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Sufficient conditions for isomorphisms between function algebras

Thomas Tonev

The University of Montana, Missoula

Göteborg, July-August, 2013

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 1 / 17

Norm-multiplicative maps Theorem 1

Norm-multiplicative maps

Let A ⊂ C(X) be a function algebra on a loc. compact Hausdorff space X, i.e. A is uniformly closed and strongly separates the points of X. We assume that X contains the Shilov boundary ∂A of A, and together

• its Choquet boundary δA.

By f = maxx∈X |f(x)| we denote the uniform norm of f ∈ A. Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If Tf Tg = fg for all f, g ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y))| for all y ∈ δB and f ∈ A.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17

Norm-multiplicative maps Theorem 1

Norm-multiplicative maps

Let A ⊂ C(X) be a function algebra on a loc. compact Hausdorff space X, i.e. A is uniformly closed and strongly separates the points of X. We assume that X contains the Shilov boundary ∂A of A, and together

• its Choquet boundary δA.

By f = maxx∈X |f(x)| we denote the uniform norm of f ∈ A. Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If Tf Tg = fg for all f, g ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y))| for all y ∈ δB and f ∈ A.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17

Norm-multiplicative maps Theorem 1

Norm-multiplicative maps

Let A ⊂ C(X) be a function algebra on a loc. compact Hausdorff space X, i.e. A is uniformly closed and strongly separates the points of X. We assume that X contains the Shilov boundary ∂A of A, and together

• its Choquet boundary δA.

By f = maxx∈X |f(x)| we denote the uniform norm of f ∈ A. Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If Tf Tg = fg for all f, g ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y))| for all y ∈ δB and f ∈ A.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17

Norm-multiplicative maps Theorem 1

Norm-multiplicative maps

Let A ⊂ C(X) be a function algebra on a loc. compact Hausdorff space X, i.e. A is uniformly closed and strongly separates the points of X. We assume that X contains the Shilov boundary ∂A of A, and together

• its Choquet boundary δA.

By f = maxx∈X |f(x)| we denote the uniform norm of f ∈ A. Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If Tf Tg = fg for all f, g ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y))| for all y ∈ δB and f ∈ A.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17

Norm-multiplicative maps Theorem 1

Norm-multiplicative maps

Let A ⊂ C(X) be a function algebra on a loc. compact Hausdorff space X, i.e. A is uniformly closed and strongly separates the points of X. We assume that X contains the Shilov boundary ∂A of A, and together

• its Choquet boundary δA.

By f = maxx∈X |f(x)| we denote the uniform norm of f ∈ A. Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If Tf Tg = fg for all f, g ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y))| for all y ∈ δB and f ∈ A.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17

Norm-multiplicative maps Theorem 1

Norm-multiplicative maps

Let A ⊂ C(X) be a function algebra on a loc. compact Hausdorff space X, i.e. A is uniformly closed and strongly separates the points of X. We assume that X contains the Shilov boundary ∂A of A, and together

• its Choquet boundary δA.

By f = maxx∈X |f(x)| we denote the uniform norm of f ∈ A. Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If Tf Tg = fg for all f, g ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y))| for all y ∈ δB and f ∈ A.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17

Norm-multiplicative maps Theorem 1

Norm-multiplicative maps

Let A ⊂ C(X) be a function algebra on a loc. compact Hausdorff space X, i.e. A is uniformly closed and strongly separates the points of X. We assume that X contains the Shilov boundary ∂A of A, and together

• its Choquet boundary δA.

By f = maxx∈X |f(x)| we denote the uniform norm of f ∈ A. Theorem 1. [T, 2009] Let T : A → B be a surjective (in general not linear) map between two function algebras A and B. If Tf Tg = fg for all f, g ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y))| for all y ∈ δB and f ∈ A.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 2 / 17

Norm-multiplicative maps Corollary 1

The theorem holds also if A, B are (uniformly) dense subalgebras of function algebras such that δA = p(A), δB = p(B) where p(A), p(B) are the sets of p-points, i.e. the strong boundary points of A, B. In particular, it holds for semisimple algebras A with δA = p(A) by the way of the Gelfand algebras A. Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. Tf = f for all f ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y)| for all y ∈ δB and f ∈ A. Indeed, in this case Tf Tg = T(fg) = fg and the result follows directly from Theorem 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17

