

a0,0 − an,n = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n−2,0 − a2n−1,1)

Toeplitz: lim

n→+∞ an,−n = h(0) exists

a0,0 − an,−n = (a1,1 − a2,0) + (a3,1 − a4,0) + · · · + (a2n−1,1 − a2n,0)

Combining with lim

n→+∞ a2n,0 = 0

a0,0 = an,n + an,−n − a2n,0 − − − − →

n→+∞ t(0) + h(0)

a0,0 = t(0) + h(0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-115
SLIDE 115
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-116
SLIDE 116
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-117
SLIDE 117
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-118
SLIDE 118
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-119
SLIDE 119
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-120
SLIDE 120
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-121
SLIDE 121
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

a0,0 − a−n,n

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-122
SLIDE 122
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

a0,0 − a−n,n = (a1,1 − a0,2) + (a1,3 − a0,4) + · · · + (a1,2n+1 − a0,2n+2)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-123
SLIDE 123
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

a0,0 − a−n,n = (a1,1 − a0,2) + (a1,3 − a0,4) + · · · + (a1,2n+1 − a0,2n+2) t(0) + h(0) = a0,0

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-124
SLIDE 124
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

a0,0 − a−n,n = (a1,1 − a0,2) + (a1,3 − a0,4) + · · · + (a1,2n+1 − a0,2n+2) t(0) + h(0) = a0,0 = an+1,n+1 + a−n,n − a0,2n+2

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-125
SLIDE 125
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

a0,0 − a−n,n = (a1,1 − a0,2) + (a1,3 − a0,4) + · · · + (a1,2n+1 − a0,2n+2) t(0) + h(0) = a0,0 = an+1,n+1 + a−n,n − a0,2n+2 − − − − →

n→+∞ t(0) + h′(0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-126
SLIDE 126
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

a0,0 − a−n,n = (a1,1 − a0,2) + (a1,3 − a0,4) + · · · + (a1,2n+1 − a0,2n+2) t(0) + h(0) = a0,0 = an+1,n+1 + a−n,n − a0,2n+2 − − − − →

n→+∞ t(0) + h′(0)

Hence h(0) = h′(0),

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-127
SLIDE 127
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof (n ≤ 0)

In a similar manner one shows the existence of limits: limn→+∞ a−n,−n = t′(0) limn→+∞ a−n,n = h′(0) limn→+∞ an,n = t(0) limn→+∞ an,−n = h(0)

Hankel: ak,k − ak+1,k+1 = a2k,0 − a2k+1,1

a0,0 − an+1,n+1 = (a0,0 − a1,1) + (a2,0 − a3,1) + · · · + (a2n,0 − a2n+1,1)

Toeplitz: a−k,k − a−(k+1),k+1 = a1,2k+1 − a0,2k+2

a0,0 − a−n,n = (a1,1 − a0,2) + (a1,3 − a0,4) + · · · + (a1,2n+1 − a0,2n+2) t(0) + h(0) = a0,0 = an+1,n+1 + a−n,n − a0,2n+2 − − − − →

n→+∞ t(0) + h′(0)

Hence h(0) = h′(0), and, similarly, t(0) = t′(0)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-128
SLIDE 128
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof

Toeplitz and Hankel parts

tj,k := t(k − j), hj,k = h(k + j)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-129
SLIDE 129
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof

Toeplitz and Hankel parts

tj,k := t(k − j), hj,k = h(k + j)

Decomposition

aj,k = t(k − j) + h(k + j) = tj,k + hj,k

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-130
SLIDE 130
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof

Toeplitz and Hankel parts

tj,k := t(k − j), hj,k = h(k + j)

Decomposition

aj,k = t(k − j) + h(k + j) = tj,k + hj,k Hence decomposition ϕ = t + h

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-131
SLIDE 131
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra on Z – proof

Toeplitz and Hankel parts

tj,k := t(k − j), hj,k = h(k + j)

Decomposition

aj,k = t(k − j) + h(k + j) = tj,k + hj,k Hence decomposition ϕ = t + h

Remark

H0(Z) ∩ F0(Z) = {0}

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-132
SLIDE 132
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel

Hecke algebra in B(l2(Z))

Theorem (with E. Ricard)

The commutant H(Z)′ in B(l2(Z)) of the Hecke algebra H(Z) decomposes into the direct sum of the Toeplitz and Hankel

  • perators in B(l2(Z)):

H(Z)′ = T (Z) ⊕ H(Z)

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices

slide-133
SLIDE 133
  • Homogeneous trees
  • Algebraic MASA: deg(X) > 2
  • Hecke MASA: general case.
  • Integers: Toeplitz and Hankel
  • T. Pytlik, Radial functions on free groups and a

decomposition of the regular representation into irreducible

  • components. J. Reine Angew. Math. 326 (1981), 124–135.
  • J. Wysocza´

nski, Hecke algebra on homogeneous trees and relations with Toeplitz and Hankel operators, Proc.

  • Amer. Math. Soc. 122 (1994), no. 4, 1203–1210.

Janusz Wysocza´ nski: Mathematical Institute, Wroc law University Hecke algebras on homogeneous trees and relation with Hankel and Toeplitz matrices