A -algebra of singular integral operators with shifts admitting distinct fixed points

J. Math. Anal. Appl. 413 (2014) 502–524

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A C∗-algebra of singular integral operators with shiftsadmitting distinct fixed points ✩

M.A. Bastos a, C.A. Fernandes b, Yu.I. Karlovich c,∗a Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugalb Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica,Portugalc Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Av. Universidad 1001, Col. Chamilpa, C.P. 62209 Cuernavaca, Morelos,México

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 December 2012Available online 4 December 2013Submitted by R. Curto

Keywords:Singular integral operator with shiftsPiecewise slowly oscillating functionC∗-algebraFaithful representationFredholmness

Representations on Hilbert spaces for a nonlocal C∗-algebra B of singular integraloperators with piecewise slowly oscillating coefficients extended by a group of unitaryshift operators are constructed. The group of unitary shift operators U g in the C∗-algebraB is associated with a discrete amenable group G of orientation-preserving piecewisesmooth homeomorphisms g : T → T that acts topologically freely on T and admits distinctfixed points for different shifts. A C∗-algebra isomorphism of the quotient C∗-algebraB/K, where K is the ideal of compact operators, onto a C∗-algebra of Fredholm symbolsis constructed by applying the local-trajectory method, spectral measures and a liftingtheorem. As a result, a Fredholm symbol calculus for the C∗-algebra B or, equivalently,a faithful representation of the quotient C∗-algebra B/K on a suitable Hilbert space isconstructed and a Fredholm criterion for the operators B ∈B is established.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

In this paper we deal with a nonlocal C∗-algebra B generated by a C∗-algebra of singular integral operators withpiecewise slowly oscillating coefficients and by a group of unitary shift operators U G associated with a discrete amenablegroup G of orientation-preserving piecewise smooth homeomorphisms. The aim is to construct a Fredholm symbol map forthe C∗-algebra B, that is, a representation ΨB :B→ B(HB) of B on a convenient Hilbert space HB such that an operatorB ∈ B is Fredholm if and only if ΨB(B) is invertible on HB. Thus, the map ΨB produces a faithful representation of thequotient C∗-algebra B/K on the Hilbert space HB, where K is the ideal of compact operators.

The more complicated the structure of the set of fixed points of shifts is, the harder a Fredholm symbol calculus for B

is constructed and the more complicated the structure of symbols becomes. In the present paper we study the C∗-algebrasB with groups G acting topologically freely. Then shifts g ∈ G can admit both common and non-common fixed points. Thetypical example of such groups is the solvable group G of affine mappings

g :R →R, x �→ kg x + hg (kg > 0, hg ∈R),

where all shifts g ∈ G have the common fixed point x = ∞ and, if kg �= 1, the shifts g ∈ G \ {e} have, in general, distinctfixed points x = hg/(1 − kg).

✩ All the authors were partially supported by FCT project PEst_OE/MAT/UI4032/2011 (Portugal). The third author was also supported by the SEP-CONACYTProject No. 168104 (México) and by PROMEP (México) via “Proyecto de Redes”.

* Corresponding author.E-mail addresses: [email protected] (M.A. Bastos), [email protected] (C.A. Fernandes), [email protected] (Yu.I. Karlovich).

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The existence of non-common fixed points for shifts g ∈ G \ {e} leads to essential difficulties in constructing a Fredholmsymbol for operators B ∈ B as compared with the case of common fixed points for all g ∈ G studied in [4,5]. Moreover,this forced us to apply a new methodology different from those used in the previous papers [4–7] and based on the local-trajectory method and its generalizations using spectral measures (see [15,18,1,2]), which are especially effective if all shiftsg ∈ G \ {e} have the same set of fixed (or periodic) points. The new tools that allowed us to construct the desired symbolmap for the C∗-algebra B are a C∗-algebra modification of the lifting theorem (cf. [14, Theorem 1.8], [21, Theorem 3.3]and [25, Section 6.3]) and ideas borrowed from [20,17,22] and related to Banach algebras of singular integral operators withpiecewise continuous coefficients and discrete subexponential groups of shifts (in particular, the crucial idea of dealing withfinite subsets of orbits instead of whole orbits).

Let B(L2(T)) be the C∗-algebra of all bounded linear operators acting on the space L2(T), where T is the unit circle in C

oriented anticlockwise; PSO(T) be the C∗-algebra of piecewise slowly oscillating functions on T defined in Subsection 2.1;G be a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms g : T → T possessing deriva-tives g′ with at most finite sets of discontinuities, and let G act on T topologically freely, that is, for every finite setG0 ⊂ G \ {e}, where e is the unit of G , and every open set V ⊂ T there exists a point τ ∈ V such that g(τ ) �= τ for everyg ∈ G0 (cf. [1,18]). Suppose G acts on T from the right: (g1 g2)(t) = g2(g1(t)) for all g1, g2 ∈ G and all t ∈ T.

We study the C∗-subalgebra

B := alg(PSO(T), ST, U G

)(1.1)

of B(L2(T)) generated by all multiplication operators aI with a ∈ PSO(T), by the Cauchy singular integral operator STdefined by

(STϕ)(t) := limε→0

1

π i

∫T\T(t,ε)

ϕ(τ )

τ − tdτ , T(t, ε) = {

τ ∈ T: |τ − t| < ε}, t ∈ T, (1.2)

and by the group U G := {U g : g ∈ G} of unitary weighted shift operators U g given by

(U gϕ)(t) := ∣∣g′(t)∣∣1/2

ϕ(

g(t))

for t ∈ T. (1.3)

The paper is organized as follows. Section 2 contains preliminaries: definition of the commutative C∗-algebra PSO(T) ofpiecewise slowly oscillating functions on T and a description of its maximal ideal space M(PSO(T)), as well as a Fredholmsymbol and a Fredholm criterion for the C∗-algebra

A := alg(PSO(T), ST

) ⊂ B (1.4)

generated by the operator ST and all operators aI with a ∈ PSO(T). Section 2 also contains a description of a spectralmeasure associated with a central subalgebra of the C∗-algebra B/K.

Section 3 contains the main results of the paper: a representation ΨB of the C∗-algebra B on a Hilbert space HB withKerΨB =K, where

ΨB =( ⊕

w∈WT

Ψw

)⊕

( ⊕w∈W 0

T

Ψ 0w

), HB :=

( ⊕w∈WT

Hw

)⊕

( ⊕w∈W 0

T

H0w

), (1.5)

WT is the set of all G-orbits of points t ∈ T and W 0T

is the set of G-orbits containing more than one point. As a result, aFredholm symbol map for B is obtained (Theorem 3.1) and a Fredholm criterion for the operators B ∈ B in terms of theirFredholm symbols is established (Theorem 3.2).

In Section 4 we give an example of a Fredholm singular integral operator with shift, which illustrates the conditions ofTheorem 3.2 in an explicit form.

Section 5 contains the main tools for studying the C∗-algebra B: a suitable version of the local-trajectory method(Theorem 5.1), a lifting theorem for C∗-algebras (Theorem 5.2) and its corollary providing sufficient Fredholm conditions(Theorem 5.3).

In Section 6, applying the local-trajectory method, we study the invertibility of functional operators that constitute theC∗-algebra

A := alg(PSO(T), U G

) ⊂ B. (1.6)

In Section 7, making use of results for the C∗-algebra (1.6), we prove that every operator B ∈ B for considered groups Gis of the form

B = A+ P+T

+ A− P−T

+ H B , (1.7)

where A± ∈ A, P±T

:= (I ± ST)/2, H B ∈ H, and H is the closed two-sided ideal in B generated by all commutators[aI, ST] and [U g, ST], where a ∈ PSO(T) and g ∈ G (Theorem 7.2). Using this decomposition we prove in Section 7 that

504 M.A. Bastos et al. / J. Math. Anal. Appl. 413 (2014) 502–524

the ∗-homomorphisms Ψ 0w (w ∈ W 0

T) characterizing the invertibility of functional operators A± in (1.7) and defined initially

on a dense subalgebra B0 of B are continuous (Corollary 7.3).Finally, in Section 8, applying results of Section 2 for the C∗-algebra (1.4) and a crucial relation for the essential norms of

products of operators N ∈ B by operators with fixed singularities (Lemma 8.2), which was obtained by using finite subsetsof G-orbits, we prove the continuity of the mappings Ψw : B → B(Hw) associated with G-orbits w ∈ WT (Theorem 8.3),check the fulfillment of all conditions of the lifting theorem presented in Subsection 5.2 and prove the main results of thepaper.

2. Preliminaries

Given a Hilbert space H, we denote by B(H) the C∗-algebra of all bounded linear operators on H, by K(H) the idealof all compact operators in B(H), and by I ∈ B(H) the identity operator on H. If S, T ∈ B(H) and S − T ∈K(H), we writeS � T . For A ∈ B(H), let Aπ := A +K(H) and

|A| = ∥∥Aπ∥∥ = inf

{‖A + K‖: K ∈ K(H)}.

Given two C∗-algebras A and B, we write A ∼= B if these C∗-algebras are ∗-isomorphic and hence isometric. Given acommutative unital C∗-algebra A, we denote by M(A) the maximal ideal space of A.

2.1. The C∗-algebra of PSO(T) functions

Let C(T), P C(T) and SO(T) denote the C∗-subalgebras of L∞(T) consisting, respectively, of all continuous functions onT, all piecewise continuous functions on T, that is, the functions having one-sided limits at each point t ∈ T, and all slowlyoscillating functions on T, that is (cf. [24]), the functions f that are slowly oscillating at each point λ ∈ T:

limε→0

ess sup{∣∣ f (z1) − f (z2)

∣∣: z1, z2 ∈ Tε(λ)} = 0,

where Tε(λ) := {z ∈ T: ε/2 � |z − λ| � ε}. Denoting by S O λ(T) the C∗-subalgebra of L∞(T) consisting of the continuousfunctions on T \ {λ} that are slowly oscillating at λ ∈ T, we deduce that SO(T) is the smallest C∗-subalgebra of L∞(T)

containing all C∗-algebras S O λ(T) for λ ∈ T.Let PSO(T) := alg(SO(T), P C(T)) be the C∗-subalgebra of L∞(T) generated by the C∗-algebras SO(T) and P C(T). Obvi-

ously, PSO(T) is the closure in L∞(T) of the set P S O 0(T) consisting of all bounded functions on T admitting piecewiseslowly oscillating discontinuities at finite subsets of T and being continuous at all other points of T.

Since C(T) ⊂ SO(T) ⊂ PSO(T), it follows from [3] that

M(SO(T)

) =⋃t∈T

Mt(SO(T)

), M

(PSO(T)

) =⋃

ξ∈M(SO(T))

Mξ

(PSO(T)

), (2.1)

where the fibers of the maximal ideal spaces M(SO(T)) and M(PSO(T)) are given for t ∈ T and ξ ∈ M(SO(T)) by

Mt(SO(T)

) = {ξ ∈ M

(SO(T)

): ξ |C(T) = t

},

Mξ

(PSO(T)

) = {y ∈ M

(PSO(T)

): y|SO(T) = ξ

}, (2.2)

and t( f ) = f (t) for f ∈ C(T). The fibers Mξ (PSO(T)) for ξ ∈ M(SO(T)) can be characterized as follows.

Theorem 2.1. (See [3, Theorem 4.6].) If ξ ∈ Mt(SO(T)) with t ∈ T, then

Mξ

(PSO(T)

) = {(ξ,0), (ξ,1)

}, (2.3)

where, for μ ∈ {0,1}, (ξ,μ)|SO(T) = ξ , (ξ,μ)|C(T) = t, (ξ,μ)|P C(T) = (t,μ); and (t,0)a = a(t − 0) and (t,1)a = a(t + 0) are theleft and right one-sided limits of a function a ∈ P C(T) at the point t ∈ T.

By (2.1) and (2.3), M(PSO(T)) = M(SO(T))×{0,1}. The Gelfand topology on M(PSO(T)) is defined as follows. If t ∈ T andξ ∈ Mt(SO(T)), a base of neighborhoods of (ξ,μ) ∈ M(PSO(T)) consists of all open sets

U (ξ,μ) ={

(Uξ,t × {0}) ∪ (U−ξ,t × {0,1}) if μ = 0,

(Uξ,t × {1}) ∪ (U+ξ,t × {0,1}) if μ = 1,

(2.4)

where Uξ,t = Uξ ∩ Mt(SO(T)), Uξ is an open neighborhood of ξ ∈ M(SO(T)), and U−ξ,t , U+

ξ,t consists of all ζ ∈ Uξ such that

τ = ζ |C(T) belong, respectively, to the open arcs (te−iε, t) and (t, teiε) of T for some ε ∈ (0,2π).


2.2. The C∗-algebra A

Consider the C∗-algebra A of singular integral operators on L2(T) with PSO(T) coefficients, which is given by (1.4). WithA we associate the set

M := M(SO(T)

) ×R, (2.5)

where R = [−∞,+∞]. Let B(M,C2×2) be the C∗-algebra of all bounded matrix functions f : M → C2×2. According to [9,

Section 7] and [4, Theorem 5.1] we have the following symbol calculus for the C∗-algebra A.

