On Isometries on Some Banach Spaces: Part I

Home , Isometry

Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections

On isometries on some Banach spaces – something old, something new, something borrowed, something blue, Part I

Dijana Iliˇsevi´c

University of Zagreb, Croatia

Recent Trends in Operator Theory and Applications Memphis, TN, USA, May 3–5, 2018

Recent work of D.I. has been fully supported by the Croatian Science Foundation under the project

IP-2016-06-1046.

Dijana Iliˇsevi´c On isometries on some Banach spaces Introduction Isometries on some important Banach spaces Hermitian projections Generalized bicircular projections Generalized n-circular projections

Something old, something new,

something borrowed, something blue

is referred to the collection of items that

helps to guarantee fertility and prosperity.

Isometries are maps between metric spaces which preserve distance between elements.

Deﬁnition Let (X , | · |) and (Y, k · k) be two normed spaces over the same ﬁeld. A linear map ϕ: X → Y is called a linear isometry if

kϕ(x)k = |x|, x ∈ X .

We shall be interested in surjective linear isometries on Banach spaces.

One of the main problems is to give explicit description of isometries on a particular space.

Isometries are maps between metric spaces which preserve distance between elements.

Deﬁnition Let (X , | · |) and (Y, k · k) be two normed spaces over the same ﬁeld. A linear map ϕ: X → Y is called a linear isometry if

kϕ(x)k = |x|, x ∈ X .

We shall be interested in surjective linear isometries on Banach spaces.

One of the main problems is to give explicit description of isometries on a particular space.

Richard J. Fleming, James E. Jamison, Isometries on Banach spaces: function spaces, Chapman & Hall/CRC, 2003 (208 pp.) Isometries on Banach spaces: vector-valued function spaces, Chapman & Hall/CRC, 2008 (248 pp.)

This talk is dedicated to the memory of Professor James Jamison.

Trivial isometries are isometries of the form λI for some λ ∈ T, where T = {λ ∈ F : |λ| = 1}.

The spectrum of a surjective linear isometry is contained in T.

For any Banach space X (real or complex) there is a norm k · k on X , equivalent to the original one, such that (X , k · k) has only trivial isometries (K. Jarosz, 1988).

Trivial isometries are isometries of the form λI for some λ ∈ T, where T = {λ ∈ F : |λ| = 1}.

The spectrum of a surjective linear isometry is contained in T.

For any Banach space X (real or complex) there is a norm k · k on X , equivalent to the original one, such that (X , k · k) has only trivial isometries (K. Jarosz, 1988).

Trivial isometries are isometries of the form λI for some λ ∈ T, where T = {λ ∈ F : |λ| = 1}.

The spectrum of a surjective linear isometry is contained in T.

For any Banach space X (real or complex) there is a norm k · k on X , equivalent to the original one, such that (X , k · k) has only trivial isometries (K. Jarosz, 1988).

Let V be a ﬁnite dimensional vector space over F ∈ {R, C}, equipped with the norm k · k induced by the inner product

hx, yi = tr (xy ∗) = y ∗x

(Frobenius norm). Then U is a linear isometry on (V, k · k) if and only if the following holds. If F = C: U is a unitary operator on V, that is,

U∗U = UU∗ = I .

If F = R: U is an orthogonal operator on V, that is,

Ut U = UUt = I .

Theorem (I. Schur, 1925)

Linear isometries of Mn(C) equipped with the spectral norm (operator norm) have one of the following forms:

X 7→ UXV or X 7→ UX t V ,

where U, V ∈ Mn(C) are unitaries.

[C.K. Li, N.K. Tsing, 1990]

Let G be the group of all linear operators of the form X 7→ UXV for some ﬁxed unitary (orthogonal) U, V ∈ Mn(F).

A norm k · k on Mn(F) is called a unitarily invariant norm if kg(X )k = kX k for all g ∈ G, X ∈ Mn(F).

If k · k is a unitarily invariant norm (which is not a multiple of the Frobenius norm) on Mn(F) 6= M4(R) then its isometry group is hG, τi, where τ : Mn(F) → Mn(F) is the transposition operator.