Norm-multiplicative maps Corollary 1

The theorem holds also if A, B are (uniformly) dense subalgebras of function algebras such that δA = p(A), δB = p(B) where p(A), p(B) are the sets of p-points, i.e. the strong boundary points of A, B. In particular, it holds for semisimple algebras A with δA = p(A) by the way of the Gelfand algebras A. Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. Tf = f for all f ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y)| for all y ∈ δB and f ∈ A. Indeed, in this case Tf Tg = T(fg) = fg and the result follows directly from Theorem 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17

Norm-multiplicative maps Corollary 1

The theorem holds also if A, B are (uniformly) dense subalgebras of function algebras such that δA = p(A), δB = p(B) where p(A), p(B) are the sets of p-points, i.e. the strong boundary points of A, B. In particular, it holds for semisimple algebras A with δA = p(A) by the way of the Gelfand algebras A. Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. Tf = f for all f ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y)| for all y ∈ δB and f ∈ A. Indeed, in this case Tf Tg = T(fg) = fg and the result follows directly from Theorem 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17

Norm-multiplicative maps Corollary 1

The theorem holds also if A, B are (uniformly) dense subalgebras of function algebras such that δA = p(A), δB = p(B) where p(A), p(B) are the sets of p-points, i.e. the strong boundary points of A, B. In particular, it holds for semisimple algebras A with δA = p(A) by the way of the Gelfand algebras A. Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. Tf = f for all f ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y)| for all y ∈ δB and f ∈ A. Indeed, in this case Tf Tg = T(fg) = fg and the result follows directly from Theorem 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17

Norm-multiplicative maps Corollary 1

The theorem holds also if A, B are (uniformly) dense subalgebras of function algebras such that δA = p(A), δB = p(B) where p(A), p(B) are the sets of p-points, i.e. the strong boundary points of A, B. In particular, it holds for semisimple algebras A with δA = p(A) by the way of the Gelfand algebras A. Corollary 1. If T : A → B is a surjective multiplicative map between two function algebras A and B which preserves the norms, i.e. Tf = f for all f ∈ A then there is a homeomorphism ψ: δB → δA so that |(Tf)(y)| = |f(ψ(y)| for all y ∈ δB and f ∈ A. Indeed, in this case Tf Tg = T(fg) = fg and the result follows directly from Theorem 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 3 / 17

Norm-multiplicative maps

Note that any weighted composition operator, i.e. T = α(f ◦ ψ) on δB with |α| = 1, is composition operator in modulus. In general, the equality |(Tf)(y)| = |f(ψ(y))| does not necessarily imply that T is a composition (or, weighted composition) operator, nor that it is an algebra isomorphism. An immediate counterexample is the conjugacy map T : f → f. In order T to be a composition (or, weighted composition) operator stronger conditions are needed for T. We’ll consider Mólnar-type multiplicative spectral conditions.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 4 / 17

Norm-multiplicative maps

Note that any weighted composition operator, i.e. T = α(f ◦ ψ) on δB with |α| = 1, is composition operator in modulus. In general, the equality |(Tf)(y)| = |f(ψ(y))| does not necessarily imply that T is a composition (or, weighted composition) operator, nor that it is an algebra isomorphism. An immediate counterexample is the conjugacy map T : f → f. In order T to be a composition (or, weighted composition) operator stronger conditions are needed for T. We’ll consider Mólnar-type multiplicative spectral conditions.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 4 / 17

Norm-multiplicative maps

Note that any weighted composition operator, i.e. T = α(f ◦ ψ) on δB with |α| = 1, is composition operator in modulus. In general, the equality |(Tf)(y)| = |f(ψ(y))| does not necessarily imply that T is a composition (or, weighted composition) operator, nor that it is an algebra isomorphism. An immediate counterexample is the conjugacy map T : f → f. In order T to be a composition (or, weighted composition) operator stronger conditions are needed for T. We’ll consider Mólnar-type multiplicative spectral conditions.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 4 / 17

Norm-multiplicative maps

Note that any weighted composition operator, i.e. T = α(f ◦ ψ) on δB with |α| = 1, is composition operator in modulus. In general, the equality |(Tf)(y)| = |f(ψ(y))| does not necessarily imply that T is a composition (or, weighted composition) operator, nor that it is an algebra isomorphism. An immediate counterexample is the conjugacy map T : f → f. In order T to be a composition (or, weighted composition) operator stronger conditions are needed for T. We’ll consider Mólnar-type multiplicative spectral conditions.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 4 / 17