Theorem 2.2. The map Sym : {aI: a ∈ PSO(T)} ∪ {ST} → B(M,C2×2) given by the matrix functions[Sym(aI)

](ξ, x) =

(a(ξ,1) 0

0 a(ξ,0)

), [Sym ST](ξ, x) =

(u(x) −v(x)v(x) −u(x)

), (2.6)

where a(ξ,μ) is the Gelfand transform of a function a ∈ PSO(T) at the point (ξ,μ) ∈ M(PSO(T)) and

u(x) := tanh(πx), v(x) := −i/ cosh(πx) for x ∈ R, (2.7)

extends to a C∗-algebra homomorphism Sym :A → B(M,C2×2) whose kernel consists of all compact operators on L2(T). An operatorA ∈A is Fredholm on the space L2(T) if and only if det([Sym A](ξ, x)) �= 0 for all (ξ, x) ∈M.

To each point t ∈ T we assign the operator Vt ∈ B(L2(T)) with fixed singularity at t , which is given for z ∈ T by

(Vtϕ)(z) := χ+t (z)

π i

∫T

ϕ(y)χ+t (y)

y + z − 2tdy − χ−

t (z)

π i

∫T

ϕ(y)χ−t (y)

y + z − 2tdy, (2.8)

where χ±t are the characteristic functions of arcs γ ±

t such that γt := γ +t ∪ γ −

t is a neighborhood of t separated from −t ,γ +

t ∩ γ −t = {t}, and γ +

t ∩ (−t, t) = ∅, γ −t ∩ (t,−t) = ∅. The operators Vt for all t ∈ T belong to the C∗-algebra A (see, e.g.,

[4, Lemma 5.3]).Let P consist of all polynomials

∑nk=0 akuk (ak ∈C, n = 0,1, . . .), and

Z := alg{

aI, H P ,t : a ∈ SO(T), P ∈ P, t ∈ T} ⊂ B

(L2(T)

)(2.9)

be the C∗-subalgebra of A generated by the operators aI (a ∈ SO(T)) and

H P ,t := P(χ+

t STχ+t I − χ−

t STχ−t I

)Vt ∈A (P ∈ P, t ∈ T).

By [4, (4.10)–(4.11)] and [22, (5.24)], we get

aH P ,t � H P ,taI, STH P ,t � H P ,t ST, U g H P ,t � H P ,g−1(t)U g (2.10)

for all a ∈ PSO(T), t ∈ T and g ∈ G . Moreover, because

bST � STbI for all b ∈ SO(T), (2.11)

we conclude that Zπ := (Z +K)/K is a central C∗-subalgebra of the C∗-algebra Aπ := A/K, where K := K(L2(T)). Giventhe set

.M := M

(SO(T)

) × .R (2.12)

with.R = R ∪ {∞}, we infer from [4, Theorem 6.3] that Zπ ∼= C(

.M), where

.M is the compact Hausdorff space equipped

with the Gelfand topology whose neighborhood base of a point (ξ, x) ∈ .M consists of all open sets of the form

W (ξ,x) ={

Uξ,t × (x − ε, x + ε) if (ξ, x) ∈ M(SO(T)) ×R,

(Uξ × .R) \ (Uξ,t × [−ε, ε]) if (ξ, x) ∈ M(SO(T)) × {∞},

ε > 0, Uξ is an open neighborhood of a point ξ ∈ M(SO(T)), and Uξ,t = Uξ ∩ Mt(SO(T)) for t = ξ |C(T) ∈ T. As usual,a(ξ) := ξ(a) for a ∈ SO(T).

Below we need the next result obtained by analogy with [4, Lemma 5.4].

Lemma 2.3. If g is an orientation-preserving piecewise smooth homeomorphism on T with a fixed point t ∈ T, then U g Vt � Vt U g ∈ A

and [Sym(U g Vt)

](ξ, x) :=

{diag{eix ln g′(t+0)v(x), eix ln g′(t−0)v(x)} if (ξ, x) ∈ Mt(SO(T)) ×R,

02×2 if (ξ, x) ∈M \ (Mt(SO(T)) ×R),

where v(x) = −i/ cosh(πx) for x ∈ R and M is given by (2.5).


2.3. The spectral measure associated with the C∗-algebra Zπ

Let B = alg(PSO(T), ST, U G) be the C∗-subalgebra of B(L2(T)) defined by (1.1)–(1.3), and let Bπ := B/K. Consider anisometric representation

ϕ : Bπ → B(Hϕ), Bπ �→ ϕ(

Bπ)

(2.13)

of the C∗-algebra Bπ on an abstract Hilbert space Hϕ . Let R(.M) be the σ -algebra of all Borel subsets of the compact

.M

given by (2.12), and let

Pϕ : R(.M) → B(Hϕ) (2.14)

be the spectral measure associated to the representation (2.13) restricted to the commutative unital C∗-algebra Zπ , whereZ is defined by (2.9). Since all shifts g ∈ G are piecewise smooth homeomorphisms, we have the relations

U gaU∗g = (a ◦ g)I ∈A, U g STU∗

g ∈A for all a ∈ PSO(T) and g ∈ G (2.15)

(see, e.g., [3, Lemma 4.2] and [7, Theorem 2.4]). Thus, for all g ∈ G the mappings αg : Aπ �→ Uπg Aπ (Uπ

g )−1 are∗-automorphisms of the C∗-algebras Aπ and Zπ . These ∗-automorphisms induce on M(Zπ ) = .

M the group of homeo-morphisms βg : .

M → .M, (ξ, x) �→ (g(ξ), x), where ξ �→ g(ξ) is the homeomorphism on M(SO(T)) given by

a(

g(ξ)) = (a ◦ g)(ξ) for all a ∈ SO(T) and all ξ ∈ M

(SO(T)

). (2.16)

Taking the G-invariant subset of R(.M) given by

RG(.M) := {

� ∈R(.M): βg(�) = � for all g ∈ G

},

we conclude from [18] that, for each � ∈ RG(.M) and each operator B ∈B,

Pϕ(�)ϕ(

Bπ) = ϕ

(Bπ

)Pϕ(�).

For every point t ∈ T we introduce the open subset of.M given by

M◦t := Mt

(SO(T)

) ×R. (2.17)

If t is a fixed point for all g ∈ G , for every function a ∈ SO(T) it follows that a(g(ξ)) = a(ξ) for all ξ ∈ Mt(SO(T)) (see, e.g.,[4, Theorem 6.4]), and therefore M◦

t is a set of fixed points for all homeomorphisms βg (g ∈ G). Consequently, M◦t ∈ RG(

.M),

while M◦t /∈RG(

.M) if g(t) �= t for some g ∈ G .

For each g ∈ G , the homeomorphism ξ �→ g(ξ) defined by (2.16) sends the fibers Mt(SO(T)) onto the fibers Mg(t)(SO(T)).Hence, similarly to [7, Lemma 4.2], we obtain the following.

Lemma 2.4. For every t ∈ T and every g ∈ G,

Pϕ

(M◦

t

)ϕ

(Uπ

g

) = ϕ(Uπ

g

)Pϕ

(M◦

g(t)

). (2.18)

Below we will also use the following result.

Lemma 2.5. (See [4, Corollary 9.3].) For any t ∈ T, the map

Sym◦t : Pϕ

(M◦

t

)ϕ

(Aπ

) → B(l2

(M◦

t ,C2)), Pϕ

(M◦

t

)ϕ

(Aπ

) �→ (Sym A)|M◦t

I

is an isometric C∗-algebra homomorphism. An operator Pϕ(M◦t )ϕ(Aπ ) for A ∈A is invertible on the space Pϕ(M◦

t )Hϕ if and only if

det([Sym A](ξ, x)

) �= 0 for all (ξ, x) ∈ Mt(SO(T)

) ×R.

3. Main results

Let TG denote the set of common fixed points for all g ∈ G . The set WT of all G-orbits G(t) = {g(t): g ∈ G} of pointst ∈ T has the form

WT = WTG ∪ W 0T, (3.1)

where WTG is the set of all one-point G-orbits on T and W 0 := WT \ WTG .
T


Given t, τ ∈ T, we define the set

Yt,τ := {g ∈ G: g(t) = τ

}. (3.2)

Fix tw ∈ w for every w ∈ WT and, for each τ ∈ w , fix gτ ∈ Ytw ,τ such that gtw = e, the unit of G . For every g ∈ Yt,τ witht, τ ∈ w , we deduce that

gt,τ := gt gg−1τ ∈ Ytw ,tw (3.3)

because

gt,τ (tw) = (gt gg−1

τ

)(tw) = g−1

τ

[g(

gt(tw))] = tw .

If t ∈ TG , then G(t) = {t}, Yt,t = G , and gt,t = g for all g ∈ Yt,t .With the C∗-algebra B we associate the Hilbert space

HB :=( ⊕

w∈WT

Hw

)⊕

( ⊕w∈W 0

T

H0w

)(3.4)

where the Hilbert spaces

Hw := l2(Mtw

(SO(T)

) ×R, l2(

w,C2)) (w ∈ WT),

H0w := l2

(Mtw

(SO(T)

) × {±∞}, l2(G,C2)) (

w ∈ W 0T

)(3.5)

consist of l2(w,C2)-valued functions defined on the set Mtw (SO(T)) × R and of l2(G,C2)-valued functions defined on theset Mtw (SO(T)) × {±∞}, where all these functions have at most countable sets of non-zero values. In its turn, l2(X,C2) forX ∈ WT∪{G} is the Hilbert space of vectors f = ( fτ )τ∈X with at most countable sets of non-zero entries fτ = ( fτ ,i)

2i=1 ∈C

2

and the norm

‖ f ‖l2(X,C2) :=( ∑

τ∈X

‖ fτ ‖2C2

)1/2

, ‖ fτ ‖2C2 := | fτ ,1|2 + | fτ ,2|2.

Thus, the norms of vector functions

F : Mtw

(SO(T)

) ×R→ l2(

w,C2), (ξ, x) �→ F (ξ, x) = (Fτ (ξ, x)

)τ∈w ,

Φ : Mtw

(SO(T)

) × {±∞} → l2(G,C2), (ξ, x) �→ Φ(ξ, x) = (

Φg(ξ, x))

g∈G

in the Hilbert spaces Hw and H0w are given, respectively, by

‖F‖Hw :=( ∑

(ξ,x)∈Mtw (SO(T))×R

∑τ∈w

∥∥Fτ (ξ, x)∥∥2C2

)1/2

,

‖Φ‖H0w

:=( ∑

(ξ,x)∈Mtw (SO(T))×{±∞}

∑g∈G

∥∥Φg(ξ, x)∥∥2C2

)1/2

.

We now construct a representation

ΨB : B→ B(HB), B �→( ⊕

w∈WT

Ψw(B)

)⊕

( ⊕w∈W 0

T

Ψ 0w(B)

)(3.6)

of the C∗-algebra B on the Hilbert space (3.4). A Fredholm criterion for the operators B ∈ B will be described in terms ofinvertibility of the operators ΨB(B) on the space HB. Hence, the representation ΨB : B → B(HB) can be referred to asthe Fredholm symbol map for the C∗-algebra B, and ΨB can be considered as the direct sum of the following C∗-algebrahomomorphisms

Ψw : B→ B(Hw), B �→ Ψw(B) = Symw(B)I (w ∈ WT),

Ψ 0w : B→ B

(H0

w

), B �→ Ψ 0

w(B) = Sym0w(B)I

(w ∈ W 0

T

), (3.7)

defined initially on the generators of the C∗-algebra B.In (3.7), Ψw(B) are operators of multiplication by finite or infinite matrix functions Symw(B) : Mtw (SO(T)) × R →

l2(w,C2) whose values at the points (ξ, x) ∈ Mtw (SO(T))×R define bounded linear operators on the Hilbert space l2(w,C2)

and are given on the generators of the C∗-algebra B by


[Symw(aI)

](ξ, x) := diag

{((a ◦ gt)(ξ,1) 0

0 (a ◦ gt)(ξ,0)

)}t∈w

(a ∈ PSO(T)

),

[Symw(ST)

](ξ, x) := diag

{(tanh(πx) i/ cosh(πx)

−i/ cosh(πx) − tanh(πx)

)}t∈w

,

[Symw(U g)

](ξ, x) :=

((δg(t, τ )eixk+

g,t,τ 0

0 δg(t, τ )eixk−g,t,τ

))t,τ∈w

(g ∈ G), (3.8)

where δg(t, τ ) = 1 if g ∈ Yt,τ and δg(t, τ ) = 0 if g /∈ Yt,τ , k±g,t,τ := ln[g′

t,τ (tw ± 0)] and g′t,τ (tw ± 0) > 0.

Further, Ψ 0w(B) are operators of multiplication by finite or infinite matrix functions Sym0

w(B) given on Mtw (SO(T)) ×{±∞}, where the values of these matrix functions at the points (ξ, x) ∈ Mtw (SO(T))×{±∞} define bounded linear operatorson the space l2(G,C2) and are given on the generators of the C∗-algebra B as follows:[

Sym0w(aI)

](ξ, x) := diag

{((a ◦ h)(ξ,1) 0

0 (a ◦ h)(ξ,0)

)}h∈G

(a ∈ PSO(T)

),

[Sym0

w(ST)](ξ, x) := diag

{(tanh(πx) i/ cosh(πx)

−i/ cosh(πx) − tanh(πx)

)}h∈G

,

[Sym0

w(U g)](ξ, x) :=

((δhg,s 0

0 δhg,s

))h,s∈G

(g ∈ G), (3.9)

where δh,s = 1 if h = s and δh,s = 0 if h �= s.Identifying the Hilbert space l2(w,C2) with C

2 for every G-orbit w ∈ WTG , we will prove below the following mainresults of the paper.