In the case of M4(R) the isometry group is hG, τi or hG, τ, αi, with α: M4(R) → M4(R) deﬁned by

α(X ) = (X + B1XC1 + B2XC2 + B3XC3)/2, where

1 0 0 −1 1 0 0 1 B = ⊗ , C = ⊗ , 1 0 1 1 0 1 0 −1 −1 0 0 −1 1 0 0 1 1 0 B = ⊗ , C = ⊗ , 2 1 0 0 −1 2 −1 0 0 1 0 −1 0 1 0 1 0 1 B = ⊗ , C = ⊗ . 3 1 0 1 0 3 1 0 −1 0

[C.K. Li, N.K. Tsing, 1990-1991]

Let G be the group of all linear operators of the form X 7→ Ut XU for some ﬁxed unitary (orthogonal) U ∈ Mn(F).

A norm k · k on V ∈ {Sn(C), Kn(F)} is called a unitary congruence invariant norm if kg(X )k = kX k for all g ∈ G, X ∈ V .

If k · k is a unitary congruence invariant norm on Sn(C), which is not a multiple of the Frobenius norm, then its isometry group is G.

If k · k is a unitary congruence invariant norm on Kn(C), which is not a multiple of the Frobenius norm, then its isometry group is G if n 6= 4, and hG, γi if n = 4, where γ(X ) is obtained from X by interchanging its (1, 4) and (2, 3) entries, and interchanging its (4, 1) and (3, 2) entries accordingly.

If k · k is a unitary congruence invariant norm on Kn(R), which is not a multiple of the Frobenius norm, then its isometry group is hG, τi if n 6= 4, and hG, τ, γi if n = 4.

Surjective linear isometries of C0(Ω)

Let C0(Ω) be the algebra of all continuous complex-valued functions on a locally compact Hausdorﬀ space Ω, vanishing at inﬁnity. Theorem (Banach–Stone)

Let T : C0(Ω1) → C0(Ω2) be a surjective linear isometry. Then there exist a homeomorphism ϕ:Ω2 → Ω1 and a continuous unimodular function u :Ω2 → C such that T (f )(ω) = u(ω)f ϕ(ω) , f ∈ C0(Ω1), ω ∈ Ω2.

The ﬁrst (Banach’s) version of this theorem (1932): for real-valued functions on compact metric spaces. Stone (1937): for real-valued functions on compact Hausdorﬀ spaces.

Surjective linear isometries of C0(Ω)

Let C0(Ω) be the algebra of all continuous complex-valued functions on a locally compact Hausdorﬀ space Ω, vanishing at inﬁnity. Theorem (Banach–Stone)

The ﬁrst (Banach’s) version of this theorem (1932): for real-valued functions on compact metric spaces. Stone (1937): for real-valued functions on compact Hausdorﬀ spaces.

A C ∗-algebra is a complex Banach ∗-algebra (A, k . k) such that ka∗ak = kak2 for all a ∈ A.

Example C = complex numbers, B(H) = all bounded linear operators on a complex Hilbert space H, K(H) = all compact operators on a complex Hilbert space H, C(Ω) = all continuous complex-valued functions on a compact Hausdorﬀ space Ω,

C0(Ω) = all continuous complex-valued functions on a locally compact Hausdorﬀ space Ω, vanishing at inﬁnity.

Theorem (R. Kadison, 1951) Let A and B be unital C*-algebras and T : A → B a surjective linear isometry. Then T = UJ, where J : A → B is a Jordan ∗-isomorphism (that is, a linear map satisfying J(a2) = J(a)2 and J(a∗) = J(a)∗ for every a ∈ A) and a unitary element U ∈ B.

Theorem (A. Paterson, A. Sinclair, 1972) Let A and B be C*-algebras and T : A → B a surjective linear isometry. Then T = UJ, where J : A → B is a Jordan ∗-isomorphism, and U on B is unitary such that there exists V on B satisfying aU(b) = V (a)b for all a, b ∈ B.

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. Throughout we ﬁx an orthonormal basis {eλ : λ ∈ Λ} of H.

Let T ∈ B(H). If S ∈ B(H) is such that hTeλ, eµi = hSeµ, eλi for all λ, µ ∈ Λ, then S is called the transpose of T associated to the t basis {eλ : λ ∈ Λ} and it is denoted by T .

Theorem Let T : B(H) → B(H) be a surjective linear isometry. Then there exist unitary U, V ∈ B(H) such that T has one of the following forms: X 7→ UXV or X 7→ UX t V .