Norm-multiplicative maps

Note that any weighted composition operator, i.e. T = α(f ◦ ψ) on δB with |α| = 1, is composition operator in modulus. In general, the equality |(Tf)(y)| = |f(ψ(y))| does not necessarily imply that T is a composition (or, weighted composition) operator, nor that it is an algebra isomorphism. An immediate counterexample is the conjugacy map T : f → f. In order T to be a composition (or, weighted composition) operator stronger conditions are needed for T. We’ll consider Mólnar-type multiplicative spectral conditions.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 4 / 17

Spectral multiplicative conditions Theorem 2

Spectral multiplicative conditions

Given an f ∈ A, we denote by σπ(f) the peripheral spectrum of f, i.e. σπ(f) is the set of values of f with maximum modulus, namely, σπ(f) = {f(x): |f(x)| = f, x ∈ X}. Clearly, σπ(f) ⊂ {z ∈ C: |z| = f}. Theorem 2. [Johnson-T, 2012] If T : A → B is a surjective map between function algebras such that σπ(Tf Tg) ⊂ σπ(fg) [or, σπ(fg) ⊂ σπ(Tf Tg)] for all f, g ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)) for all f ∈ A and y ∈ δB. Hence, T is a weighted composition operator on δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 5 / 17

Spectral multiplicative conditions Theorem 2

Spectral multiplicative conditions

Given an f ∈ A, we denote by σπ(f) the peripheral spectrum of f, i.e. σπ(f) is the set of values of f with maximum modulus, namely, σπ(f) = {f(x): |f(x)| = f, x ∈ X}. Clearly, σπ(f) ⊂ {z ∈ C: |z| = f}. Theorem 2. [Johnson-T, 2012] If T : A → B is a surjective map between function algebras such that σπ(Tf Tg) ⊂ σπ(fg) [or, σπ(fg) ⊂ σπ(Tf Tg)] for all f, g ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)) for all f ∈ A and y ∈ δB. Hence, T is a weighted composition operator on δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 5 / 17

Spectral multiplicative conditions Theorem 2

Spectral multiplicative conditions

Given an f ∈ A, we denote by σπ(f) the peripheral spectrum of f, i.e. σπ(f) is the set of values of f with maximum modulus, namely, σπ(f) = {f(x): |f(x)| = f, x ∈ X}. Clearly, σπ(f) ⊂ {z ∈ C: |z| = f}. Theorem 2. [Johnson-T, 2012] If T : A → B is a surjective map between function algebras such that σπ(Tf Tg) ⊂ σπ(fg) [or, σπ(fg) ⊂ σπ(Tf Tg)] for all f, g ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)) for all f ∈ A and y ∈ δB. Hence, T is a weighted composition operator on δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 5 / 17

Spectral multiplicative conditions Theorem 2

Spectral multiplicative conditions

Given an f ∈ A, we denote by σπ(f) the peripheral spectrum of f, i.e. σπ(f) is the set of values of f with maximum modulus, namely, σπ(f) = {f(x): |f(x)| = f, x ∈ X}. Clearly, σπ(f) ⊂ {z ∈ C: |z| = f}. Theorem 2. [Johnson-T, 2012] If T : A → B is a surjective map between function algebras such that σπ(Tf Tg) ⊂ σπ(fg) [or, σπ(fg) ⊂ σπ(Tf Tg)] for all f, g ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)) for all f ∈ A and y ∈ δB. Hence, T is a weighted composition operator on δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 5 / 17

Spectral multiplicative conditions Theorem 2

Spectral multiplicative conditions

Given an f ∈ A, we denote by σπ(f) the peripheral spectrum of f, i.e. σπ(f) is the set of values of f with maximum modulus, namely, σπ(f) = {f(x): |f(x)| = f, x ∈ X}. Clearly, σπ(f) ⊂ {z ∈ C: |z| = f}. Theorem 2. [Johnson-T, 2012] If T : A → B is a surjective map between function algebras such that σπ(Tf Tg) ⊂ σπ(fg) [or, σπ(fg) ⊂ σπ(Tf Tg)] for all f, g ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)) for all f ∈ A and y ∈ δB. Hence, T is a weighted composition operator on δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 5 / 17