Theorem 3.1. The map ΨB defined on the generators of the C∗-algebra B by formulas (3.6)–(3.9) extends to a C∗-algebra homo-morphism ΨB of B into the C∗-algebra B(HB) such that ‖ΨB(B)‖ � |B| for all B ∈ B, and KerΨB coincides with the ideal of allcompact operators in the C∗-algebra B(L2(T)).

Theorem 3.2. An operator B ∈ B is Fredholm on the space L2(T) if and only if the operator ΨB(B) is invertible on the space HB,that is, if the following three conditions hold:

(i) for every w ∈ WTG and every (ξ, x) ∈ Mtw (SO(T)) ×R the 2 × 2 matrix [Symw(B)](ξ, x) is invertible and

infw∈WTG

inf(ξ,x)∈Mtw (SO(T))×R

∣∣det([

Symw(B)](ξ, x)

)∣∣ > 0;

(ii) for every w ∈ W 0T

and every (ξ, x) ∈ Mtw (SO(T))×R the operator [Symw(B)](ξ, x)I is invertible on the Hilbert space l2(w,C2)

and

supw∈W 0

T

sup(ξ,x)∈Mtw (SO(T))×R

∥∥([Symw(B)

](ξ, x)I

)−1∥∥B(l2(w,C2))

< ∞;

(iii) for every w ∈ W 0T

and every (ξ, x) ∈ Mtw (SO(T)) × {±∞} the operator [Sym0w(B)](ξ, x)I is invertible on the Hilbert space

l2(G,C2) and

supw∈W 0

T

sup(ξ,x)∈Mtw (SO(T))×{±∞}

∥∥([Sym0

w(B)](ξ, x)I

)−1∥∥B(l2(G,C2))

< ∞.

4. Example of a Fredholm singular integral operator with shift

Consider an example that illustrates the conditions of Theorem 3.2 in an explicit form. Let the group G consist of theshifts

gk,h : T → T, t �→ ki(1 + t) + (h − i)(1 − t)

ki(1 + t) + (h + i)(1 − t)(k > 0, h ∈R),

which are similar to affine mappings x �→ kx + h on R. Then, according to (3.1), TG = {1}, the sets WTG and W 0T

consist ofthe G-orbits w1 = {1} and w−1 = T \ {1}, respectively. If k > 0 and k �= 1, then for every h ∈ R the shift gk,h has two fixedpoints on T: the point tk,h = [h − i(1 − k)]/[h + i(1 − k)] and the point 1, at which g′

k,h(tk,h) = k and g′k,h(1) = k−1.

Fix a shift g = gk,h with k �= 1 and put t0 := tk,h . Consider the operator

B := (aI − bU g)P+ + (cI − dU g)P− ∈B⊂ B(L2(T)

), (4.1)

T T


where a,b, c,d ∈ PSO(T), the unitary weighted shift operator U g is given by (1.3) and P±T

= (I ± ST)/2. Let us establishexplicit Fredholm conditions for the operator (4.1) by applying Theorem 3.2.

By (3.8)–(3.9) and (4.1), we obtain[Symw1

(B)](ξ, x) = (

Bn,m(ξ, x))2

n,m=1 for (ξ, x) ∈ M1(SO(T)

) ×R,

where the entries of the 2 × 2 matrix function Symw1(B) are given by

B1,1(ξ, x) = [a(ξ,1) − b(ξ,1)k−ix]P+(x) + [

c(ξ,1) − d(ξ,1)k−ix]P−(x),

B1,2(ξ, x) = −([a(ξ,1) − b(ξ,1)k−ix] − [

c(ξ,1) − d(ξ,1)k−ix])v(x)/2,

B2,1(ξ, x) = ([a(ξ,0) − b(ξ,0)k−ix] − [

c(ξ,0) − d(ξ,0)k−ix])v(x)/2,

B2,2(ξ, x) = [a(ξ,0) − b(ξ,0)k−ix]P−(x) + [

c(ξ,0) − d(ξ,0)k−ix]P+(x),

and P±(x) = [1 ± tanh(πx)]/2, v(x) = −i/ cosh(πx).Hence, condition (i) of Theorem 3.2 takes the form:

inf(ξ,x)∈M1(SO(T))×R

∣∣det([

Symw1(B)

](ξ, x)

)∣∣ > 0, (4.2)

where

det([

Symw1(B)

](ξ, x)

) = [a(ξ,1) − b(ξ,1)k−ix][c(ξ,0) − d(ξ,0)k−ix]P+(x)

+ [a(ξ,0) − b(ξ,0)k−ix][c(ξ,1) − d(ξ,1)k−ix]P−(x).

Condition (iii) of Theorem 3.2 takes the form: for every ξ ∈ M−1(SO(T)) the operators [Sym0w−1

(B)](ξ,±∞)I are invert-

ible on the space l2(G,C2) and

supξ∈M−1(SO(T))

∥∥([Sym0

w−1(B)

](ξ,±∞)I

)−1∥∥B(l2(G,C2))

< ∞, (4.3)

where [Sym0w−1

(B)](ξ,±∞) = (diag{B+

h,s(ξ,±∞), B−h,s(ξ,±∞)})h,s∈G ,

B+h,s(ξ,+∞) = (a ◦ h)(ξ,1)δh,s − (b ◦ h)(ξ,1)δhg,s,

B+h,s(ξ,−∞) = (c ◦ h)(ξ,1)δh,s − (d ◦ h)(ξ,1)δhg,s,

B−h,s(ξ,+∞) = (c ◦ h)(ξ,0)δh,s − (d ◦ h)(ξ,0)δhg,s,

B−h,s(ξ,−∞) = (a ◦ h)(ξ,0)δh,s − (b ◦ h)(ξ,0)δhg,s,

with δh,s = 1 for h = s and δh,s = 0 for h �= s.In fact (see Theorem 6.2 and relations (7.24) below), condition (iii) of Theorem 3.2 is equivalent to the invertibility of

both the functional operators A+ = aI − bU g and A− = cI − dU g on the space L2(T), which in its turn is equivalent to theinvertibility of these operators on the spaces L2(l1) and L2(l2), where l1 = [1, t0] ⊂ T, l2 = [t0,1] ⊂ T and {1, t0} is the setof fixed points of g on T. Hence, by analogy with [19, Theorem 1.2] one can prove that the operator A+ is invertible on thespace L2(l1) if and only if either

(C1) minξ∈M1(SO(T))

(∣∣a(ξ,1)∣∣ − ∣∣b(ξ,1)

∣∣) > 0, minξ∈Mt0 (SO(T))

(∣∣a(ξ,0)∣∣ − ∣∣b(ξ,0)

∣∣) > 0,

and a(ξ,μ) �= 0 for every (ξ,μ) ∈ ⋃t∈(1,t0) Mt(SO(T)) × {0,1}; or

(C2) maxξ∈M1(SO(T))

(∣∣a(ξ,1)∣∣ − ∣∣b(ξ,1)

∣∣) < 0, maxξ∈Mt0 (SO(T))

(∣∣a(ξ,0)∣∣ − ∣∣b(ξ,0)

∣∣) < 0,

and b(ξ,μ) �= 0 for every (ξ,μ) ∈ ⋃t∈(1,t0) Mt(SO(T))×{0,1}. Analogously, the operator A+ is invertible on the space L2(l2)

if and only if either

(C3) minξ∈Mt0 (SO(T))

(∣∣a(ξ,1)∣∣ − ∣∣b(ξ,1)

∣∣) > 0, minξ∈M1(SO(T))

(∣∣a(ξ,0)∣∣ − ∣∣b(ξ,0)

∣∣) > 0,

and a(ξ,μ) �= 0 for every (ξ,μ) ∈ ⋃t∈(t0,1) Mt(SO(T)) × {0,1}; or

(C4) maxξ∈Mt0 (SO(T))

(∣∣a(ξ,1)∣∣ − ∣∣b(ξ,1)

∣∣) < 0, maxξ∈M1(SO(T))

(∣∣a(ξ,0)∣∣ − ∣∣b(ξ,0)

∣∣) < 0,

and b(ξ,μ) �= 0 for every (ξ,μ) ∈ ⋃t∈(t ,1) Mt(SO(T)) × {0,1}.

0


Similarly, the operator A− = cI − dU g is invertible on the space L2(T) if and only if one of the conditions (C1) or (C2)and one of the conditions (C3) or (C4) hold, with a and b replaced by c and d, respectively.

Fix tw−1 = −1. Then condition (ii) of Theorem 3.2 takes the form: for every (ξ, x) ∈ M−1(SO(T)) × R the operator[Symw−1

(B)](ξ, x)I is invertible on the space l2(w−1,C2) and

sup(ξ,x)∈M−1(SO(T))×R

∥∥([Symw−1

(B)](ξ, x)I

)−1∥∥B(l2(w−1,C2))

< ∞, (4.4)

where [Symw−1(B)](ξ, x) = C(ξ, x) −D(ξ, x),

C(ξ, x) = diag{Ct(ξ, x)

}t∈w−1

, Ct(ξ, x) = (Ct,n,m(ξ, x)

)2n,m=1, (4.5)

Ct,1,1(ξ, x) = (a ◦ gt)(ξ,1)P+(x) + (c ◦ gt)(ξ,1)P−(x),

Ct,1,2(ξ, x) = −[(a ◦ gt)(ξ,1) − (c ◦ gt)(ξ,1)

]v(x)/2,

Ct,2,1(ξ, x) = [(a ◦ gt)(ξ,0) − (c ◦ gt)(ξ,0)

]v(x)/2,

Ct,2,2(ξ, x) = (a ◦ gt)(ξ,0)P−(x) + (c ◦ gt)(ξ,0)P+(x);

D(ξ, x) = (Dt,τ (ξ, x)

)t,τ∈w−1

, Dt,τ (ξ, x) = (Dt,τ ,n,m(ξ, x)

)2n,m=1,

Dt,τ ,1,1(ξ, x) = [(b ◦ gt)(ξ,1)P+(x) + (d ◦ gt)(ξ,1)P−(x)

]δg(t, τ )eixk+

g,t,τ ,

Dt,τ ,1,2(ξ, x) = −[(b ◦ gt)(ξ,1) − (d ◦ gt)(ξ,1)

]v(x)δg(t, τ )eixk+

g,t,τ /2,

Dt,τ ,2,1(ξ, x) = [(b ◦ gt)(ξ,0) − (d ◦ gt)(ξ,0)

]v(x)δg(t, τ )eixk−

g,t,τ /2,

Dt,τ ,2,2(ξ, x) = [(b ◦ gt)(ξ,0)P−(x) + (d ◦ gt)(ξ,0)P+(x)

]δg(t, τ )eixk−

g,t,τ .

Changing the order of rows and columns, the matrix function C−D can be represented as a block diagonal matrix function,where the blocks correspond to different G-orbits of points t ∈ w−1 and G = {gn: n ∈ Z} is a cyclic subgroup of G . If, forexample, conditions (C1) and (C3) hold for the operators A± , then condition (ii) of Theorem 3.2 in view of (4.4) can berewritten in the following equivalent form by analogy with conditions (iii) and (i):

inft∈w−1\{t0} inf

(ξ,x)∈M−1(SO(T))×R

∣∣detCt(ξ, x)∣∣ > 0, (4.6)

inf(ξ,x)∈Mt0 (SO(T))×R

∣∣[a(ξ,1) − b(ξ,1)kix][c(ξ,0) − d(ξ,0)kix]P+(x)

+ [a(ξ,0) − b(ξ,0)kix][c(ξ,1) − d(ξ,1)kix]P−(x)

∣∣ > 0, (4.7)

where from (4.5) it follows that

detCt(ξ, x) = (a ◦ gt)(ξ,1)(c ◦ gt)(ξ,0)P+(x) + (a ◦ gt)(ξ,0)(c ◦ gt)(ξ,1)P−(x).

Thus, for example, the operator (4.1) is Fredholm on the space L2(T) if all the conditions (4.2), (C1) and (C3) for A± ,(4.6) and (4.7) are fulfilled.