A JB*-triple is a complex Banach space A together with a continuous triple product {· · ·}: A × A × A → A such that (i) {xyz} is linear in x and z and conjugate linear in y; (ii) {xyz} = {zyx}; (iii) for any x ∈ A, the operator δ(x): A → A deﬁned by δ(x)y = {xxy} is hermitian with nonnegative spectrum; (iv) δ(x){abc} = {δ(x)a, b, c} − {a, δ(x)b, c} + {a, b, δ(x)c}; (v) for every x ∈ A, k{xxx}k = kxk3.

Example 1 complex Hilbert spaces: {xyz} = 2 (hx, yiz + hz, yix) 1 ∗ ∗ C*-algebras, S(H), A(H): {xyz} = 2 (xy z + zy x), where S(H) = {T ∈ B(H): T t = T } symmetric operators, A(H) = {T ∈ B(H): T t = −T } antisymmetric operators.

Theorem (W. Kaup, 1983) Let A be a JB*-triple. Then every surjective linear isometry T : A → A satisﬁes

T ({xyz}) = {T (x)T (y)T (z)}, x, y, z ∈ A.

In particular, if A is a C*-algebra then

T (xy ∗x) = T (x)T (y)∗T (x), x, y ∈ A.

Theorem (W. Kaup, 1983) Let A be a JB*-triple. Then every surjective linear isometry T : A → A satisﬁes

T ({xyz}) = {T (x)T (y)T (z)}, x, y, z ∈ A.

In particular, if A is a C*-algebra then

T (xy ∗x) = T (x)T (y)∗T (x), x, y ∈ A.

Every surjective linear isometry T : A → A, where A is S(H) or A(H), satisﬁes

T (XY ∗X ) = T (X )T (Y )∗T (X )

for all X , Y ∈ A.

The following theorem gives an explicit formula for T .

Theorem (A. Foˇsnerand D. I., 2011) Let A be S(H) or A(H) and let T : A → A be a surjective linear isometry. Then there exists a unitary U ∈ B(H) such that T has the form X 7→ UXUt .

Every surjective linear isometry T : A → A, where A is S(H) or A(H), satisﬁes

T (XY ∗X ) = T (X )T (Y )∗T (X )

for all X , Y ∈ A.

The following theorem gives an explicit formula for T .

Theorem (A. Foˇsnerand D. I., 2011) Let A be S(H) or A(H) and let T : A → A be a surjective linear isometry. Then there exists a unitary U ∈ B(H) such that T has the form X 7→ UXUt .

A minimal norm ideal (I, ν) consists of a two-sided proper ideal I in B(H) together with a norm ν on I satisfying the following: the set of all ﬁnite rank operators on H is dense in I, ν(X ) = kX k for every rank one operator X , ν(UXV ) = ν(X ) for every X ∈ I and all unitary U, V ∈ B(H).

Theorem (A. Sourour, 1981) If I is diﬀerent from the Hilbert-Schmidt class then every surjective linear isometry on I has the form X 7→ UXV or X 7→ UX t V for some unitary U, V ∈ B(H).

Deﬁnition Let X be a complex Banach space. A bounded linear operator T : X → X is said to be hermitian if eiϕT is an isometry for all ϕ ∈ R.

Example C 1[0, 1], the space of continuously diﬀerentiable complex-valued 0 functions on [0, 1] with kf k = kf k∞ + kf k∞, admits only trivial hermitian operators, that is, real multiples of I (E. Berkson, A. Sourour, 1974).

Example Hermitian operators on a C*-algebra A have the form x 7→ ax + xb for some self-adjoint a, b ∈ M(A).

Deﬁnition Let X be a complex Banach space. A bounded linear operator T : X → X is said to be hermitian if eiϕT is an isometry for all ϕ ∈ R.

Example Hermitian operators on a C*-algebra A have the form x 7→ ax + xb for some self-adjoint a, b ∈ M(A).

By a projection on a complex Banach space we mean a linear operator P such that P2 = P. Theorem (J. Jamison, 2007) A projection P on a complex Banach space is a hermitian projection if and only if P + λ(I − P) is an isometry for all λ ∈ T, where T = {λ ∈ C : |λ| = 1}.

Example C 1[0, 1] admits only trivial hermitian projections (0 and I ).

Example Every orthogonal projection on a complex Hilbert space is hermitian.

Example C 1[0, 1] admits only trivial hermitian projections (0 and I ).

