We study the structure C∗-algebras generated by a system of unilateral weighted shifts. Finally the obtained results are applied to a class of integral equations.

1. Introduction

The structure of C*-algebras generated by isometry is determined in [1, 2]. The structure is the same with the structure of C*-algebras generated by unilateral weighted shift operators, that is, the structure of C*-algebras generated by multiplication operators with the independent variable in the Hardy space on the unit disc. The analogue of the unit disc on ℂN is the polydisc(1.1)ΔN={z=(z1,z2,…,zN)∈ℂN:|zi|<1,1≤i≤N}
or the unit ball
(1.2)BN={z=(z1,z2,…,zN)∈ℂN:|z1|2+|z2|2+⋯+|zN|2<1}.
The structures of C*-algebras generated by multiplication operators with the independent variable in the Hardy space on the unit ball and polydisc are different. To understand this difference we study the structure of C*-algebras generated by system of unilateral weighted shifts.

Let I be a multiindex (i1,…,iN) of integers and I±εj denotes (i1,…,ij±1,…,iN) for the multi-index I. Here εj is another multi-index (δ1j,…,δNj), where δijis the Kronocker symbol.

Let {eI}I≥0 be an orthonormal basis of a separable complex Hilbert space H and let {ωI,j:I≥0,1≤j≤N} be a bounded net of complex numbers. Denote by Aj the bounded linear operators whose effect on the elements of basis {eI}I≥0 of H is given as AjeI=ωI,jeI+εj,1≤j≤N. A family of N operators, denoted by A=(A1,A2,…,AN), is called a system of unilateral weighted shifts, and the numbers of {ωI,j:I≥0,1≤j≤N} are called the weights of the system. It is known from [3, page 209, Corollary 2] that for shifts with nonzero weights {ωI,j}, without loss of generality we may always assume the weights are a set of positive real numbers, that is, the system A positive. It is possible to show that if there exists a solution for the multivariable moment problem for the net {βI2}I≥0 where βI+εj=ωI,jβI, β0=1, that is, if there exists the probability measure νA on [0,1]N=[0,1]×⋯×[0,1](N times) such that
(1.3)βI2=∫[0,1]Nr12i1r22i2⋯rN2iNdν(r1,r2,…,rN),
then the system A is unitarily equivalent to the system of multiplication operators by the independent variables zj, 1≤j≤N, on the space
(1.4)H2(ΔN,μA)={f=∑I≥0fIzI:z∈ΔN,∑I≥0|fI|2βI2<∞},
where μA=νAdθ1dθ2⋯dθN, 0≤θi≤2π.

The reader can find for more details of such operators in the article by Jewell and Lubin [3] and Ergezen and Sadik [4]. Furthermore the papers of Curto and Yoon [5] and Curto and Yan [6] are closely related to our study.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M44"><mml:mtext /><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mi>*</mml:mi></mml:mrow></mml:msup><mml:mtext /></mml:math></inline-formula>-Algebras Generated by a System of Unilateral Weighted Shifts

Let Ω denote the family of the systems A which satisfy the functional model defined above. Moreover, let Ω1 be a subset of Ω defined by(2.1)Ω1={[0,1]NA∈Ω:νA(U(1,1,…,1))>0for arbitrary neighborhoodU(1,1,…,1)of the point(1,1,…,1)∈[0,1]N}.

Theorem 2.1 (see [<xref ref-type="bibr" rid="B7">4</xref>, page 25, Theorem 2]).

Let A∈Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isomorphic to the polydisc algebra is that A belongs to Ω1.

This theorem will be helpful in studying the structure of C*-algebra C*(A) generated by A∈Ω1.

Let P denote the orthogonal projection of L2(ΔN,μA) onto H2(ΔN,μA) and let ψ lie in C(suppμA). Then the Toeplitz operator Tψf=P(ψf) for f in H2(ΔN,μA).

Without loss of generality we may take N=2. The following theorems for N=1 were given by Sadikov [7].

Theorem 2.2.

Let A∈Ω. If the algebra generated by the system A is polydisc algebra then the commutator ideal 𝒥 of C*(A) contains properly the ideal of compact operators 𝒦 and the quotient space 𝒥/𝒦 is isometrically isomorphic to C(T×{0,1})⊕𝒦 and C*(A)/𝒥=C(T×T), where T is unit circle and {0,1} is the two point space.

Corollary 2.3.

Let Tψ∈C*(A). Then necessary and sufficient condition for Tψ to be Fredholm is that ψ(z) is nonvanishing for z∈T2 and ψ|T2is homotopic to constant.

Theorem 2.2 and Corollary 2.3 are proved by using the methods of Douglas and Howe in their study [8] and Curto and Muhly in [9].

Let
(2.2)S1={(r1,r2)∈[0,1]×[0,1]:r12+r22≤1},S1~={(r1,r2)∈[0,1]×[0,1]:r12+r22=1},
and let Ω2 be subset of Ω defined by
(2.3)Ω2={S1~A∈Ω:νA(U(a))>0for arbitrary neighborhoodU(a)of the arbitrary pointa∈S1~}.

