On C*-algebras of singular integral operators with PQC coefficients and shifts with fixed points

Abstract

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the $C^{\ast }$-algebra $\mathfrak{B}$ generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator $S_{\mathbb{T}}$ and by the unitary weighted shift operators $U_g:\phi \mapsto |g^{\prime }{|}^{1/2}(\phi \circ g)$, $g\in G,$ acting on the space $L^2(\mathbb{T})$ over the unit circle $\mathbb{T}\subset \mathbb{C}$. Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms $g:\mathbb{T}\to \mathbb{T}$ with finite sets of discontinuities for their derivatives $g^{\prime }$, which acts topologically freely on $\mathbb{T}\setminus {\mathrm{\Lambda }}^{\circ }$, where ${\mathrm{\Lambda }}^{\circ }$ is the interior of the nonempty closed set $\mathrm{\Lambda }\subset \mathbb{T}$ composed by all common fixed points for all shifts $g\in G$, with boundary $\mathrm{\partial }\mathrm{\Lambda }$ of zero Lebesgue measure. A Fredholm symbol calculus for the $C^{\ast }$-algebra $\mathfrak{B}$ is constructed and a Fredholm criterion for the operators $B\in \mathfrak{B}$ is established.

Keywords:

• Singular integral operator with shifts
• piecewise quasicontinuous function
• fixed points
• representation of a $C^{\ast }$-algebra
• Fredholmness

AMS Subject Classifications:

• 45E05
• 47A53
• 47A67
• 47B33
• 47L15

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UIDB/04721/2020 (Centro de Análise Funcional, Estruturas Lineares e Aplicações) and UID/MAT/00297/2019 (Centro de Matemática e Aplicações). The third author was also supported by the SEP-CONACYT projects A1-S-8793 and A1-S-9201 (México).