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In this work we examine C*-algebras of Toeplitz
operators over the unit ball in
C^n and the unit polydisc in C^2. Toeplitz
operators are interesting examples of non-normal operators that
generate non-commutative C*-algebras. Moreover, in the nice
cases (depending on the geometry of the domain) of algebras of
Toeplitz operators we can recover some analogues of the spectral
theorem up to compact operators. In this setting, we can capture
the index of a Fredholm operator which is a fundamental numerical
invariant in Operator Theory.