We describe C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebras generated by Toeplitz operators with homogeneous symbols acting on polyharmonic Bergman spaces of the upper half-plane Π\documentclass[12pt]{minimal}
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\begin{document}$$\Pi $$\end{document}. The symbols considered here have finite limits at the points 0 and π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}. Under these conditions on the family of symbols, a Toeplitz operator acting on the true polyharmonic space H(n)2(Π)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{(n)}^{2}(\Pi )$$\end{document} is unitarily equivalent to a 2×2\documentclass[12pt]{minimal}
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\begin{document}$$2\times 2$$\end{document} matrix-valued function defined on R¯\documentclass[12pt]{minimal}
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\begin{document}$$\overline{{\mathbb {R}}}$$\end{document}. The C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra generated by these matrix-valued functions turns out to be isomorphic to the algebra C:=f=(fij)∈M2(C(R¯)):f(±∞)is diagonal,f11(±∞)=f22(∓∞).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathfrak {C}} := \left\{ f=(f_{ij})\in M_{2}(C(\overline{{\mathbb {R}}})) : f(\pm \infty ) \text { is diagonal}, f_{11}(\pm \infty )=f_{22}(\mp \infty ) \right\} . \end{aligned}$$\end{document}Besides, we prove that the C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^*$$\end{document}-algebra generated by Toeplitz operators with homogeneous symbols, acting on the polyharmonic Bergman space Hn2(Π)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {H}}_{n}^{2}(\Pi )$$\end{document}, is isomorphic to the C∗\documentclass[12pt]{minimal}
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\begin{document}$$C^{*}$$\end{document}-subalgebra of M2n(C(R¯))\documentclass[12pt]{minimal}
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\begin{document}$$M_{2n}(C(\overline{{\mathbb {R}}}))$$\end{document} consisting of all matrix-valued functions f=(fij)\documentclass[12pt]{minimal}
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\begin{document}$$f=(f_{ij})$$\end{document} such that f(-∞)=λ1I0I0Iλ2I,f(+∞)=λ2I0I0Iλ1I,λ1,λ2∈C,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} f(-\infty )=\left( \begin{array}{cc} \lambda _1 I &{} 0I \\ 0I &{} \lambda _2 I \end{array} \right) , \quad f(+\infty )=\left( \begin{array}{cc} \lambda _2 I &{} 0I \\ 0I &{} \lambda _1 I \end{array} \right) , \ \lambda _1,\lambda _2 \in {\mathbb {C}}, \end{aligned}$$\end{document}where I is the n×n\documentclass[12pt]{minimal}
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\begin{document}$$n\times n$$\end{document} identity matrix.