Since for every real-valued function f ∈ PSO(T) and every t ∈ T we have

minξ∈Mt (SO(T))

f (ξ,1) = lim infε→+0

f(teiε), min

ξ∈Mt (SO(T))f (ξ,0) = lim inf

ε→+0f(te−iε),

we can rewrite conditions (C1) and (C3) for A± in the next equivalent form:

a−1 ∈ L∞(T), lim infε→±0

(∣∣a(eiε)∣∣ − ∣∣b(

eiε)∣∣) > 0, lim infε→±0

(∣∣a(t0eiε)∣∣ − ∣∣b(

t0eiε)∣∣) > 0,

c−1 ∈ L∞(T), lim infε→±0

(∣∣c(eiε)∣∣ − ∣∣d(eiε)∣∣) > 0, lim inf

ε→±0

(∣∣c(t0eiε)∣∣ − ∣∣d(t0eiε)∣∣) > 0. (4.8)

Analogously, conditions (4.2), (4.7) and (4.6) can be rewritten, respectively, in the form:

infx∈R lim inf

ε→+0

∣∣[a(eiε) − b

(eiε)k−ix][c

(e−iε) − d

(e−iε)k−ix]P+(x)

+ [a(e−iε) − b

(e−iε)k−ix][c

(eiε) − d

(eiε)k−ix]P−(x)

∣∣ > 0, (4.9)

infx∈R lim inf

ε→+0

∣∣[a(t0eiε) − b

(t0eiε)kix][c

(t0e−iε) − d

(t0e−iε)kix]P+(x)

+ [a(t0e−iε) − b

(t0e−iε)kix][c

(t0eiε) − d

(t0eiε)kix]P−(x)

∣∣ > 0, (4.10)

inft∈w \{t } min lim inf

∣∣a(teiε)c

(te−iε)μ + a

(te−iε)c

(teiε)(1 − μ)

∣∣ > 0. (4.11)
−1 0 μ∈[0,1] ε→+0


Suppose now that the coefficients a,b, c,d of the operator (4.1) are continuous functions on the set T \ {1, t0, t1, . . . , tn},where t1, t2, . . . , tn are distinct points in T \ {1, t0}, and coincide at sufficiently small semi-neighborhoods u±

τ of the pointsτ ∈ T := {1, t0, t1, . . . , tn} with functions which are slowly oscillating at τ and given for ε > 0, respectively, by

a(τe±iε) = λ±

a,τ + δ±a,τ sin

(ln(− lnε)

), b

(τe±iε) = λ±

b,τ + δ±b,τ cos

(ln(− lnε)

),

c(τe±iε) = λ±

c,τ + δ±c,τ cos

(ln(− lnε)

), d

(τe±iε) = λ±

d,τ + δ±d,τ sin

(ln(− lnε)

), (4.12)

where λ±f ,τ and δ±

f ,τ are complex constants for all f ∈ {a,b, c,d}.Observe that for any differentiable function f : R→C bounded with its first derivative, the function h(x) = f (ln(− ln |x|))

is continuous on the set (−1,1) \ {0} and slowly oscillating at 0 because limx→0

xh′(x) = 0, which ensures according to (4.12)

that the functions a,b, c,d are in fact in PSO(T).With the functions a,b, c,d ∈ PSO(T) chosen as above, one can see that conditions (4.8) are fulfilled if the parameters in

(4.12) are such that∣∣λ±a,τ

∣∣ >∣∣δ±

a,τ

∣∣, ∣∣λ±c,τ

∣∣ >∣∣δ±

c,τ

∣∣ for all τ ∈ {t1, . . . , tn}, (4.13)∣∣λ±a,τ

∣∣ >∣∣λ±

b,τ

∣∣ + ∣∣δ±a,τ

∣∣ + ∣∣δ±b,τ

∣∣, ∣∣λ±c,τ

∣∣ >∣∣λ±

d,τ

∣∣ + ∣∣δ±c,τ

∣∣ + ∣∣δ±d,τ

∣∣ (4.14)

for all τ ∈ {1, t0}, and

a(t)c(t) �= 0 for all t ∈ T \⋃τ∈T

(u−

τ ∪ {τ } ∪ u+τ

). (4.15)

Indeed, for every τ ∈ T and every f ∈ {a,b, c,d}, the graphs of the functions ε �→ f (τe±iε) for all sufficiently small ε > 0lie in the closed discs

D±f ,τ := {

z ∈C:∣∣z − λ±

f ,τ

∣∣ � ∣∣δ±f ,τ

∣∣},respectively. Inequalities (4.13) and (4.14) imply that the closed discs D±

a,τ and D±c,τ are separated from the origin for all

τ ∈ T , which together with (4.15) ensures the invertibility of the functions a and b in L∞(T). Inequalities (4.14) mean thatthe closed discs D±

a,τ and D±c,τ are separated, respectively, from the closed discs D±

b,τ and D±d,τ for each τ ∈ {1, t0}, and

hence, by (4.14),∣∣a(τe±iε)∣∣ − ∣∣b(

τe±iε)∣∣ � (∣∣λ±a,τ

∣∣ − ∣∣δ±a,τ

∣∣) − (∣∣λ±b,τ

∣∣ + ∣∣δ±b,τ

∣∣) > 0,∣∣c(τe±iε)∣∣ − ∣∣d(τe±iε)∣∣ � (∣∣λ±

c,τ

∣∣ − ∣∣δ±c,τ

∣∣) − (∣∣λ±d,τ

∣∣ + ∣∣δ±d,τ

∣∣) > 0,

which completes the proof of (4.8) on the basis of (4.13)–(4.15).Conditions (4.9)–(4.11) are fulfilled if, in addition to the fulfillment of (4.8), for every τ ∈ {1, t0} both the sectors

Sτ ,1 := {z = reiϕ : r > 0,

∣∣ϕ − arg(λ+

a,τ λ−c,τ

)∣∣ � ν+a,τ + ν−

c,τ

},

Sτ ,2 := {z = reiϕ : r > 0,

∣∣ϕ − arg(λ−

a,τ λ+c,τ

)∣∣ � ν−a,τ + ν+

c,τ

}with excluded vertices lie in the same open complex half-plane whose boundary contains the origin, where, by (4.14),

ν±a,τ := arcsin

((∣∣λ±b,τ

∣∣ + ∣∣δ±a,τ

∣∣ + ∣∣δ±b,τ

∣∣)/∣∣λ±a,τ

∣∣) ∈ (0,π/2),

ν±c,τ := arcsin

((∣∣λ±d,τ

∣∣ + ∣∣δ±c,τ

∣∣ + ∣∣δ±d,τ

∣∣)/∣∣λ±c,τ

∣∣) ∈ (0,π/2);and for every τ ∈ {t1, . . . , tn} both the sectors

S◦τ ,1 := {

z = reiϕ : r > 0,∣∣ϕ − arg

(λ+

a,τ λ−c,τ

)∣∣ �μ+a,τ + μ−

c,τ

},

S◦τ ,2 := {

z = reiϕ : r > 0,∣∣ϕ − arg

(λ−

a,τ λ+c,τ

)∣∣ �μ−a,τ + μ+

c,τ

}with excluded vertices lie in the same open complex half-plane whose boundary contains the origin, where, in view of(4.13),

μ±a,τ := arcsin

(∣∣δ±a,τ

∣∣/∣∣λ±a,τ

∣∣) ∈ [0,π/2), μ±c,τ := arcsin

(∣∣δ±c,τ

∣∣/∣∣λ±c,τ

∣∣) ∈ [0,π/2).

Indeed, if the mentioned conditions are fulfilled, then for every τ ∈ {1, t0} the graphs of the functions (ε, x) �→ a(τe±iε) −b(τe±iε)kix and (ε, x) �→ c(τe±iε) − d(τe±iε)kix for all sufficiently small ε > 0 and all x ∈ R lie, respectively, in the closeddiscs


D±a,τ := {

z ∈C:∣∣z − λ±

a,τ

∣∣ � ∣∣λ±b,τ

∣∣ + ∣∣δ±a,τ

∣∣ + ∣∣δ±b,τ

∣∣},D±

c,τ := {z ∈C:

∣∣z − λ±c,τ

∣∣ � ∣∣λ±d,τ

∣∣ + ∣∣δ±c,τ

∣∣ + ∣∣δ±d,τ

∣∣},which are separated from the origin due to (4.14). Then for every τ ∈ {1, t0} the closed sets{

z1z2 ∈C: z1 ∈ D+a,τ , z2 ∈ D−

c,τ

}and

{z1z2 ∈C: z1 ∈ D−

a,τ , z2 ∈ D+c,τ

}lie in the sectors Sτ ,1 and Sτ ,2, respectively. Analogously, we infer that for every τ ∈ {t1, . . . , tn} the closed sets{

z1z2 ∈C: z1 ∈ D+a,τ , z2 ∈ D−

c,τ

}and

{z1z2 ∈C: z1 ∈ D−

a,τ , z2 ∈ D+c,τ

}lie in the sectors S◦

τ ,1 and S◦τ ,2, respectively. Since the sets{

z1μ + z2(1 − μ): z1 ∈ Sτ ,1, z2 ∈ Sτ ,2, μ ∈ [0,1]} for τ ∈ {1, t0},{z1μ + z2(1 − μ): z1 ∈ S◦

τ ,1, z2 ∈ S◦τ ,2, μ ∈ [0,1]} for τ ∈ {t1, . . . , tn}

lie in open half-planes whose boundaries contains the origin, we conclude that conditions (4.9)–(4.11) are fulfilled.In particular, the operator (4.1) with coefficients a,b, c,d ∈ PSO(T) described above and satisfying (4.12) is Fredholm on

the space L2(T) if

a−1, c−1 ∈ L∞(T), arg((

λ+a,τ λ−

c,τ

)/(λ−

a,τ λ+c,τ

))< π/3 for all τ ∈ T ,∣∣δ±

a,τ

∣∣ <∣∣λ±

a,τ

∣∣/2,∣∣δ±

c,τ

∣∣ <∣∣λ±

c,τ

∣∣/2 for all τ ∈ {t1, . . . , tn},(∣∣λ±b,τ

∣∣ + ∣∣δ±a,τ

∣∣ + ∣∣δ±b,τ

∣∣) <∣∣λ±

a,τ

∣∣/2,(∣∣λ±

d,τ

∣∣ + ∣∣δ±c,τ

∣∣ + ∣∣δ±d,τ

∣∣) <∣∣λ±

c,τ

∣∣/2 for τ ∈ {1, t0},which implies that μ±

a,τ ,μ±c,τ , ν±

a,τ , ν±c,τ ∈ [0,π/6).

5. Local methods for nonlocal C∗-algebras

5.1. The local-trajectory method

To study the C∗-algebra (1.6) of functional operators we will use the following simplified version of the local-trajectorymethod (cf. [18,4]) based on the Allan–Douglas local principle [11,10].

Let D be a commutative C∗-algebra with unit I , G a discrete group with unit e, u : g �→ ug a homomorphism of thegroup G onto a group uG = {ug : g ∈ G} of unitary elements such that ug1 g2 = ug1 ug2 and ue = I . Suppose D and uG arecontained in a C∗-algebra Y and assume that

(A1) for every g ∈ G, the mapping αg : d �→ ugdu∗g is a ∗-automorphism of the commutative C∗-algebra D;

(A2) G is an amenable discrete group.

By [13, §1.2], a discrete group G is called amenable if the C∗-algebra l∞(G) of all bounded complex-valued functions onG with sup-norm has an invariant mean, that is, a positive linear functional ρ of norm 1 such that

ρ( f ) = ρ(s f ) = ρ( f s) for all s ∈ G and all f ∈ l∞(G),

where (s f )(g) = f (s−1 g), ( f s)(g) = f (gs), g ∈ G .Let Q := alg(D, uG) be the minimal C∗-algebra containing the C∗-algebra D and the group uG . By virtue of (A1), Q is

the closure of the set Q0 of elements q = ∑dg ug , where dg ∈D and g runs through finite subsets of G .

Let M := M(D) be the maximal ideal space of the commutative C∗-algebra D. By the Gelfand–Naimark theorem[23, §16], D ∼= C(M). Under assumption (A1), identifying characters ϕm of the C∗-algebra D and the maximal idealsm = Kerϕm ∈ M , we obtain the homomorphism g �→ βg(·) of the group G into the homeomorphism group of M accordingto the rule

d(βg(m)

) = (αg(d)

)(m), d ∈ D, m ∈ M, g ∈ G,

where d(·) ∈ C(M) is the Gelfand transform of the element d ∈D. The set G(m) := {βg(m): g ∈ G} is called the G-orbit of apoint m ∈ M .

Let the next version of topologically free action of G hold (see [15,4]):

(A3) there is a set M0 ⊂ M(D) such that for every finite set G0 ⊂ G \ {e} and every nonempty open set V ⊂ M(D) there exists a pointm0 ∈ V ∩ G(M0) such that βg(m0) �= m0 for all g ∈ G0.


For every m ∈ M , we take the representation πm : D → B(C), d �→ d(m). Given M0 ⊂ M , let Ω(M0) be the set ofG-orbits of all points m ∈ M0. Fix a point m = mω in each G-orbit ω ∈ Ω(M0), and let l2(G) be the Hilbert space of allfunctions f : G �→C such that f (g) �= 0 for at most countable set of points g ∈ G and ‖ f ‖ := (

∑ | f (g)|2)1/2 < ∞. For everyω ∈ Ω(M0) we consider the representation πω :Q→ B(l2(G)) defined by[

πω(d) f](g) = πm

(αg(d)

)f (g),

[πω(uh) f

](g) = f (gh)

for all d ∈D, all g,h ∈ G and all f ∈ l2(G).We infer the following nonlocal version of the Allan–Douglas local principle from [18, Theorems 4.1, 4.12] and [4, Theo-

rem 3.1].

Theorem 5.1. If assumptions (A1)–(A3) are satisfied, then an element q ∈Q is invertible in Q if and only if for every orbit ω ∈ Ω(M0)

the operator πω(q) is invertible on the space l2(G) and, in the case of infinite set Ω(M0),

sup{∥∥(

πω(q))−1∥∥

B(l2(G)): ω ∈ Ω(M0)

}< ∞.

5.2. Lifting theorem

Let B := B(H) be the C∗-algebra of all bounded linear operators on a Hilbert space H and let B be a C∗-subalgebra ofB containing the identity operator I ∈ B. Suppose the ideal K :=K(H) is contained in B.

To investigate the Fredholmness of operators B ∈ B, we will apply the following analogue of the lifting theorem from[14, Theorem 1.8] (see also [25, Section 6.3]), which is a C∗-algebra modification of [21, Theorem 3.3].

