Originally posted on What's new:

One of the basic problems in the field of operator algebras is to develop a functional calculus for either a single operator $latex {A}&fg=000000$, or a collection $latex {A_1, A_2, \ldots, A_k}&fg=000000$ of operators. These operators could in principle act on any function space, but typically one either considers complex matrices (which act on a complex finite dimensional space), or operators (either bounded or unbounded) on a complex Hilbert space. (One can of course also obtain analogous results for real operators, but we will work throughout with complex operators in this post.)

Roughly speaking, a functional calculus is a way to assign an operator $latex {F(A)}&fg=000000$ or $latex {F(A_1,\ldots,A_k)}&fg=000000$ to any function $latex {F}&fg=000000$ in a suitable function space, which is linear over the complex numbers, preserve the scalars (i.e. $latex {c(A) = c}&fg=000000$ when $latex {c \in {\bf C}}&fg=000000$), and should be either an exact or approximate homomorphism in…

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