Spectral multiplicative conditions Theorem 3

As a consequence we see that there is a clopen set E ⊂ δB so that Tf = f on E, and Tf = −f on δB \ E. Clearly, in this case, the operator αT = f ◦ ψ is an algebraic isomorphism. The condition σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A is a more general than σπ(Tf Tg) ⊂ σπ(fg). However, it alone does not imply that T a composition (or, weighted composition) operator, unless X is a metric space. Theorem 3. [Johnson-T, 2013] If a surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅ for all f, g ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 6 / 17

Spectral multiplicative conditions Theorem 3

As a consequence we see that there is a clopen set E ⊂ δB so that Tf = f on E, and Tf = −f on δB \ E. Clearly, in this case, the operator αT = f ◦ ψ is an algebraic isomorphism. The condition σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A is a more general than σπ(Tf Tg) ⊂ σπ(fg). However, it alone does not imply that T a composition (or, weighted composition) operator, unless X is a metric space. Theorem 3. [Johnson-T, 2013] If a surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅ for all f, g ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 6 / 17

Spectral multiplicative conditions Theorem 3

As a consequence we see that there is a clopen set E ⊂ δB so that Tf = f on E, and Tf = −f on δB \ E. Clearly, in this case, the operator αT = f ◦ ψ is an algebraic isomorphism. The condition σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A is a more general than σπ(Tf Tg) ⊂ σπ(fg). However, it alone does not imply that T a composition (or, weighted composition) operator, unless X is a metric space. Theorem 3. [Johnson-T, 2013] If a surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅ for all f, g ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 6 / 17

Spectral multiplicative conditions Theorem 3

As a consequence we see that there is a clopen set E ⊂ δB so that Tf = f on E, and Tf = −f on δB \ E. Clearly, in this case, the operator αT = f ◦ ψ is an algebraic isomorphism. The condition σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A is a more general than σπ(Tf Tg) ⊂ σπ(fg). However, it alone does not imply that T a composition (or, weighted composition) operator, unless X is a metric space. Theorem 3. [Johnson-T, 2013] If a surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅ for all f, g ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 6 / 17

Spectral multiplicative conditions Theorem 3

As a consequence we see that there is a clopen set E ⊂ δB so that Tf = f on E, and Tf = −f on δB \ E. Clearly, in this case, the operator αT = f ◦ ψ is an algebraic isomorphism. The condition σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A is a more general than σπ(Tf Tg) ⊂ σπ(fg). However, it alone does not imply that T a composition (or, weighted composition) operator, unless X is a metric space. Theorem 3. [Johnson-T, 2013] If a surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅ for all f, g ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 6 / 17

Spectral multiplicative conditions Theorem 3

As a consequence we see that there is a clopen set E ⊂ δB so that Tf = f on E, and Tf = −f on δB \ E. Clearly, in this case, the operator αT = f ◦ ψ is an algebraic isomorphism. The condition σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A is a more general than σπ(Tf Tg) ⊂ σπ(fg). However, it alone does not imply that T a composition (or, weighted composition) operator, unless X is a metric space. Theorem 3. [Johnson-T, 2013] If a surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅ for all f, g ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 6 / 17

Spectral multiplicative conditions Theorem 4

It is not known if Theorem 3 holds for non-metric spaces without additional conditions. Theorem 4. [Johnson-T, 2012] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg) = ∅, f, g ∈ A and σπ(Tf) is a singleton whenever σπ is a singleton, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Note that if, in addition dist(σπ(Tf), σπ(f)) < 2, then α = 1, i.e. (Tf)(y) = f(ψ(y)) in all previous theorems. Therefore, T is a composition operator on δB, and consequently, an isometric algebra isomorphism.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 7 / 17

Spectral multiplicative conditions Theorem 4

It is not known if Theorem 3 holds for non-metric spaces without additional conditions. Theorem 4. [Johnson-T, 2012] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg) = ∅, f, g ∈ A and σπ(Tf) is a singleton whenever σπ is a singleton, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Note that if, in addition dist(σπ(Tf), σπ(f)) < 2, then α = 1, i.e. (Tf)(y) = f(ψ(y)) in all previous theorems. Therefore, T is a composition operator on δB, and consequently, an isometric algebra isomorphism.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 7 / 17

Spectral multiplicative conditions Theorem 4

It is not known if Theorem 3 holds for non-metric spaces without additional conditions. Theorem 4. [Johnson-T, 2012] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg) = ∅, f, g ∈ A and σπ(Tf) is a singleton whenever σπ is a singleton, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Note that if, in addition dist(σπ(Tf), σπ(f)) < 2, then α = 1, i.e. (Tf)(y) = f(ψ(y)) in all previous theorems. Therefore, T is a composition operator on δB, and consequently, an isometric algebra isomorphism.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 7 / 17