Theorem 2.4 (see [<xref ref-type="bibr" rid="B6">10</xref>, p. 1932, Theorem 2]).

Let A∈Ω. A necessary and sufficient condition for the operator algebra generated by the system A to be isometrically isometric to the ball algebra is that A belongs to Ω2.

Theorem 2.5.

Let A∈Ω. If the algebra generated by the system A is ball algebra then C*(A) contains 𝒦 ideal of compact operators and C*(A)={Tψ+K:ψ∈C(suppμA),K∈𝒦}. The quotient C*(A)/𝒦 is naturally identified with C(S3) by a map σ(Tψ+K)=ψ|S3~.

It is enough to show compactness of the operators Ai*Ai-AiAi*, i=1,2 for proving that commutant of the algebra C*(A) is 𝒦. For this, we just study the case i=2. Then the case i=1 is similarly showed, as well. With the basic computations, we obtain (A2*A2-A2A2*)e(m,n)=(ω(m,n),22-ω(m,n-1),22)e(m,n). If we show ω(m,n),22-ω(m,n-1),22 converges the zero when |I|→∞ then the proof is completed. For this, we need the following lemma.

Lemma 2.6.

Let νA be a measure determined by the system A∈Ω2 and I=(m,n). then the expression
(*)∫S1r12mr22n+2dν(r1,r2)∫S1r12mr22ndν(r1,r2)-∫S1r12mr22ndν(r1,r2)∫S1r12mr22n-2dν(r1,r2)
converges to zero when |I|→∞.

In view of the following process, the lemma is proved. Without loss of generality, we can take n=1. Hence the second fractional of the expression (*) becomes I1/I2:=∫S1r12mr22dν(r1,r2)/∫S1r12mdν(r1,r2). It is enough to show I1/I2 goes to zero when m→∞. Take r20=ε/2 and r10=1-(ε/2)=1-r202 for given ε>0. Consider S10:={(r1,r2):r10<r1≤1}∩S1 and S11:=S1/S10. We have I1=I10+I11, where I10=∫S10r12mr22dν(r1,r2). It easily shows that I10≤(ε/2)I2 and I11≤r102mνA(S1) for all m. Moreover, take r11=(1+r10)/2; then we have I2≥r112mνA(S101) for all m, where S101={(r1,r2):r11<r1<1}∩S1. Hence, there exists M>0 such that for all m>M it is obtained I1/I2<ε.

It follows from Lemma 2.6 and a well-known result in C*-algebras [2, page 212, Proposition 1] that 𝒦⊂C*(A).

3. An Application

Throughout this section, we follow the notations and definitions in the preceding section.

Let H be separable complex Hilbert space, letB(H) denote the algebra of linear bounded operators on H and let I be identity operator. Consider an operator(3.1)T=B1+B2S+K,
where B1 and B2 are the elements of a subalgebra Λ of B(H) such that the image τ(Λ) is a commutative subalgebra of the algebra B(H)/𝒦 under the natural quotient map τ from B(H) to B(H)/𝒦 and S denotes an automorphism in the algebra τ(Λ), that is, τ(S)τ(B)τ(S-1)=τ(B'), where B and B' belong to Λ. It is obvious that if B∈Λ then SBS-1=B'+K, where B'∈Λ and K∈𝒦.

Theorem 3.1 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

If the operator τ(B1)τ(B1′)-τ(B2)τ(B2′) has an inverse in τ(Λ) then T=B1+B2S+K is Fredholm operator.

Using the Theorem 3.1 we take A∈Ω2 and consider the operator Tψ1+Tψ2S+K, where Tψ1 and Tψ2 are Toeplitz operators in C*(A) and the operators S and T satisfy the conditions given above.

Moreover if we take into account the orthonormal projection P has the form
(3.2)(Pf)(z1,z2)=∫Δ2K(z,l)f(l1,l2)dμA(l1,l2),
where K is the reproducing Bergman kernel of the functional space H2(Δ2,μA), then the equation Tf=φ is written in the form
(3.3)∫Δ2K(z,l)ψ1(l1,l2)f(l1,l2)dμA(l1,l2)+∫Δ2K(z,l)ψ2(l1,l2)f(l1,l2)dμA(l1,l2)+(Kf)(z1,z2)=φ(z1,z2).

Hence we have the following theorem.

Theorem 3.2.

If the function ψ1(z1,z2)ψ1(z2,z1)-ψ2(z1,z2)ψ2(z2,z1) does not vanish in S3, then all of Noether’s theorems is true for (3.3). In particular, if we take ωI,1=(m+1)/(2+m+n), ωI,2=(n+1)/(2+m+n) and Sf(l1,l2)=f(l2,l1) then (3.3) has the form
(3.4)∫S3ψ1(l1,l2)f(l1,l2)(1-z1e-1-z2e-2)2ds+∫S3ψ2(l1,l2)f(l2,l1)(1-z1e-1-z2e-2)2ds+(Kf)(z1,z2)=φ(z1,z2),
where ds is the surface measure in S3.

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