Theorem 5.2. Let Λ be an index set and suppose that, for each λ ∈ Λ, we are given a unital C∗-algebra Lλ , a ∗-homomorphismΨλ : B→Lλ , and a closed two-sided ideal Hλ ⊂ B such that:

(i) K ⊂ Hλ ∩ KerΨλ and Hμ ⊂ KerΨλ for all μ ∈ Λ \ {λ};(ii) the restriction of the quotient homomorphism

B/K → Lλ, B +K �→ Ψλ(B)

(which is well defined by (i) and denoted again by Ψλ) onto the ideal Hλ/K is a ∗-isomorphism of Hλ/K onto the closed two-sidedideal Rλ := Ψλ(Hλ) of the C∗-algebra Bλ := Ψλ(B) ⊂Lλ .

Let H be the smallest closed two-sided ideal of B containing all ideals Hλ (λ ∈ Λ). Then an operator B ∈ B is Fredholm if and only ifthe coset B +H is invertible in B/H and for every λ ∈ Λ the element Ψλ(B) is invertible in Lλ .

In addition to the previous conditions of this subsection on the C∗-algebra B⊂ B = B(H), we assume that B containsa unital C∗-algebra A (with unit I ∈ B). Let P± ∈ B \A, P+ + P− = I , and let H0 be the smallest closed two-sided ideal inB containing the ideal K, the operator P+ P− and all commutators [A, P±] = A P± − P± A for A ∈A. Consider an operator

B = A+ P+ + A− P− + H0 ∈B (5.1)

where A± ∈ A and H0 ∈ H0. If the operators A± are invertible in B and therefore (A±)−1 ∈ A, then the coset B + H0is invertible in the C∗-algebra B/H0 with (B + H0)

−1 = (A+)−1 P+ + (A−)−1 P− + H0. Hence, Theorem 5.2 implies thefollowing sufficient Fredholm conditions for the operator (5.1).

Theorem 5.3. If the conditions of Theorem 5.2 are fulfilled with H = H0 , then an operator B ∈ B of the form (5.1) is Fredholm on thespace H if the operators A± are invertible in the C∗-algebraB and for every λ ∈ Λ the element Ψλ(B) is invertible in the C∗-algebra Lλ .

Theorem 5.3 becomes a Fredholm criterion for the operator (5.1) under an addition condition on A, P± and H whichimplies the invertibility of A± from the Fredholmness of B (see, e.g., [20,16] and [21, Theorems 2.1, 3.5]).

6. The C∗-algebra A of functional operators

Using the local-trajectory method we will obtain here an invertibility criterion for the operators in the C∗-algebra A =alg(PSO(T), U G) ⊂ B(L2(T)).

Since U gaU−1g = (a ◦ g)I where a ◦ g ∈ PSO(T) for each g ∈ G and each a ∈ PSO(T) (see, e.g., [3, Lemma 4.2]), the

C∗-algebra A is the closure of the algebra A0 consisting of the functional operators∑

g∈F ag U g , where ag ∈ P S O 0(T) andg runs through finite subsets F ⊂ G .

Consider the central C∗-subalgebra Z := {aI: a ∈ PSO(T)} of A. Then Z ∼= C(M(PSO(T))). For each g ∈ G , the∗-automorphism αg : aI �→ (a ◦ g)I of Z induces on M(PSO(T)) = M(SO(T)) × {0,1} the homeomorphism


βg : (ξ,μ) �→ (g(ξ),μ

)(6.1)

where ξ �→ g(ξ) is given by (2.16). For any g ∈ G , let Tg denote the set of all fixed points of g on T. If t ∈ Tg andξ ∈ Mt(SO(T)), then g(ξ) = ξ by the proof of [4, Theorem 6.4], which in view of (6.1) gives the following.

Lemma 6.1. For each g ∈ G, the set Mg := ⋃t∈Tg

(Mt(SO(T)) × {0,1}) is the set of all fixed points of βg on M(PSO(T)).

Since G acts topologically freely on T, we easily deduce from Lemma 6.1 and the Gelfand topology (2.4) on M(PSO(T))

that the group G acts topologically freely on M(PSO(T)) as well. Moreover, since the set

M0 :=⋃

t∈T\TG

(Mt

(SO(T)

) × {0,1})is dense in M(PSO(T)), we see that for every nonempty open set V ⊂ M(PSO(T)) and every finite set G0 ⊂ G there existsa point (ξ0,μ0) ∈ V ∩ M0 such that βg(ξ0,μ0) �= (ξ0,μ0) for all g ∈ G0 \ {e}. Due to this fact and the amenability of thegroup G , we infer that all conditions of the local-trajectory method (see Subsection 5.1) for the C∗-algebra A are fulfilled.

Clearly, from (6.1) and (2.16) it follows that the set

Ω :=⋃

w∈W 0T

(Mtw

(SO(T)

) × {0,1}) (6.2)

contains exactly one point in each G-orbit of every point in M0. Consider the Hilbert space l2(G) consisting of all complex-valued functions defined on G and having at most countable sets of non-zero values, and with every point (ξ,μ) ∈ Ω weassociate the representation

Πξ,μ : A → B(l2(G)

), A �→ Aξ,μ (6.3)

given on the operator A = ∑g∈F ag U g ∈A0 by

(Aξ,μ f )(h) =∑g∈F

[(ag ◦ h)(ξ,μ)

]f (hg) for all f ∈ l2(G) and h ∈ G. (6.4)

Then we immediately obtain an invertibility criterion from Theorem 5.1.

Theorem 6.2. A functional operator A ∈ A is invertible on the space L2(T) if and only if the operators Aξ,μ for all (ξ,μ) ∈ Ω are

invertible on the space l2(G) and sup(ξ,μ)∈Ω ‖A−1ξ,μ‖ < ∞.

Applying Theorem 6.2 to the operator A A∗ ∈ A and using the relation ‖A‖ = ‖A A∗‖1/2 = [r(A A∗)]1/2, where r(A A∗) isthe spectral radius of the operator A A∗ , we conclude that

‖A‖B(L2(T)) = [r(

A A∗)]1/2 = sup(ξ,μ)∈Ω

[r(

Aξ,μ A∗ξ,μ

)]1/2 = sup(ξ,μ)∈Ω

‖Aξ,μ‖B(l2(G)), (6.5)

whence we obtain the following.

Corollary 6.3. The representation

A → B( ⊕

(ξ,μ)∈Ω

l2(G)

), A �→

⊕(ξ,μ)∈Ω

Πξ,μ(A),

where Ω and Πξ,μ are given by (6.2) and (6.3), respectively, is an isometric C∗-algebra homomorphism.

7. General form of singular integral operators with shifts

We establish here a general form of the operators in the C∗-algebra

B= alg(PSO(T), ST, U G

) ⊂ B(L2(T)

).

Obviously, B= alg(A, U G), where the C∗-algebra A is given by (1.4).Let A0 and B0 be the non-closed algebras of operators of the form

n∑Ti,1Ti,2 . . . Ti, ji (n, ji ∈N) (7.1)

i=1


where Ti,k are, respectively, the generators aI (a ∈ P S O 0(T)) and ST of the C∗-algebra A and the generators aI(a ∈ PSO0(T)), ST and U g (g ∈ G) of the C∗-algebra B. Clearly, A0 is a dense subalgebra of A, B0 is a dense subalge-bra of B, and the non-closed algebra A0 ⊂ B0 consisting of all operators of the form A = ∑

g∈F ag U g with coefficients

ag ∈ P S O 0(T) and finite sets F ⊂ G is dense in the C∗-algebra A of functional operators.Let H be the closed two-sided ideal of B generated by all commutators [aI, ST] = aST − STaI , where a ∈ P S O 0(T), and

by all operators U g STU∗g − ST , where g ∈ G . Then H is the closure of the set

H0 :={

n∑i=1

Bi Hi Ci: Bi, Ci ∈B0, Hi ∈H, n ∈N

}, (7.2)

where, in view of (2.15),

H := {[aI, ST], U g STU∗g − ST: a ∈ P S O 0(T), g ∈ G

} ⊂ A. (7.3)

The ideal K of all compact operators on the space L2(T) is contained in H (see, e.g., [12]). Furthermore, by (7.2) and (7.3),[A, ST] ∈H for all A ∈A. Thus, H can be viewed as the ideal H0 defined in Subsection 5.2.

Given w ∈ WT , let Hw be the closed two-sided ideal of the C∗-algebra B generated by Vtw and K, and let Hπw := Hw/K.

Applying Lemma 2.3 and (2.10), we obtain the following characterization of the ideal Hπ := H/K of the C∗-algebra Bπ byanalogy with [22, Lemma 5.4] and [4, Lemma 10.4].

Lemma 7.1. Every coset Hπ ∈Hπ is represented in the form

Hπ = limm→∞

∑ω∈Λm

Hπω,m (7.4)

where Λm are finite subsets of WT and Hπω,m ∈ Hπ

ω . For every w ∈ WT , the cosets Hπw ∈Hπ

w have the form

Hπw =

{limn→∞(Aπ

n,w V πtw

) if w ∈ WTG ,

limn→∞(∑

t∈Tn

∑g∈Fn

Aπt,g,n V π

t Uπg ) if w ∈ W 0

T,

(7.5)

An,w , At,g,n ∈ A0 , and Tn and Fn are finite subsets of w and G, respectively.

Let A be the C∗-algebra of 2 × 2 diagonal matrices with A-valued entries, where A is the C∗-algebra of functionaloperators considered in Section 6. As in [22, Theorem 5.1] and [4, Theorem 10.3], for B we obtain the following.

Theorem 7.2. Every operator B ∈ B is uniquely represented in the form

B = A+ P+T

+ A− P−T

+ H B , (7.6)

where A± are functional operators in the C∗-algebra A, P±T

= (I ± ST)/2 are the orthogonal projections associated with the Cauchysingular integral operator ST , and H B ∈ H. Moreover, the mapping B �→ diag{A+, A−} is a C∗-algebra homomorphism of B onto Awhose kernel is H, and∥∥A±∥∥� inf

H∈H‖B + H‖� |B| = infK∈K‖B + K‖. (7.7)

Proof. Since U gaU∗g = (a ◦ g)I and U g STU∗

g − ST ∈ H0 for every a ∈ P S O 0(T) and every g ∈ G , we infer from (2.11)

and (2.15) that every operator B ∈ B0 has the form (7.6) where A± ∈ A0 and H B ∈ H0. Moreover, the mapping B �→diag{A+, A−} is an algebraic ∗-homomorphism of the non-closed algebra B0 into A whose kernel is H0. This map is givenon the generators of B by

aI �→ diag{aI,aI}, U g �→ diag{U g, U g}, S �→ diag{I,−I}.It remains to prove (7.7) for all B ∈ B0. Then, by continuity, we will obtain (7.7) for all operators B ∈ B and hence decom-position (7.6) for these B .

Fix B = A+ P+T

+ A− P−T

+ H B ∈B0, where

A± :=∑g∈F

a±g U g ∈ A0, a±

g ∈ P S O 0(T), H B ∈H0,

and F is a finite subset of G . Suppose without loss of generality that the set F is symmetric, that is, g−1 ∈ F for each g ∈ F .In view of (6.5) we have

∥∥A±∥∥B(L2(T))
= sup(ξ,μ)∈Ω

∥∥A±ξ,μ

∥∥B(l2(G))

, (7.8)

where Ω is given by (6.2) and the operators A±ξ,μ are defined as in (6.4). Thus, by (7.8), it suffices to prove that, for all

H ∈ H and all (ξ,μ) ∈ Ω ,∥∥A±ξ,μ

∥∥B(l2(G))

� ‖B + H‖B(L2(T)). (7.9)

Fix H ∈ H0. Then we get

B + H = A+ P+T

+ A− P−T

+ H0, with H0 = H B + H ∈H0. (7.10)

For every function a ∈ P S O 0(T) and every function c ∈ C(T) whose support supp c does not contain the discontinuities ofa, it follows that c[aI, ST] � 0. Further, since the functions g′ ∈ P C(T) have finite sets Λg of discontinuities on T, that is,g′ ∈ P C0(T), it follows that, by analogy with (7.5),(

U g STU∗g − ST

)π =∑

t∈Λg

limn→∞

(Aπ

n,t V πt

) (An,t ∈A0).