Spectral multiplicative conditions Theorem 4

It is not known if Theorem 3 holds for non-metric spaces without additional conditions. Theorem 4. [Johnson-T, 2012] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg) = ∅, f, g ∈ A and σπ(Tf) is a singleton whenever σπ is a singleton, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Note that if, in addition dist(σπ(Tf), σπ(f)) < 2, then α = 1, i.e. (Tf)(y) = f(ψ(y)) in all previous theorems. Therefore, T is a composition operator on δB, and consequently, an isometric algebra isomorphism.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 7 / 17

Spectral multiplicative conditions Theorem 4

It is not known if Theorem 3 holds for non-metric spaces without additional conditions. Theorem 4. [Johnson-T, 2012] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg) = ∅, f, g ∈ A and σπ(Tf) is a singleton whenever σπ is a singleton, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Note that if, in addition dist(σπ(Tf), σπ(f)) < 2, then α = 1, i.e. (Tf)(y) = f(ψ(y)) in all previous theorems. Therefore, T is a composition operator on δB, and consequently, an isometric algebra isomorphism.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 7 / 17

Spectral multiplicative conditions Theorem 5

Theorem 5. [Johnson-T, 2013] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A and σπ(Tf) ⊂ σπ(f), f ∈ A, then (Tf)(y) = f(ψ(y)) for all y ∈ δB and f ∈ A. Hence, T is a composition operator on δB, and therefore, an algebra isomorphism. All previous theorems hold also in the case when the uniform closures A, B are function algebras with δA = p(A), δB = p(B). In particular, they hold for semisimple algebras A with δA = p(A). For instance, Theorem 3 holds for any Lipschitz algebra.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 8 / 17

Spectral multiplicative conditions Theorem 5

Theorem 5. [Johnson-T, 2013] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A and σπ(Tf) ⊂ σπ(f), f ∈ A, then (Tf)(y) = f(ψ(y)) for all y ∈ δB and f ∈ A. Hence, T is a composition operator on δB, and therefore, an algebra isomorphism. All previous theorems hold also in the case when the uniform closures A, B are function algebras with δA = p(A), δB = p(B). In particular, they hold for semisimple algebras A with δA = p(A). For instance, Theorem 3 holds for any Lipschitz algebra.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 8 / 17

Spectral multiplicative conditions Theorem 5

Theorem 5. [Johnson-T, 2013] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A and σπ(Tf) ⊂ σπ(f), f ∈ A, then (Tf)(y) = f(ψ(y)) for all y ∈ δB and f ∈ A. Hence, T is a composition operator on δB, and therefore, an algebra isomorphism. All previous theorems hold also in the case when the uniform closures A, B are function algebras with δA = p(A), δB = p(B). In particular, they hold for semisimple algebras A with δA = p(A). For instance, Theorem 3 holds for any Lipschitz algebra.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 8 / 17

Spectral multiplicative conditions Theorem 5

Theorem 5. [Johnson-T, 2013] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A and σπ(Tf) ⊂ σπ(f), f ∈ A, then (Tf)(y) = f(ψ(y)) for all y ∈ δB and f ∈ A. Hence, T is a composition operator on δB, and therefore, an algebra isomorphism. All previous theorems hold also in the case when the uniform closures A, B are function algebras with δA = p(A), δB = p(B). In particular, they hold for semisimple algebras A with δA = p(A). For instance, Theorem 3 holds for any Lipschitz algebra.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 8 / 17

Spectral multiplicative conditions Theorem 5

Theorem 5. [Johnson-T, 2013] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A and σπ(Tf) ⊂ σπ(f), f ∈ A, then (Tf)(y) = f(ψ(y)) for all y ∈ δB and f ∈ A. Hence, T is a composition operator on δB, and therefore, an algebra isomorphism. All previous theorems hold also in the case when the uniform closures A, B are function algebras with δA = p(A), δB = p(B). In particular, they hold for semisimple algebras A with δA = p(A). For instance, Theorem 3 holds for any Lipschitz algebra.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 8 / 17

