In this talk, we present an algebraic and categorical approach to local quantum physics. We revisit C*-algebraic quantum theory since Segal and Haag-Kastler and formulate measurement theory in C*-algebraic quantum theory. For this purpose, we use central subspace and C*-$L^1$ space. The former, introduced in a previous study by the author, is an invariant closed subspace of the dual space of a C*-algebra. We analyze several categories of central subspaces in order to compare central subspaces in different ways. This analysis is a variant of the investigations by Fell, Haag-Kastler, and others. A C*-$L^1$ space is a pair of a C*-algebra and a central subspace of its dual space. Next, we define completely positive (CP) instrument, a central concept in quantum measurement theory. A CP instrument is a measure that takes values in CP maps between two C*-$L^1$ spaces and is used to describe the dynamical state changes including measuring processes. Based on CP instruments, we develop local measurements in local quantum physics.