Example Every orthogonal projection on a complex Hilbert space is hermitian.

Theorem (L.L. Stach´oand B. Zalar, 2004)

(i) Let P : B(H) → B(H) be a hermitian projection. Then P has the form X 7→ QX or X 7→ XQ for some Q ∈ B(H) such that Q = Q∗ = Q2. (ii) Let P : S(H) → S(H) be a hermitian projection. Then either P = 0 or P = I. (iii) Let P : A(H) → A(H) be a hermitian projection. Then P or I − P has the form X 7→ QX + XQt with Q = x ⊗ x for some unit vector x ∈ H.

Theorem (M. Foˇsnerand D. I., 2005) Let A be a C*-algebra and let P : A → A be a hermitian projection. Then there exist a ∗-ideal I of A and p = p∗ = p2 ∈ M(I⊥ ⊕ I⊥⊥) such that P(x) = px for all x ∈ I⊥ and P(x) = xp for all x ∈ I⊥⊥.

Corollary Let Ω be a locally compact Hausdorﬀ space. Then P : C0(Ω) → C0(Ω) is a hermitian projection if and only if Pf = 1Y f , where 1Y f is the indicator function on a proper component Y of Ω. In particular, if Ω is connected then C0(Ω) admits only trivial hermitian projections.

Corollary Let A be K(H) or B(H) and let P : A → A be a hermitian projection. Then there exists p = p∗ = p2 ∈ B(H) such that P has the form x 7→ px or x 7→ xp.

Theorem (J. Jamison, 2007) Let I be a minimal norm ideal in B(H), diﬀerent from the Hilbert-Schmidt class, and let P : I → I be a hermitian projection. Then P has the form X 7→ QX or X 7→ XQ for some Q = Q∗ = Q2 ∈ B(H).

Corollary Let A be K(H) or B(H) and let P : A → A be a hermitian projection. Then there exists p = p∗ = p2 ∈ B(H) such that P has the form x 7→ px or x 7→ xp.

Recall that a projection P on X is a hermitian projection if and only if the map

P + λ(I − P) is an isometry for all λ ∈ T.

These projections are also known as bicircular projections.

We can also study projections P such that

P + λ(I − P) is an isometry for some λ ∈ T \{1}.

These projections are also known as generalized bicircular projections (GBPs).

Let A be Sn(C) or Kn(C). A norm k · k on A is said to be a unitary congruence invariant norm if

kUXUt k = kX k

for all unitary U ∈ Mn(C) and all X ∈ A.

Theorem (M. Foˇsner,D. I. and C.K. Li, 2007)

Let k · k be a unitary congruence invariant norm on Sn(C), which is not a multiple of the Frobenius norm. Suppose P : Sn(C) → Sn(C) is a nontrivial projection and λ ∈ T \{1}. Then P + λ(I − P) is an isometry of (Sn(C), k · k) if and only if λ = −1 and there exists ∗ 2 Q = Q = Q ∈ Mn(C) such that P or I − P has the form X 7→ QXQt + (I − Q)X (I − Qt ).

Let A be Sn(C) or Kn(C). A norm k · k on A is said to be a unitary congruence invariant norm if

kUXUt k = kX k

for all unitary U ∈ Mn(C) and all X ∈ A.

Theorem (M. Foˇsner,D. I. and C.K. Li, 2007)

Let k · k be a unitary congruence invariant norm on Kn(C), which is not a multiple of the Frobenius norm. Suppose P : Kn(C) → Kn(C) is a nontrivial projection and λ ∈ T \{1}. Then P + λ(I − P) is an isometry of (Kn(C), k · k) if and only if one of the following holds. ∗ n (i) There exists Q = vv for a unit vector v ∈ C such that P or I − P has the form X 7→ QX + XQt . ∗ 2 (ii) λ = −1, K = G and there exists Q = Q = Q ∈ Mn(C) such that P or I − P has the form X 7→ QXQt + (I − Q)X (I − Qt ).

(iii) (λ, n) = (−1, 4), ψ ∈ K, and there is a unitary U ∈ M4(C), t ∗ satisfying ψ(U XU) = Uψ(X )U for all X ∈ K4(C), such that P or I − P has the form X 7→ (X + ψ(Ut XU))/2 = (X + Uψ(X )U∗)/2.