Spectral multiplicative conditions Theorem 5

Theorem 5. [Johnson-T, 2013] If a surjection T : A → B between function algebras is such that σπ(Tf Tg) ∩ σπ(fg)) = ∅, f, g ∈ A and σπ(Tf) ⊂ σπ(f), f ∈ A, then (Tf)(y) = f(ψ(y)) for all y ∈ δB and f ∈ A. Hence, T is a composition operator on δB, and therefore, an algebra isomorphism. All previous theorems hold also in the case when the uniform closures A, B are function algebras with δA = p(A), δB = p(B). In particular, they hold for semisimple algebras A with δA = p(A). For instance, Theorem 3 holds for any Lipschitz algebra.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 8 / 17

Corollaries Corollary 2

Corollaries

As direct corollaries of the above theorems we obtain: Corollary 2. If a surjective multiplicative map T : A → B between two function algebras is such that σπ(Tf) ⊂ σπ(f) for all f ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)), for all f ∈ A, y ∈ δB. Indeed, in this case σπ(Tf Tg) = σπ(T(fg)) ⊂ σπ(fg) and the result follows from Theorem 2.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 9 / 17

Corollaries Corollary 2

Corollaries

As direct corollaries of the above theorems we obtain: Corollary 2. If a surjective multiplicative map T : A → B between two function algebras is such that σπ(Tf) ⊂ σπ(f) for all f ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)), for all f ∈ A, y ∈ δB. Indeed, in this case σπ(Tf Tg) = σπ(T(fg)) ⊂ σπ(fg) and the result follows from Theorem 2.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 9 / 17

Corollaries Corollary 2

Corollaries

As direct corollaries of the above theorems we obtain: Corollary 2. If a surjective multiplicative map T : A → B between two function algebras is such that σπ(Tf) ⊂ σπ(f) for all f ∈ A, then there is a function α ∈ C(δB) with α2 = 1 so that (Tf)(y) = α(y) f(ψ(y)), for all f ∈ A, y ∈ δB. Indeed, in this case σπ(Tf Tg) = σπ(T(fg)) ⊂ σπ(fg) and the result follows from Theorem 2.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 9 / 17

Corollaries Corollary 3

By the same reasoning, Corollary 3. If a multiplicative surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf) ∩ σπ(f)) = ∅ for all f ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 10 / 17

Corollaries Corollary 3

By the same reasoning, Corollary 3. If a multiplicative surjection T : A → B between function algebras A ⊂ C(X), B ⊂ C(Y) on metric spaces X, Y is such that σπ(Tf) ∩ σπ(f)) = ∅ for all f ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 10 / 17

Corollaries Corollary 4

Corollary 4. If a multiplicative surjection T : A → B between function algebras maps peaking functions of A to peaking functions on B and is such that σπ(Tf) ∩ σπ(f)) = ∅ for all f ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Indeed, in this case σπ(Tf Tg) ∩ σπ(fg) = σπ(T(fg)) ∩ σπ(fg) = ∅, and,

• bviously, T preserves the singleton peripheral spectra.
• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 11 / 17

Corollaries Corollary 4

Corollary 4. If a multiplicative surjection T : A → B between function algebras maps peaking functions of A to peaking functions on B and is such that σπ(Tf) ∩ σπ(f)) = ∅ for all f ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Indeed, in this case σπ(Tf Tg) ∩ σπ(fg) = σπ(T(fg)) ∩ σπ(fg) = ∅, and,

• bviously, T preserves the singleton peripheral spectra.
• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 11 / 17

Corollaries Corollary 4

Corollary 4. If a multiplicative surjection T : A → B between function algebras maps peaking functions of A to peaking functions on B and is such that σπ(Tf) ∩ σπ(f)) = ∅ for all f ∈ A, then (Tf)(y) = α(y) f(ψ(y)) for all y ∈ δB and f ∈ A, where α ∈ C(δB) with α2 = 1. Indeed, in this case σπ(Tf Tg) ∩ σπ(fg) = σπ(T(fg)) ∩ σπ(fg) = ∅, and,

• bviously, T preserves the singleton peripheral spectra.
• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 11 / 17

Almost isomorphisms Theorem 6

Almost isomorphisms [to appear in the PAMS]

Theorem 6. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective map with Tf Tg = fg, f, g ∈ A, such that there is an ǫ, 0 ≤ ǫ < 2/3, so that σπ(Tf Tg) ⊂ Oǫ fg(σπ(fg)) for all f, g ∈ A, f = 1. Then there is a homeomorphism ψ: δB → δA and a continuous function α: δB → {±1} such that |(Tf)(y) − α(y) f(ψ(y))| ≤ 2ǫ |f(ψ(y))|, for all f ∈ A and y ∈ δB. Hence T is an almost weighted composition operator.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 12 / 17