Hence, for any operator H0 ∈H0 there is a finite set Λ(H0) ⊂ T such that

χ H0χ I � 0 (7.11)

for all characteristic functions χ of Borel sets on T with suppχ ⊂ T \ Λ(H0).Consider the subgroup F ∞ of G generated by the finite set F and choose a subset G∞ of G containing exactly one

element in each left coset g F ∞ of G with respect to the subgroup F ∞ . Since for every g ∈ G∞ the Hilbert spaces l2(g F ∞)

are invariant with respect to all operators A±ξ,μ with (ξ,μ) ∈ Ω , we infer that every such operator A±

ξ,μ can be represented

as the direct sum of its restrictions to the subspaces l2(g F ∞), where g runs the set G∞ . Thus,∥∥A±ξ,μ

∥∥B(l2(G))

= supg∈G∞

∥∥A±ξ,μ

∥∥B(l2(g F ∞))

. (7.12)

Fix w ∈ W 0T

, (ξ,μ) ∈ Mtw (SO(T)) × {0,1} ⊂ Ω and ε > 0. Then, from (7.12) it follows that there exists a shift g ∈ G∞ suchthat ∥∥A±

ξ,μ

∥∥B(l2(G))

�∥∥A±

ξ,μ

∥∥B(l2(g F ∞))

+ ε. (7.13)

Without loss of generality assume that (7.13) holds for g = e.For each n ∈N, let F n ⊂ F ∞ be the finite set of words of the length � n which are constituted by the elements g ∈ F and

let Πn ∈ B(l2(F ∞)) be the multiplication operator by the characteristic function of the set F n . Obviously, s- limn→∞ Πn = Ion l2(F ∞), and consequently∥∥A±

ξ,μ

∥∥B(l2(F ∞))

� lim infn→∞

∥∥Πn A±ξ,μΠn

∥∥B(l2(F ∞))

. (7.14)

Since for every g ∈ F the functions a±g and g′ belong to P S O 0(T) and P C0(T), respectively, and therefore have finite

sets of discontinuities, we conclude that in each neighborhood of the point tw ∈ T there exists an open arc γ ⊂ T \ {tw }such that all functions a±

g and g′ (g ∈ F ) are continuous on the sets h(γ ) for all h ∈ F n+1. In view of the topologically freeaction of group G on T and due to the continuity of the shifts h ∈ G on T, there exists an open arc u ⊂ γ such that theclosures h(u) of h(u) are disjoint for all h ∈ F n+1, do not intersect the set Λ(H0) and∥∥Πn A±

ξ,μΠn∥∥B(l2(F ∞))

�∥∥Πn A±

ζ,μΠn∥∥B(l2(F ∞))

+ ε

= ∥∥Πn A±ζ,μΠn

∥∥B(l2(F n))

+ ε (7.15)

for every (ζ, μ) ∈ Mu(PSO(T)) := ⋃τ∈u Mτ (SO(T)) × {0,1}.

Let m = mn be the cardinality of the set F n . Since the Hilbert space l2(F n) is isometrically isomorphic to Cm , the Hilbert

space of m-dimensional complex vectors, we infer that for all (ζ, μ) ∈ Mu(PSO(T)) the operators Πn A±ζ,μΠn ∈ B(l2(F n))

admit matrix representations of the form

A±(n)(ζ, μ) := ((

a±h−1s

◦ h)(ζ, μ)

)h,s∈F n = ((

a±h−1s

◦ h)(t)

)h,s∈F n ,

for all (ζ, μ) ∈ Mt(SO(T)) × {0,1} with t ∈ u.Let χn be the characteristic function of the open set un := ⋃

h∈F n h(u). Then suppχn ∩ Λ(H0) = ∅, and therefore from(7.11) we obtain

χn H0χn I � 0. (7.16)


Furthermore, taking a function χn ∈ C(T) such that χn = 1 on un and⋃g∈F

(supp

(χn ◦ g−1)) ∩ (supp χn \ suppχn) = ∅,

and applying the equality (χn ◦ g−1)(χn − χn)I = 0, we deduce that(χn ◦ g−1)P±

Tχn I � (

χn ◦ g−1)χn P±Tχn I = (

χn ◦ g−1)χn P±Tχn I. (7.17)

Consequently, we infer from (7.10), (7.16) and (7.17) that

χn(B + H)χn I � χn A+ P+Tχn I + χn A− P−

Tχn I

= χn

∑g∈F

[a+

g χnU g P+Tχn + a−

g χnU g P−Tχn

]I

= χn

∑g∈F

[a+

g U g(χn ◦ g−1)P+

Tχn + a−

g U g(χn ◦ g−1)P−

Tχn

]I

� χn

∑g∈F

[a+

g U g(χn ◦ g−1)χn P+

Tχn + a−

g U g(χn ◦ g−1)χn P−

Tχn

]I

= χn

∑g∈F

[a+

g U gχn P+Tχn + a−

g U gχn P−Tχn

]I

= χn A+χn P+Tχn I + χn A−χn P−

Tχn I. (7.18)

The continuity of the derivatives g′ on the sets h(γ ) for all h ∈ F n+1 implies the continuity of the derivatives h′ on γ ⊃ ufor all h ∈ F n . Hence, applying the isometric isomorphism

σn : L2(un) → L2m(u), (σnϕ)(t) = (

(Uhϕ)(t))

h∈F n , t ∈ u,

we infer that σn(χn P±Tχn I)σ−1

n � diag{P±u }m

n=1, where P±u = (I ± Su)/2 and Su is given by (1.2) with T replaced by u ⊂ T.

Therefore, we obtain

σn(χn A+χn P+

Tχn I + χn A−χn P−

Tχn I

)σ−1

n � A+(n) P+

u +A−(n) P−

u ∈ B(L2

m(u)). (7.19)

Assuming without loss of generality that u is an interval of the real line R, with operator (7.19) we associate the operator

T := χA+(n)χ P+

Rχ I + χA−

(n)χ P−Rχ I ∈ B

(L2

m(R)),

where χ is the characteristic function of u ⊂ R, P±R

= (I ± SR)/2, SR = F−1sign(·)F , and F is the Fourier transform,(F f )(x) = ∫

Re−ixt f (t)dt , x ∈ R.

Thus, from (7.19) and (7.18) it follows that

|T | = ∣∣χn(B + H)χn I∣∣. (7.20)

Applying [8, Lemma 10.1], we obtain

s- limt→±∞(e−t K et I) = 0 for all K ∈ K

(L2

m(R)),

s- limt→±∞(e−t SRet I) = s- lim

t→±∞(F−1sign

((·) + t

)F

) = ±I, (7.21)

where e±t(x) = e±itx for x, t ∈R. Then we infer from (7.21) that

s- limt→±∞

(e−t(T + K )et I

) = s- limt→±∞(e−t T et I) + s- lim

t→±∞(e−t K et I) = χA±(n)χ I,

for every K ∈K(L2m(R)), and hence∥∥χA±

(n)χ I∥∥� lim inf

t→±∞∥∥e−t(T + K )et I

∥∥� ‖T + K‖.

Consequently, ‖χA±(n)

χ I‖� |T | and, due to the fact that

sup˜∥∥Πn A±

ζ,μΠn∥∥B(l2(F ∞))

= ∥∥A±(n) I

∥∥B(L2

m(u))= ∥∥χA±

(n)χ I∥∥B(L2(R))

,
(ζ,μ)∈Mu(PSO(T))


from (7.20) it follows that, for all (ζ, μ) ∈ Mu(PSO(T)),∥∥Πn A±ζ,μΠn

∥∥B(l2(F ∞))

� |T | = ∣∣χn(B + H)χn I∣∣� ‖B + H‖. (7.22)

Finally, taking into account (7.13) with g = e, (7.14), (7.15) and (7.22), we infer that, for every (ξ,μ) ∈ Mtw (SO(T)) ×{0,1} ⊂ Ω and every w ∈ W 0

T,∥∥A±

ξ,μ

∥∥B(l2(G))

� lim infn→∞

∥∥Πn A±ζ,μΠn

∥∥B(l2(F ∞))

+ 2ε � ‖B + H‖ + 2ε.

This implies (7.9) for all (ξ,μ) ∈ Ω and all H ∈ H due to the arbitrariness of ε > 0, and hence completes the proofof (7.7). �

For every w ∈ W 0T

, we consider the algebraic ∗-homomorphism

Ψ 0w : B0 → B

(H0

w

), B �→ Ψ 0

w(B) = Sym0w(B)I (7.23)

on the dense subalgebra B0 of the C∗-algebra B, where the Hilbert space H0w is defined in (3.5) and the matrix functions

(ξ, x) �→ [Sym0w(B)](ξ, x) for (ξ, x) ∈ Mtw (SO(T)) × {±∞} are defined for operators B ∈ B0 by (3.9).

It is easily seen that for every w ∈ W 0T

there exists a matrix D w such that D w I ∈ B(l2(G,C2)), D w has exactlyone non-zero entry in each row and each column, all these entries equal 1, and the similarity transform Sym0

w(B) �→D w Sym0

w(B)D−1w := (Ai, j)

2i, j=1 changing positions of rows and columns sends odd diagonal entries into A1,1 and even diag-

onal entries into A2,2. Then for any operator B = A+ P+T

+ A− P−T

+ H B ∈ B0, every w ∈ W 0T

and every point ξ ∈ Mtw (SO(T)),we infer from (3.9) and (6.4) that

D w([

Sym0w(B)

](ξ,+∞)

)D−1

w I = diag{

A+ξ,1, A−

ξ,0

},

D w([

Sym0w(B)

](ξ,−∞)

)D−1

w I = diag{

A−ξ,1, A+

ξ,0

}. (7.24)

Hence, applying (7.8) and estimate (7.7), we deduce from (7.24) that∥∥[Sym0

w(B)](ξ,+∞)

∥∥B(l2(G,C2))

= max{∥∥A+

ξ,1

∥∥B(l2(G))

, ‖A−ξ,0‖B(l2(G))

}� max

{∥∥A±∥∥B(L2(T))

}� |B|,∥∥[

Sym0w(B)

](ξ,−∞)

∥∥B(l2(G,C2))

= max{∥∥A−

ξ,1

∥∥B(l2(G))

,∥∥A+

ξ,0

∥∥B(l2(G))

}� max

{∥∥A±∥∥B(L2(T))

}� |B|. (7.25)

Relations (7.25) immediately imply the following.

Corollary 7.3. For every w ∈ W 0T

, the algebraic ∗-homomorphisms Ψ 0w : B0 → B(H0

w) defined on the generators of the C∗-algebraB by formulas (3.7) and (3.9), extends by continuity to representations Ψ 0

w : B → B(H0w) such that ‖Ψ 0

w(B)‖ � |B| for all B ∈ B

and hence KerΨ 0w ⊃K.

8. Homomorphisms into local algebras

8.1. Algebraic homomorphisms Ψw

For every G-orbit w ∈ WT , taking the dense subalgebra B0 of B composed by the operators of the form (7.1), weconsider the algebraic ∗-homomorphism

Ψw : B0 → B(Hw), Ψw(B) = Symw(B)I, (8.1)

where the Hilbert space Hw is given in (3.5) and the matrix function (ξ, x) �→ [Symw(B)](ξ, x) for (ξ, x) ∈ Mtw (SO(T)) ×R

is defined for operators B ∈B0 by formulas (3.8).Given any set α ⊂ T, we define the sets

Mα

(SO(T)

) :=⋃t∈α

Mt(SO(T)

), M◦

α := Mα

(SO(T)

) ×R. (8.2)

By (2.17) and (8.2), M◦{t} = M◦t for all t ∈ T. For any finite set α ⊂ T we introduce the operator

Vα :=∑t∈α

Vt ∈ H, (8.3)

where the operators Vt for t ∈ T are given by (2.8) and H is the closed two-sided ideal being the closure of (7.2). For anyset Y ⊂ T \TG , let

ΠY := diag{

E2χY (t)}

t∈T\TG, (8.4)

where E2 := diag{1,1} and χY is the characteristic function of the set Y .


Lemma 8.1. If N ∈ B0 and w ∈ WTG , then

|N Vtw | = ∥∥Ψw(N)v I∥∥B(Hw )

, (8.5)

where the function v is given by (2.7).

Proof. Fix a G-orbit w = {tw } ∈ WTG . Since U gAU−1g = A for all g ∈ G , we infer that every operator N ∈ B0 has the form

N =∑g∈F

Ag U g (8.6)

where Ag ∈ A and F is a finite subset of G . Let ϕ be the isometric representation of the C∗-algebra Bπ on a Hilbert spaceHϕ considered in (2.13). Then

|N Vtw | = ∥∥ϕ([N Vtw ]π )∥∥B(Hϕ)

. (8.7)

It follows from (8.6), Lemma 2.3 and formulas (3.8) that the operator N Vtw = ∑g∈F Ag U g Vtw belongs to the C∗-algebra A,

and [Sym(N Vtw )

](ξ, x) =

∑g∈F

([SymAg](ξ, x))

diag{

eixk+g,w , eixk−

g,w}

v(x) = [Ψw(N)

](ξ, x)v(x)

if (ξ, x) ∈ Mtw

(SO(T)

) ×R,[Sym(N Vtw )

](ξ, x) = 02×2 if (ξ, x) ∈ M \ (

Mtw

(SO(T)

) ×R), (8.8)

where k±g,w = ln g′(tw ± 0) and g′(tw ± 0) > 0. Hence, applying the spectral projection given by (2.14), we infer from

Lemma 2.5 that∥∥Pϕ

(M◦

tw

)ϕ

([N Vtw ]π )∥∥B(Hϕ)

= ∥∥(Sym(N Vtw )

)∣∣M◦

twI∥∥B(Hw )

= ∥∥Ψw(N)v I∥∥B(Hw )

. (8.9)

Taking the open set MT\{tw }(SO(T)) × .R in

.M, we infer from (8.8) by analogy with [4, Subsection 8.1] that

Pϕ

(MT\{tw }

(SO(T)

) × .R

)ϕ

([N Vtw ]π ) = 0, (8.10)

and, by [4, Lemma 10.5], for the closed set Mtw (SO(T)) × {∞} ⊂ .M,

Pϕ

(Mtw

(SO(T)

) × {∞})ϕ([N Vtw ]π ) = 0. (8.11)

Applying the equalities Pϕ(.M) = I and

Pϕ(.M) = Pϕ

(MT\{tw }

(SO(T)

) × .R

) + Pϕ

(M◦

tw

) + Pϕ

(Mtw

(SO(T)