Theorem (F. Botelho and J. Jamison, 2008) Let I be a minimal norm ideal in B(H), diﬀerent from the Hilbert-Schmidt class, and let P : A → A be a projection. Then P + λ(I − P) is an isometry for some λ ∈ T \{1} if and only if one of the following holds: (i) P has the form X 7→ QX or X 7→ XQ for some Q = Q∗ = Q2 ∈ B(H), (ii) λ = −1 and P has one of the following forms: 1 X 7→ 2 (X + UXV ) for some unitary U, V ∈ B(H) such that U2 = µI,V 2 = µI for some µ ∈ C, |µ| = 1, 1 t X 7→ 2 (X + UX V ) for some unitary U, V ∈ B(H) such that V = ±(Ut )∗.

Theorem (P.-K. Lin, 2008) Let X be a complex Banach space and let P : X → X be a projection. Then P + λ(I − P) is an isometry for some λ ∈ T \{1} if and only if one of the following holds: (i) P is hermitian, 2πi (ii) λ = e n for some integer n ≥ 2. 2πi Furthermore, if n is any integer such that n ≥ 2, then for λ = e n there is a complex Banach space X and a nontrivial projection P on X such that P + λ(I − P) is an isometry.

Theorem (D. I., 2010) Let A be a JB*-triple and let P : A → A be a projection. Then P + λ(I − P) is an isometry for some λ ∈ T \{1} if and only if one of the following holds: (i) P is hermitian, 1 (ii) λ = −1 and P = 2 (I + T ) for some linear isometry T : A → A satisfying T 2 = I .

Corollary Let A = B(H) or A = K(H), and let P : A → A be a nonhermitian projection. Then P + λ(I − P) is an isometry for some λ ∈ T \{1} if and only if λ = −1 and P has one of the following forms: 1 X 7→ 2 (X + UXV ) for unitary U, V ∈ B(H) such that 2 2 U = µI , V = µI for some µ ∈ C, |µ| = 1, 1 t X 7→ 2 (X + UX V ) for unitary U, V ∈ B(H) such that V = ±(Ut )∗.

Generalized bicircular projections on C0(Ω)

Theorem (F. Botelho, 2008) Let Ω be a connected compact Hausdorﬀ space and let P : C(Ω) → C(Ω) be a nontrivial projection. Then P + λ(I − P) is an isometry for some λ ∈ T \{1} if and only if λ = −1 and there exist a homeomorphism ϕ:Ω → Ω satisfying 2 ϕ = I and a continuous unimodular function u :Ω → C satisfying u(ϕ(ω)) = u(ω) for every ω ∈ Ω, such that

1 P(f )(ω) = f (ω) + u(ω)f ϕ(ω) , f ∈ C (Ω), ω ∈ Ω. 2 0

Generalized bicircular projections on C0(Ω)

Theorem (D. I., 2010) Let Ω be a locally compact Hausdorﬀ space and let P : C0(Ω) → C0(Ω) be a projection. Then P + λ(I − P) is an isometry for some λ ∈ T \{1} if and only if one of the following holds. (i) P is hermitian, (ii) λ = −1 and there exist a homeomorphism ϕ:Ω → Ω satisfying ϕ2 = I and a continuous unimodular function u :Ω → C satisfying u(ϕ(ω)) = u(ω) for every ω ∈ Ω, such that 1 P(f )(ω) = f (ω) + u(ω)f ϕ(ω) , f ∈ C (Ω), ω ∈ Ω. 2 0

Corollary (A. Foˇsnerand D. I., 2011) Let P : S(H) → S(H) be a nontrivial projection and λ ∈ T \{1}. Then P + λ(I − P) is an isometry if and only if λ = −1 and there exists Q = Q∗ = Q2 ∈ B(H) such that P or I − P has the form X 7→ QXQt + (I − Q)X (I − Qt ).

Corollary (A. Foˇsnerand D. I., 2011) Let P : A(H) → A(H) be a nontrivial projection and λ ∈ T \{1}. Then P + λ(I − P) is an isometry if and only if one of the following holds: (i) P or I − P has the form X 7→ QX + XQt , where Q = x ⊗ x for some norm one x ∈ H, (ii) λ = −1 and there exists Q = Q∗ = Q2 ∈ B(H) such that P or I − P has the form X 7→ QXQt + (I − Q)X (I − Qt ).