Almost isomorphisms Theorem 6

Almost isomorphisms [to appear in the PAMS]

Theorem 6. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective map with Tf Tg = fg, f, g ∈ A, such that there is an ǫ, 0 ≤ ǫ < 2/3, so that σπ(Tf Tg) ⊂ Oǫ fg(σπ(fg)) for all f, g ∈ A, f = 1. Then there is a homeomorphism ψ: δB → δA and a continuous function α: δB → {±1} such that |(Tf)(y) − α(y) f(ψ(y))| ≤ 2ǫ |f(ψ(y))|, for all f ∈ A and y ∈ δB. Hence T is an almost weighted composition operator.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 12 / 17

Almost isomorphisms Theorem 6

Almost isomorphisms [to appear in the PAMS]

Theorem 6. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective map with Tf Tg = fg, f, g ∈ A, such that there is an ǫ, 0 ≤ ǫ < 2/3, so that σπ(Tf Tg) ⊂ Oǫ fg(σπ(fg)) for all f, g ∈ A, f = 1. Then there is a homeomorphism ψ: δB → δA and a continuous function α: δB → {±1} such that |(Tf)(y) − α(y) f(ψ(y))| ≤ 2ǫ |f(ψ(y))|, for all f ∈ A and y ∈ δB. Hence T is an almost weighted composition operator.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 12 / 17

Almost isomorphisms Theorem 6

Almost isomorphisms [to appear in the PAMS]

Theorem 6. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective map with Tf Tg = fg, f, g ∈ A, such that there is an ǫ, 0 ≤ ǫ < 2/3, so that σπ(Tf Tg) ⊂ Oǫ fg(σπ(fg)) for all f, g ∈ A, f = 1. Then there is a homeomorphism ψ: δB → δA and a continuous function α: δB → {±1} such that |(Tf)(y) − α(y) f(ψ(y))| ≤ 2ǫ |f(ψ(y))|, for all f ∈ A and y ∈ δB. Hence T is an almost weighted composition operator.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 12 / 17

Almost isomorphisms Theorem 6

Note that if f(ψ(y)) = 0, the condition in Theorem 6 can be rewritten as

• (Tf)(y)

α(y) f(ψ(y)) − 1

• ≤ 2ǫ.

Analogous to Theorem 6 result holds if the hypothesis is replaced by its symmetric σπ(fg) ⊂ Oǫ fg(σπ(Tf Tg)), f, g ∈ A. Note that with ε = 0 Theorem 6 implies Theorem 2.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 13 / 17

Almost isomorphisms Theorem 6

Note that if f(ψ(y)) = 0, the condition in Theorem 6 can be rewritten as

• (Tf)(y)

α(y) f(ψ(y)) − 1

• ≤ 2ǫ.

Analogous to Theorem 6 result holds if the hypothesis is replaced by its symmetric σπ(fg) ⊂ Oǫ fg(σπ(Tf Tg)), f, g ∈ A. Note that with ε = 0 Theorem 6 implies Theorem 2.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 13 / 17

Almost isomorphisms Theorem 6

Note that if f(ψ(y)) = 0, the condition in Theorem 6 can be rewritten as

• (Tf)(y)

α(y) f(ψ(y)) − 1

• ≤ 2ǫ.

Analogous to Theorem 6 result holds if the hypothesis is replaced by its symmetric σπ(fg) ⊂ Oǫ fg(σπ(Tf Tg)), f, g ∈ A. Note that with ε = 0 Theorem 6 implies Theorem 2.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 13 / 17

Almost isomorphisms Theorem 7

Theorem 7. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective with Tf Tg = fg, f, g ∈ A such that (i) There is an ǫ, 0 ≤ ǫ < 2/3, so that σπ(Tf Tg) ⊂ Oǫ fg(σπ(fg)) (ii) There is an η, 0 ≤ η < 2/3 so that d(σπ(Tf), σπ(f)) ≤ η for all f, g ∈ A with f = 1. Then there is a homeomorphism ψ: δB → δA so that |(Tf)(y) − f(ψ(y))| ≤ 2ε |f(ψ(y))| for all f ∈ A and y ∈ δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 14 / 17