) × {∞}),we infer from (8.10)–(8.11) that∥∥ϕ([N Vtw ]π )∥∥

B(Hϕ)= ∥∥Pϕ(

.M)ϕ


= ∥∥Pϕ

(M◦

tw

)ϕ


. (8.12)

Combining (8.7), (8.12) and (8.9), we obtain (8.5). �Lemma 8.2. If N ∈ B0 , w ∈ W 0

Tand the operator Vα is given by (8.3), then for every finite set α ⊂ w,

|N Vα | = ∥∥Ψw(N)Πα v I∥∥B(Hw )

. (8.13)

Proof. Fix a G-orbit w ∈ W 0T

and a finite set α ⊂ w and take an operator N ∈ B0 written in the form (8.6) where Ag ∈ A

and F is a finite subset of G . The set β := {g−1(t): t ∈ α, g ∈ F } is a finite subset of w along with α ⊂ w and F ⊂ G . Takingthe isometric ∗-isomorphism ϕ of the quotient C∗-algebra Bπ on a Hilbert space Hϕ from (2.13), we conclude that

|N Vα | = ∥∥ϕ([N Vα]π )∥∥B(Hϕ)

. (8.14)

Applying (8.6) and the relations U g Vt � V g−1(t)U g and Ag Vt � Vt Ag for t ∈ T and Ag ∈ A, we can represent the operatorN Vα in the form


N Vα =∑g∈F

∑t∈α

Ag U g Vt =∑g∈F

∑t∈α

V g−1(t) Ag U g + K , (8.15)

where K ∈K. Taking the symbol

[Sym Vα](ξ, x) ={

diag{v(x), v(x)} if (ξ, x) ∈ Mα(SO(T)) ×R,

0 if (ξ, x) ∈M \ (Mα(SO(T)) ×R),

of the operator Vα ∈ A (see Lemma 2.3), we infer from the second equality in (8.15) by analogy with (8.10) and (8.11) that

Pϕ

(MT\β

(SO(T)

) × .R

)ϕ

([N Vα]π ) = 0

for the open set MT\β(SO(T)) × .R⊂ .

M, and

Pϕ

(Mβ

(SO(T)

) × {∞})ϕ([N Vα]π ) = 0

for the closed set Mβ(SO(T)) × {∞} ⊂ .M. Applying then the partition

.M= M◦

β ∪ (MT\β

(SO(T)

) × .R

) ∪ (Mβ

(SO(T)

) × {∞})we conclude, using (2.18), that∥∥ϕ([N Vα]π )∥∥

B(Hϕ)= ∥∥Pϕ

(M◦

β

)ϕ

([N Vα]π )∥∥B(Hϕ)

= ∥∥Pϕ

(M◦

β

)ϕ

([N Vα]π )Pϕ

(M◦

α

)∥∥B(Hϕ)

. (8.16)

Let G N be the subgroup of G generated by the finite set F and let ΩN,α be the finite set of G N -orbits λ of all pointst ∈ α. Then αλ := α ∩ λ is a finite subset of λ ∈ ΩN,α . Since Vα = ∑

λ∈ΩN,αVαλ , Vαλ = ∑

t∈αλVt and Πα = ∑

λ∈ΩN,αΠαλ ,

and therefore

|N Vα | = maxλ∈ΩN,α

|N Vαλ |,∥∥Ψw(N)Πα v I

∥∥B(Hw )

= maxλ∈ΩN,α

∥∥Ψw(N)Παλ v I∥∥B(Hw )

,

it remains to prove (8.13) for Vα replaced by any Vαλ . Hence, in what follows we assume without loss of generality thatα ⊂ λ and λ = G N (tw). Since the group G N for F �= {e} is at most countable, so is the G N -orbit λ.

Let Hλ := ⊕t∈λ Pϕ(M◦

tw)Hϕ . Obviously, this space is isometrically isomorphic to the Hilbert space l2(M◦

tw, l2(λ,C2)).

Consider the isomorphism

ηλ : Pϕ

(M◦

λ

)Hϕ →

⊕t∈λ

Pϕ

(M◦

tw

)Hϕ, Pϕ

(M◦

λ

)f �→ (

Pϕ

(M◦

tw

)ϕ

(Uπ

gt

)f)

t∈λ, (8.17)

where f ∈ Hϕ and gt for every t ∈ λ is a fixed shift in Ytw ,t given by (3.2). Clearly, for every β ⊂ λ and every f ∈ Hϕ weget

ηλ

(Pϕ

(M◦

β

)f) = Π

βλ

(Pϕ

(M◦

tw

)ϕ

(Uπ

gt

)f)

t∈λ, (8.18)

where

Πβλ = diag

{χβ(t)

}t∈λ

I. (8.19)

Taking now the isometric C∗-algebra homomorphism

Υλ : B(Pϕ

(M◦

λ

)Hϕ

) → B(⊕

t∈λ

Pϕ

(M◦

tw

)Hϕ

), T �→ ηλTη−1

λ , (8.20)

where ηλ is given by (8.17), and applying the relations

Ag,t := U gt Ag U−1gt

∈A, U gt U g U−1gτ

= U gt,τ , U gτ V sU−1gτ

� V g−1τ (s) (s ∈ α)

for t, τ ∈ w , where gt,τ = gt gg−1τ ∈ Ytw ,tw by (3.3), we infer from (8.15), (8.18) and (8.19) that

Υλ

(Pϕ

(M◦

β

)ϕ

([N Vα]π )Pϕ

(M◦

α

))= Υλ

(Pϕ

(M◦

β

)∑g∈F

∑s∈α

ϕ([Ag U g V s]π

)Pϕ

(M◦

α

))

= Πβλ

(Pϕ

(M◦

tw

)∑g∈F

∑s∈α

ϕ([

U gt Ag U−1gt

]π [U gt U g U−1

gτ

]π [U gτ V sU−1

gτ

]π )Pϕ

(M◦

tw

))t,τ∈λ

Παω

= Πβλ

(∑δg(t, τ )Pϕ

(M◦

tw

)ϕ

([Ag,t U gt,τ Vtw ]π ))t,τ∈λ

Παλ . (8.21)

g∈F


On the other hand, from (8.20) and (8.21) it follows that∥∥Pϕ

(M◦

β

)ϕ

([N Vα]π )Pϕ

(M◦

α

)∥∥B(Hϕ)

= ∥∥Υλ

(Pϕ

(M◦

β

)ϕ

([N Vα]π )Pϕ

(M◦

α

))∥∥B(Hλ)

=∥∥∥∥Π

βλ

(∑g∈F

δg(t, τ )Pϕ

(M◦

tw

)ϕ


Παλ

∥∥∥∥B(Hλ)

. (8.22)

Since the coset [Ag,t U gt,τ Vtw ]π belongs to the quotient C∗-algebra Aπ , we have

δg(t, τ )Pϕ

(M◦

tw

)ϕ

([Ag,t U gt,τ Vtw ]π ) ∈ Pϕ

(M◦

tw

)ϕ

(Aπ

)for all t, τ ∈ λ.

Hence, taking into account the finiteness of the sets α,β ⊂ λ in (8.22) and applying entry-wise the isometric C∗-algebrahomomorphism

Sym◦tw

: Pϕ

(M◦

tw

)ϕ

(Aπ

) → B(l2

(M◦

tw,C2))

from Lemma 2.5, we infer from (3.8), (8.1), (8.4), (8.22) and Lemma 2.3 that∥∥∥∥Πβλ

(∑g∈F

δg(t, τ )Pϕ

(M◦

tw

)ϕ


Παλ

∥∥∥∥B(Hλ)

=∥∥∥∥Π

βλ

(∑g∈F

δg(t, τ )(Sym(Ag,t U gt,τ Vtw )

)∣∣M◦

twI

)t,τ∈λ

Παλ

∥∥∥∥B(l2(M◦

tw ,l2(λ,C2)))

=∥∥∥∥Πβ

(∑g∈F

δg(t, τ )(Sym

(Ag,t(U gt,τ Vtw )

))∣∣M◦

twI

)t,τ∈w

Πα I

∥∥∥∥B(Hw )

= ∥∥Ψw(N)Πα v I∥∥B(Hw )

. (8.23)

Finally, combining (8.14), (8.16), (8.22) and (8.23), we obtain (8.13). �8.2. Continuity of homomorphisms Ψw

Theorem 8.3. If N ∈B0 , then for every w ∈ WT ,∥∥Ψw(N)∥∥B(Hw )

� |N|. (8.24)

Proof. Let N ∈ B0. If w ∈ WTG , we deduce from [4, Theorem 9.5] that∥∥Ψw(N)∥∥B(Hw )

= ∥∥Pϕ

(M◦

tw

)ϕ

(Nπ

)∥∥B(Hϕ)

�∥∥ϕ(

Nπ)∥∥

B(Hϕ)= |N|,

which proves (8.24) for w ∈ WTG .Fix w ∈ W 0

T. Take an operator N ∈ B0 of the form (8.6) and consider the at most countable subgroup G N of G generated

by the finite set F ⊂ G . Let ΩN be the set of all G N -orbits λ of points t ∈ w . Using the algebraic ∗-homomorphism Ψw :B0 → B(Hw) given by (8.1) and (3.8), we obtain∥∥Ψw(N)

∥∥B(Hw )

= supλ∈ΩN

∥∥ΠλΨw(N)Πλ I∥∥B(Hw )

. (8.25)

Fix λ ∈ ΩN . Then the G N -orbit λ is at most countable. Consider a sequence {αn}n∈N of finite subsets of λ such thatαn ⊂ αn+1 and

⋃n∈N αn = λ. Since s- limn→∞(Παn I) = Πλ I on the space Hw , we conclude that

ΠλΨw(N)Πλ I = s- limn→∞

(ΠλΨw(N)Παn I

).

Let χk be the characteristic function of the segment [−k,k] ⊂ R. Because s- limk→∞(χk I) = I on the space Hw , we inferthat, for every n ∈N,

ΠλΨw(N)Παn I = s- limk→∞

(ΠλΨw(N)χkΠαn I

)on the space ΠλHw . Hence

∥∥ΠλΨw(N)Πλ I∥∥B(Hw )
� lim infn→∞

∥∥ΠλΨw(N)Παn I∥∥B(Hw )

� lim infn→∞ lim inf

k→∞∥∥ΠλΨw(N)χkΠαn I

∥∥B(Hw )

. (8.26)

Given k ∈ N, let χk be a function in C(R) such that χk(x) = 1 for |x| � k, χk(x) = 0 for |x| � k + 1, and χk(x) ∈ [0,1] for|x| ∈ [k,k + 1]. Further, we infer from [4, Lemmas 6.1 and 6.2] that for every t ∈ w , every k ∈ N and every m ∈N there is anoperator Bt,k,m ∈A0 such that

[Sym(Bt,k,m Vt)

](ξ, x) =

{diag{Pk,m(u(x))v(x), Pk,m(u(x))v(x)} if (ξ, x) ∈ Mt(SO(T)) ×R,

02×2 if (ξ, x) ∈M \ (Mt(SO(T)) ×R),

maxx∈R

∣∣Pk,m(u(x)

)v(x) − χk(x)

∣∣ < 1/m, (8.27)

where Pk,m(u) is a polynomial in u ∈ [0,1], and the functions u(·) and v(·) are given by (2.7). Taking a finite set α ⊂ w ,pairwise disjoint closed neighborhoods γt of points t ∈ α, and the operator Bα,k,m = ∑

t∈α χγt Bt,k,m ∈A0, where χγt are thecharacteristic functions of γt , we conclude that[

Sym(Bα,k,m Vα)](ξ, x) =

{diag{Pk,m(u(x))v(x), Pk,m(u(x))v(x)} if (ξ, x) ∈ Mα(SO(T)) ×R,

02×2 if (ξ, x) ∈M \ Mα(SO(T)) ×R.

Taking α = αn and applying the previous formula, we obtain[Symw(Bαn,k,m)

](ξ, x)v(x)Παn = (

Pk,m(u(x)

)v(x)E2

)t,τ∈wΠαn . (8.28)

Taking into account (8.27), we define the function νk,m ∈ C(R) for m > 1 by

νk,m(x) = [Pk,m

(u(x)

)v(x)

]−1for x ∈ [−k,k], νk,m(±x) = νk,m(±k) for x > k. (8.29)

Making use of (8.28) and (8.29), we infer that

ΠλΨw(N)χkΠαn I = ΠλΨw(N Bαn,k,m)Παn vνk,mχkΠαn I.

Consequently, applying Lemma 8.2, we get∥∥ΠλΨw(N)χkΠαn I∥∥B(Hw )

= ∥∥ΠλΨw(N Bαn,k,m)Παn vνk,mχkΠαn I∥∥B(Hw )

�∥∥Ψw(N Bαn,k,m)Παn v I

∥∥B(Hw )

‖νk,mχkΠαn I‖B(Hw )

�∣∣N Bαn,k,m Vαn

∣∣‖νk,mχkΠαn I‖B(Hw ). (8.30)

By (8.27) and (8.29), we obtain

‖νk,mχkΠαn I‖B(Hw ) = maxx∈[−k,k]

∣∣[Pk,m(u(x)

)v(x)

]−1∣∣� 1/(1 − 1/m). (8.31)

On the other hand, from Lemma 8.2, (8.28) and (8.27) it follows that∣∣Bαn,k,m Vαn

∣∣ = ∥∥Ψw(Bαn,k,m)Παn v I∥∥B(Hw )

= maxx∈R

∣∣Pk,m(u(x)

)v(x)

∣∣ � 1 + 1/m. (8.32)

Thus, according to (8.30)–(8.32), we get∥∥ΠλΨw(N)χkΠαn I∥∥B(Hw )

� |N|(1 + 1/m)/(1 − 1/m). (8.33)

Combining (8.25), (8.26) and (8.33), we conclude that∥∥Ψw(N)∥∥B(Hw )

� |N|(1 − 1/m2),which in view of the arbitrariness of m ∈ N implies the estimate (8.24) for all w ∈ W 0

Tand hence for all w ∈ WT . �

By continuity, estimate (8.24) holds for every N ∈ B and every w ∈ WT . Hence, Theorem 8.3 implies the followingcorollary.