If P is a projection such that

T def= P + λ(I − P)

is an isometry for some λ ∈ T \{1}, then T is a surjective isometry and σ(T ) = {1, λ}.

Conversely, if T is a surjective isometry with σ(T ) = {1, λ}, λ 6= 1, then |λ| = 1 and

T − λI P def= 1 − λ is a projection.

If P is a projection such that

T def= P + λ(I − P)

is an isometry for some λ ∈ T \{1}, then T is a surjective isometry and σ(T ) = {1, λ}.

Conversely, if T is a surjective isometry with σ(T ) = {1, λ}, λ 6= 1, then |λ| = 1 and

T − λI P def= 1 − λ is a projection.

Every isolated point in the spectrum σ(T ) of a surjective isometry T on a Banach space is an eigenvalue of T with a complemented eigenspace. In particular, if σ(T ) = {λ0, λ1, . . . , λn−1} then all λi ’s are eigenvalues, and the associated eigenprojections Pi ’s satisfy

P0 ⊕ P1 ⊕ · · · ⊕ Pn−1 = I and T = P0 + λ1P1 + ··· + λn−1Pn−1.

Here, we write P ⊕ Q to indicate that the Banach space projections P and Q disjoint from each other, i.e., PQ = QP = 0.

Deﬁnition

Let P0 be a nonzero projection on a Banach space X , and n ≥ 2. We call P0 a generalized n-circular projection if there exists a (surjective) isometry T : X → X with σ(T ) = {1, λ1, . . . , λn−1} consisting of n distinct (modulus one) eigenvalues such that P0 is the eigenprojection of T associated to λ0 = 1. In this case, there are nonzero projections P1,..., Pn−1 on X such that

P0 ⊕ P1 ⊕ · · · ⊕ Pn−1 = I and T = P0 + λ1P1 + ··· + λn−1Pn−1.

We also say that P0 is a generalized n-circular projection associated with (λ1, . . . , λn−1, P1,..., Pn−1).

Generalized n-circular projections on C0(Ω)

Let Ω be a locally compact Hausdorﬀ space. m Let ϕ:Ω → Ω be a homeomorphism with period m, i.e., ϕ = idΩ k and ϕ 6= idΩ for k = 1, 2,..., m − 1. Let u be a continuous unimodular scalar function on Ω such that

u(ω) ··· u(ϕm−1(ω)) = 1, ω ∈ Ω.

Then the surjective isometry T : C0(Ω) → C0(Ω) deﬁned by

Tf (ω) = u(ω)f (ϕ(ω))

satisﬁes T m = I . Therefore, the spectrum σ(T ) = {λ0, λ1, . . . , λn−1} consists of n distinct mth roots of unity. Replacing T with λ0T , we can assume that λ0 = 1.

Generalized n-circular projections on C0(Ω) This gives rise to a spectral decomposition

I = P0 ⊕ P1 ⊕ · · · ⊕ Pn−1, T = λ0P0 + λ1P1 + ··· + λn−1Pn−1. Here, the spectral projections are deﬁned by m−1 (I + λ T + ··· + λ T m−1)f (ω) P f (w) = i i i m 1 = f (ω) + λ u(ω)f ϕ(ω) + ... m i m−1 m−2 m−1 + λi u(ω) ... u ϕ (ω) f ϕ (ω)

for all f ∈ C0(Ω), ω ∈ Ω, and i = 0, 1,..., n − 1. An mth root λ of unity does not belong to σ(T ) if and only if m−1 I + λT + ··· + λ T m−1 = 0.

Generalized n-circular projections on C0(Ω)

Theorem (D. I. , C.-N. Liu and N.-C. Wong) Let Ω be a connected locally compact space. Let T be a surjective isometry of C0(Ω) with ﬁnite spectrum consisting of n points. Then all eigenvalues of T are of ﬁnite orders.

Deﬁnition

We call the generalized n-circular projection P0 periodic (resp. primitive) if it is an eigenprojection of a periodic surjective isometry T of period m ≥ n (resp. of period m = n).

Example i 2π Let n be a positive integer and let τ = e n . Let T be the unit circle in the complex plane. Then Tf (z) = f (τz) is a surjective isometry of C(T), and

σ(T ) = {1, τ, . . . , τ n−1}.