Almost isomorphisms Theorem 7

Theorem 7. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective with Tf Tg = fg, f, g ∈ A such that (i) There is an ǫ, 0 ≤ ǫ < 2/3, so that σπ(Tf Tg) ⊂ Oǫ fg(σπ(fg)) (ii) There is an η, 0 ≤ η < 2/3 so that d(σπ(Tf), σπ(f)) ≤ η for all f, g ∈ A with f = 1. Then there is a homeomorphism ψ: δB → δA so that |(Tf)(y) − f(ψ(y))| ≤ 2ε |f(ψ(y))| for all f ∈ A and y ∈ δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 14 / 17

Almost isomorphisms Theorem 7

Theorem 7. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective with Tf Tg = fg, f, g ∈ A such that (i) There is an ǫ, 0 ≤ ǫ < 2/3, so that σπ(Tf Tg) ⊂ Oǫ fg(σπ(fg)) (ii) There is an η, 0 ≤ η < 2/3 so that d(σπ(Tf), σπ(f)) ≤ η for all f, g ∈ A with f = 1. Then there is a homeomorphism ψ: δB → δA so that |(Tf)(y) − f(ψ(y))| ≤ 2ε |f(ψ(y))| for all f ∈ A and y ∈ δB.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 14 / 17

Almost isomorphisms Theorem 8

Theorem 8. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective with Tf Tg = fg, f, g ∈ A, such that (i) There is an ǫ, 0 ≤ ǫ < 1, so that d(σπ(Tf Tg), σπ(fg)) ≤ ǫ fg (ii) There is an η, 0 ≤ η < 1 so that σπ(Tf) ⊂ Oη(σπ(f)) for all f, g ∈ A with f = 1. Then there is a homeomorphism ψ: δB → δA so that |(Tf)(y) − f(ψ(y))| ≤ (ε + η) |f(ψ(y))|, for all f ∈ A and y ∈ δB. Note that with ε = η = 0 Theorem 8 implies Theorem 5.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 15 / 17

Almost isomorphisms Theorem 8

Theorem 8. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective with Tf Tg = fg, f, g ∈ A, such that (i) There is an ǫ, 0 ≤ ǫ < 1, so that d(σπ(Tf Tg), σπ(fg)) ≤ ǫ fg (ii) There is an η, 0 ≤ η < 1 so that σπ(Tf) ⊂ Oη(σπ(f)) for all f, g ∈ A with f = 1. Then there is a homeomorphism ψ: δB → δA so that |(Tf)(y) − f(ψ(y))| ≤ (ε + η) |f(ψ(y))|, for all f ∈ A and y ∈ δB. Note that with ε = η = 0 Theorem 8 implies Theorem 5.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 15 / 17

Almost isomorphisms Theorem 8

Theorem 8. A ⊂ C(X), B ⊂ C(Y) – function algebras. T : A → B – surjective with Tf Tg = fg, f, g ∈ A, such that (i) There is an ǫ, 0 ≤ ǫ < 1, so that d(σπ(Tf Tg), σπ(fg)) ≤ ǫ fg (ii) There is an η, 0 ≤ η < 1 so that σπ(Tf) ⊂ Oη(σπ(f)) for all f, g ∈ A with f = 1. Then there is a homeomorphism ψ: δB → δA so that |(Tf)(y) − f(ψ(y))| ≤ (ε + η) |f(ψ(y))|, for all f ∈ A and y ∈ δB. Note that with ε = η = 0 Theorem 8 implies Theorem 5.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 15 / 17

References

References

• T. Tonev, Weak multiplicative operators on function algebras

without units, Banach Center Publications, v. 91(2010), 411-421.

• J. Johnson, T. Tonev, Spectral conditions for weighted composition
• perators on function algebras, Communications in Math. and

Appl., 3(2012), 51-59.

• O. Hatori, T. Miura, S. Lambert, A. Luttman, T. Tonev, R. Yates,

Spectral preservers in commutative Banach algebras, Contemp. Mathematics, v. 547(2011), 103-123.

• J. Johnson, Peripherally-multiplicative spectral mpreservers

between function algebras, PhD Thesis, The University of Montana, 2013.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 16 / 17

That’s all.

Thank you.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 17 / 17

That’s all.

Thank you.

• T. Tonev (University of Montana)

Sufficient conditions for isomorphisms Göteborg, July-August, 2013 17 / 17