Corollary 8.4. For every w ∈ WT , the algebraic homomorphisms Ψw given by (8.1) and (3.8) extend to representations Ψw : B →B(Hw) such that (8.24) holds for every N ∈ B and KerΨw ⊃K.


Lemma 8.5. For every w ∈ WT , the restriction of the quotient homomorphism Ψw :Bπ → B(Hw), Bπ �→ Ψw(B) onto the ideal Hπw

of Bπ is an isometric ∗-isomorphism of Hπw onto the closed two-sided ideal Ψw(Hw) of the C∗-algebra Ψw(B).

Proof. Since the set {N Vtw : N ∈ B0} for w ∈ WTG is dense in the ideal Hw and Ψw(Vtw ) = v I , and as the set{N Vα: N ∈ B0, α runs finite subsets of w} is dense in the ideal Hw for w ∈ W 0

Tand Ψw(Vα) = Πα v I , we infer from

Lemmas 8.1 and 8.2 that for every w ∈ WT the restriction Ψw |Hw is a ∗-homomorphism of Hw into Lw := B(Hw) suchthat ‖Ψw(H w)‖ = |H w | for every H w ∈ Hw . Consequently, Ker(Ψw |Hw) =K and Ψw is a C∗-algebra isomorphism of Hw/Konto Ψw(Hw). �8.3. Proof of Theorem 3.1

Corollaries 8.4 and 7.3 imply that the map ΨB defined on the generators of the C∗-algebra B by formulas (3.6)–(3.9)extends to a C∗-algebra homomorphism of B into the C∗-algebra B(HB), and for all B ∈ B,

‖ΨB(B)‖ = max{

supw∈WT

∥∥Ψw(B)∥∥, sup

w∈W 0T

∥∥Ψ 0w(B)

∥∥}� |B|. (8.34)

By (8.34), KerΨB ⊃ K. On the other hand, if an operator B ∈ B is of the form (7.6) and ΨB(B) = 0, then Ψ 0w(B) =

Sym0w(B)I = 0 for all w ∈ W 0

T, and from (7.24) and (7.8) it follows that A± = 0, and hence B = H B ∈H. Thus,

Ψw(H B) = Ψw(B) = 0 for all w ∈ WT. (8.35)

By Lemma 7.1, there is a sequence of finite subsets Λm ⊂ WT such that

HπB = lim

m→∞∑

ω∈Λm

Hπω,m (Hω,m ∈Hω). (8.36)

By [22, Lemma 6.9], for every finite set Λ ⊂ WT and any {Hπω ∈ Hπ

ω: ω ∈ Λ},∥∥∥∥∑ω∈Λ

Hπω

∥∥∥∥ = supω∈Λ

∥∥Hπω

∥∥. (8.37)

Fix w ∈ WT . Setting H w,m = Hω,m if w ∈ Λm and w = ω, and setting H w,m = 0 otherwise, we infer from (8.36) and (8.37)that for every w ∈ WT there exists a limit Hπ

B,w = limm→∞ Hπw,m ∈Hw , Hπ

B,w �= 0 for at most countable set of w ∈ WT , and

|H B | = supw∈WT

∥∥HπB,w

∥∥. (8.38)

Applying (8.1), (3.8) and Lemma 2.3, we infer that Ψw(Vt) = 0 for all w ∈ WT and all t ∈ T \ w . This in view of (7.5)implies that

Ψw(Hω) = {0} for all w ∈ WT and all ω ∈ WT \ {w}. (8.39)

Hence, from (8.36), (8.39) and (8.35) it follows that Ψw(HπB,w) = Ψw(H B) = 0 for all w ∈ WT . Thus, we conclude from

Lemma 8.5 that HπB,w =K for all w ∈ WT . Consequently, we infer from (8.38) that

|B| = |H B | = supw∈WT

∥∥HπB,w

∥∥ = 0,

whence KerΨB =K, which completes the proof of Theorem 3.1.

8.4. Proof of Theorem 3.2

Sufficiency. Let Λ = WT and Lw = B(Hw) for all w ∈ WT . It follows from Corollary 8.4, Lemma 8.5 and relations (8.39)that all the conditions of Theorem 5.2 are fulfilled for the C∗-algebra B defined by (1.1), for the ideal H0 of B generatedby all commutators [aI, ST] (a ∈ P S O 0(T)) and by all operators U g STU∗

g − ST (g ∈ G), which coincides with the ideal H

generated by the ideals Hw (w ∈ WT), and for the representations Ψw : B→ B(Hw) given by (8.1) for all w ∈ WT , wherethe Hilbert spaces Hw are defined in (3.5). Hence, by Theorems 7.2 and 5.3, an operator B ∈ B is Fredholm on the spaceL2(T) if the functional operators A± in (7.6) are invertible in the C∗-algebra B(L2(T)) and for every w ∈ WT the operatorΨw(B) is invertible in the C∗-algebra B(Hw).

Fix an operator B = A+ P+T

+ A− P−T

+ H B in B. If conditions (iii) of Theorem 3.2 are fulfilled, then by relations (7.24)and Theorem 6.2 the functional operators A± are invertible on the space L2(T). On the other hand, the fulfillment ofconditions (i) and (ii) of Theorem 3.2 implies the invertibility of operators Ψw(B) ∈ B(Hw) for all w ∈ WTG and all w ∈ W 0

T,

respectively. Hence, we conclude from Theorem 5.3 that the operator B ∈B is Fredholm on the space L2(T).


Necessity. Let an operator B ∈ B be Fredholm on the space L2(T) or, equivalently, the coset Bπ be invertible in thequotient C∗-algebra Bπ . Since KerΨB =K by Theorem 3.1, the quotient ∗-homomorphism

ΨB : B/K → B(HB), N +K �→ ΨB(N),

is a C∗-algebra isomorphism. Consequently, ΨB(B) is invertible on the space HB and then, according to (3.6), the operatorsΨw(B) are invertible on the spaces Hw for all w ∈ WT , the operators Ψ 0

w(B) are invertible on the spaces H0w for all

w ∈ W 0T

, and the norms of their inverses are uniformly bounded. This, in view of (3.7)–(3.9), immediately implies parts(i)–(iii) of Theorem 3.2, which completes the proof.

Acknowledgments

The authors are grateful to the referee for the useful suggestion to present an example illustrating conditions of Theo-rem 3.2 in an explicit form.

References

[1] A. Antonevich, Linear Functional Equations. Operator Approach, Oper. Theory Adv. Appl., vol. 83, Birkhäuser, Basel, 1996, Russian original: UniversityPress, Minsk, 1988.

[2] A. Antonevich, A. Lebedev, Functional Differential Equations: I. C∗-Theory, Pitman Monogr. Surv. Pure Appl. Math., vol. 70, Longman, Harlow, 1994.[3] M.A. Bastos, C.A. Fernandes, Yu.I. Karlovich, C∗-algebras of integral operators with piecewise slowly oscillating coefficients and shifts acting freely,

Integral Equations Operator Theory 55 (2006) 19–67.[4] M.A. Bastos, C.A. Fernandes, Yu.I. Karlovich, Spectral measures in C∗-algebras of singular integral operators with shifts, J. Funct. Anal. 242 (2007)

86–126.[5] M.A. Bastos, C.A. Fernandes, Yu.I. Karlovich, C∗-algebras of singular integral operators with shifts having the same nonempty set of fixed points,

Complex Anal. Oper. Theory 2 (2008) 241–272.[6] M.A. Bastos, C.A. Fernandes, Yu.I. Karlovich, A C∗-algebra of functional operators with shifts having a nonempty set of periodic points, in: H.G.W.

Begehr, et al. (Eds.), Further Progress in Analysis. Proceedings of the 6th International ISAAC Congress, Ankara, Turkey, August 13–18, 2007, WorldScientific, Hackensack, NJ, 2009, pp. 111–121.

[7] M.A. Bastos, C.A. Fernandes, Yu.I. Karlovich, A nonlocal C∗-algebra of singular integral operators with shifts having periodic points, Integral EquationsOperator Theory 71 (2011) 509–534.

[8] A. Böttcher, Yu.I. Karlovich, I.M. Spitkovsky, Convolution Operators and Factorization of Almost Periodic Matrix Functions, Oper. Theory Adv. Appl.,vol. 131, Birkhäuser, Basel, 2002.

[9] A. Böttcher, S. Roch, B. Silbermann, I.M. Spitkovsky, A Gohberg–Krupnik–Sarason symbol calculus for algebras of Toeplitz, Hankel, Cauchy, and Carlemanoperators, in: L. de Branges, I. Gohberg, J. Rovnyak (Eds.), Topics in Operator Theory. Ernst D. Hellinger Memorial Volume, in: Oper. Theory Adv. Appl.,vol. 48, Birkhäuser, Basel, 1990, pp. 189–234.

[10] A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd ed., Springer, Berlin, 2006.[11] R.G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972.[12] I. Gohberg, N. Krupnik, On the algebra generated by the one-dimensional singular integral operators with piecewise continuous coefficients, Funct.

Anal. Appl. 4 (1970) 193–201.[13] F.P. Greenleaf, Invariant Means on Topological Groups and Their Representations, Van Nostrand-Reinhold, New York, 1969.[14] R. Hagen, S. Roch, B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations, Birkhäuser, Basel, 1995.[15] Yu.I. Karlovich, The local-trajectory method of studying invertibility in C∗-algebras of operators with discrete groups of shifts, Soviet Math. Dokl. 37

(1988) 407–411.[16] Yu.I. Karlovich, C∗-algebras of operators of convolution type with discrete groups of shifts and oscillating coefficients, Soviet Math. Dokl. 38 (1989)

301–307.[17] Yu.I. Karlovich, On algebras of singular integral operators with discrete groups of shifts in Lp -spaces, Soviet Math. Dokl. 39 (1989) 48–53.[18] Yu.I. Karlovich, A local-trajectory method and isomorphism theorems for nonlocal C∗-algebras, in: Ja.M. Erusalimsky, I. Gohberg, S.M. Grudsky, V. Ra-

binovich, N. Vasilevski (Eds.), Modern Operator Theory and Applications. The Igor Borisovich Simonenko Anniversary Volume, in: Oper. Theory Adv.Appl., vol. 170, Birkhäuser, Basel, 2007, pp. 137–166.

[19] A.Yu. Karlovich, Yu.I. Karlovich, A.B. Lebre, Invertibility of functional operators with slowly oscillating non-Carleman shifts, in: A. Böttcher,M.A. Kaashoek, A.B. Lebre, A.F. dos Santos, F.-O. Speck (Eds.), Singular Integral Operators, Factorization and Applications. International Workshop onOperator Theory and Applications, IWOTA 2000, Portugal, in: Oper. Theory Adv. Appl., vol. 142, Birkhäuser, Basel, 2003, pp. 147–174.

[20] Yu.I. Karlovich, V.G. Kravchenko, An algebra of singular integral operators with piecewise-continuous coefficients and piecewise-smooth shift on acomposite contour, Math. USSR-Izv. 23 (1984) 307–352.

[21] Yu.I. Karlovich, B. Silbermann, Local method for nonlocal operators on Banach spaces, in: A. Böttcher, I. Gohberg, P. Junghanns (Eds.), Toeplitz Matricesand Singular Integral Equations. The Bernd Silbermann Anniversary Volume, in: Oper. Theory Adv. Appl., vol. 135, Birkhäuser, Basel, 2002, pp. 235–247.

[22] Yu.I. Karlovich, B. Silbermann, Fredholmness of singular integral operators with discrete subexponential groups of shifts on Lebesgue spaces, Math.Nachr. 272 (2004) 55–94.

[23] M.A. Naimark, Normed Algebras, Wolters-Noordhoff, Groningen, 1972.[24] S.C. Power, Fredholm Toeplitz operators and slow oscillation, Canad. J. Math. 32 (1980) 1058–1071.[25] S. Roch, P.A. Santos, B. Silbermann, Non-Commutative Gelfand Theories. A Tool-kit for Operator Theorists and Numerical Analysts, Springer, London,

2011.

http://refhub.elsevier.com/S0022-247X(13)01079-2/bib41s1


http://refhub.elsevier.com/S0022-247X(13)01079-2/bib414Cs1

http://refhub.elsevier.com/S0022-247X(13)01079-2/bib42464B31s1

















http://refhub.elsevier.com/S0022-247X(13)01079-2/bib446Fs1

http://refhub.elsevier.com/S0022-247X(13)01079-2/bib476F68624Bs1

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http://refhub.elsevier.com/S0022-247X(13)01079-2/bib4B61726Cs1

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A -algebra of singular integral operators with shifts admitting distinct fixed points

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