Generalized bicircular and tricircular projections on C0(Ω)

Theorem Let Ω be a connected compact Hausdorﬀ space and let P0 : C0(Ω) → C0(Ω) be a projection. Then the following holds. (i) [F. Botelho, 2008] If T = P0 + λ1P1, with P0 ⊕ P1 = I , is an isometry for some λ1 ∈ T \{1} then σ(T ) = {1, −1}. (ii) [A. B. Abubaker and S. Dutta, 2011] If T = P0 + λ1P1 + λ2P2, with P0 ⊕ P1 ⊕ P2 = I , is an isometry for some distinct λ1, λ2 ∈ T \{1} then i 2π i 4π σ(T ) = {1, e 3 , e 3 }.

Generalized 4-circular projections on C0(Ω) – an example

Example

3 A = {(x, y, z) ∈ R : x, y, z ∈ [0, 1]}, 3 B = {(s, −s, 0) ∈ R : s ∈ [−1, 1]}, Ω = A ∪ B.

(y, z, x), if (x, y, z) ∈ A; ϕ(x, y, z) = (−x, −y, −z), if (x, y, z) ∈ B.

The isometry Tf def= f ◦ ϕ of period 6 has 4 eigenvalues

2 i 2π λ0 = 1, λ1 = −1, λ2 = β, λ3 = β , where β = e 3 .

2 Hence T = P0 − P1 + βP2 + β P3.

Generalized n-circular projections on C0(Ω) – the structure theorem

Theorem (D. I. , C.-N. Liu and N.-C. Wong) Let Ω be a connected locally compact Hausdorﬀ space. Let ϕ:Ω → Ω be a homeomorphism and u be a unimodular continuous scalar function deﬁned on Ω. Let P0 be a generalized n-circular projection on C0(Ω) associated to Tf = u · f ◦ ϕ with the spectral decomposition

I = P0 ⊕ P1 ⊕ · · · ⊕ Pn−1,

T = P0 + λ1P1 + ··· + λn−1Pn−1.

Assume all eigenvalues λ0 = 1, λ1, . . . , λn−1 of T have a (minimum) ﬁnite common period m ≥ n. In particular, all of them are mth roots of unity, and T m = I. Then the following holds.

Generalized n-circular projections on C0(Ω) – the structure theorem

Theorem (continuation) The homeomorphism ϕ has (minimum) period m. The cardinality k(ω) of the orbit {ω, ϕ(ω), ϕ2(ω),...} of each point ω under ϕ is not greater than n. m is the least common multiple of k(ω) for all ω in Ω.

Generalized n-circular projections on C0(Ω) – the structure theorem

Theorem (continuation) The spectrum σ(T ) of T can be written as a union of the complete set of k(ω)th roots of the modulus one scalar k(ω)−1 αω = u(ω)u(ϕ(ω)) ··· u(ϕ (ω)). More precisely,

[ 2 k(ω)−1 σ(T ) = {λω, λωηω, λωηω, . . . , λωηω }, ω∈Ω

where λω and ηω are primitive k(ω)th roots of αω and unity, respectively. We call the set in the union a complete cycle of k(ω)th roots of unity shifted by λω.

Generalized n-circular projections on C0(Ω) – the structure theorem

Theorem (continuation)

If u(ω) = 1 on Ω then we can choose all λω = 1, and thus σ(T ) consists of all k(ω)th roots of unity. If m is a prime integer, then n = m and σ(T ) consists of the complete cycle of nth roots of unity.

Generalized bicircular and tricircular projections on C0(Ω)

Corollary Let Ω be a connected locally compact Hausdorﬀ space. Then every generalized bicircular or tricircular projection P0 on C0(Ω) is primitive. In other words, P0 can only be an eigenprojection of a surjective isometry T on C0(Ω) with a spectral decomposition

T = P0 − (I − P0) for the bicircular case,

2 T = P0 + βP1 + β P2 for the tricircular case, i 2π where β = e 3 .

Generalized 4-circular projections on C0(Ω)

Corollary Let Ω be a connected locally compact Hausdorﬀ space. Let Tf = u · f ◦ ϕ be a surjective isometry on C0(Ω) with the spectral decomposition

T = P0 + λ1P1 + λ2P2 + λ3P3.

Then σ(T ) = {1, λ1, λ2, λ3} can only be one of the following:

{1, −1, i, −i}, {1, −1, β, β2}, {1, −1, −β, −β2},

{1, −β, β, β2}, {1, β, β2, −β2}.

i 2π All above cases can happen. Here β = e 3 .

Generalized 5-circular projections on C0(Ω)

Corollary Let Ω be a connected locally compact Hausdorﬀ space. Let Tf = u · f ◦ ϕ be a surjective isometry on C0(Ω) with the spectral decomposition

T = P0 + λ1P1 + λ2P2 + λ3P3 + λ4P4.

Then σ(T ) = {1, λ1, λ2, λ3, λ4} can only be one of the following:

{1, δ, δ2, δ3, δ4}, {1, −1, β, −β, β2}, {1, −1, β, −β, −β2},

{1, −1, β, β2, −β2}, {1, β, −β, β2, −β2}.

i 2π i 2π All above cases can happen. Here, β = e 3 and δ = e 5 . If u is a constant function, then only the ﬁrst case is allowed.

Generalized 5-circular projections on C0(Ω) – an example

Example

3 A1 = {(1, 0, ρ) ∈ R : ρ ∈ [0, π]}, 2π A = {(1, , ρ) ∈ 3 : ρ ∈ [0, π]}, 2 3 R 4π A = {(1, , ρ) ∈ 3 : ρ ∈ [0, π]}, 3 3 R 3 B = {(r, 0, 0) ∈ R : r ∈ [1/2, 3/2]}, 3 C = {(r, 0, π) ∈ R : r ∈ [1/2, 3/2]}, Ω = A1 ∪ A2 ∪ A3 ∪ B ∪ C.

Generalized 5-circular projections on C0(Ω) – an example

Example

(r, θ + 2π , ρ), if (r, θ, ρ) ∈ A ∪ A ∪ A ; ϕ(r, θ, ρ) = 3 1 2 3 (2 − r, θ, ρ), if (r, θ, ρ) ∈ B ∪ C.

 i 2ρ e 3 , if (r, θ, ρ) ∈ A ∪ A ;  1 2  −i 4ρ u(r, θ, ρ) = e 3 , if (r, θ, ρ) ∈ A3; 1, if (r, θ, ρ) ∈ B;   i 2π  e 3 , if (r, θ, ρ) ∈ C.

Then Tf def= u · f ◦ ϕ has period 6.

Generalized 5-circular projections on C0(Ω) – an example

Example

σ(T ) = {1, β, β2} ∪ {1, −1} ∪ {β, −β} = {1, −1, β, −β, β2}.

Non-primitive generalized n-circular projections on C0(Ω)

Theorem (D. I. , C.-N. Liu and N.-C. Wong) There exists a non-primitive generalized n-circular projection on continuous functions on a connected compact Hausdorﬀ space for each n ≥ 4.

Theorem (D. I., 2017)

Let A be a JB*-triple, and P0 : A → A be a generalized n-circular projection, n ≥ 2, associated with (λ1, . . . , λn−1, P1,..., Pn−1). Let λ0 = 1. Then one of the following holds. (i) There exist i, j, k ∈ {0, 1,..., n − 1}, k 6= i, k 6= j, such that λi λj λk ∈ {λm : m = 0, 1,..., n − 1}.

(ii) All P0,P1,...,Pn−1 are hermitian.

When n = 2: if P is not hermitian then λ2 ∈ {1, λ}, or λ ∈ {1, λ}; hence λ = −1.

When n = 3: if P, Q, R are not hermitian then λ1λ2 = 1, or 2 2 λ1 = λ2, or λ2 = λ1.

Theorem (D. I., 2017)

(ii) All P0,P1,...,Pn−1 are hermitian.

When n = 2: if P is not hermitian then λ2 ∈ {1, λ}, or λ ∈ {1, λ}; hence λ = −1.

When n = 3: if P, Q, R are not hermitian then λ1λ2 = 1, or 2 2 λ1 = λ2, or λ2 = λ1.

Theorem (D. I., 2017)

(ii) All P0,P1,...,Pn−1 are hermitian.

When n = 2: if P is not hermitian then λ2 ∈ {1, λ}, or λ ∈ {1, λ}; hence λ = −1.

When n = 3: if P, Q, R are not hermitian then λ1λ2 = 1, or 2 2 λ1 = λ2, or λ2 = λ1.

Dijana Iliˇsevi´c On isometries on some Banach spaces