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In this paper, we prove Slodkowski version of the infinite-dimensional spectral mapping theorem and Cartan-Slodkowski version of the finite-dimensional spectral mapping theorem for nilpotent operator Lie subalgebras with respect to the various noncommutative functional calculi.

In the paper we propose an operator approach to the noncommutative Taylor localization problem based on the local left invertibility for operator tuples acting on a Fréchet space. We prove that the canonical homomorphism U. g / ! O g of the universal enveloping algebra U. g / of a nilpotent Lie algebra g into its Arens-Michael envelope O g is the Taylor localization whenever g has normal growth.

In this note, we generalize Ruan's representation theorem and propose an ArvesonHanh-Banach-Webster theorem for local operator spaces. Further, we investigate the decomposability of a complete contraction acting from a unital multinormed C * -algebra to a local operator system into a product of contractions and a unital contractive * -representation, and we study injectivity in both local operator space and local operator system contexts.The known representation theorem [3, Theorem 2.3.5] for operator spaces states that each abstract operator space V can be realized as a subspace of the space B(H) of all bounded linear operators on a Hilbert space H. By realization we mean a matrix isometry Φ :. This result plays a central role in the operator space theory and provides an abstract characterization of a linear space of bounded linear operators on a Hilbert space. Physically well motivated, operator spaces can be thought of as quantized normed spaces, where we have replaced functions with operators, thus regarding classical normed spaces as abstract function spaces. Another motivation is predicted by the domination property observed in a noncommutative functional calculus problem [1, Sec. 4], which suggests that (joint) spectral properties of elements in an operator algebra might be expressed in terms of matrices over the original algebra. The implementation of this proposal would lead to a reasonable joint spectral theory in an operator algebra. To have a more solid justification of quantum physics and noncommutative function theory, it is necessary to consider operator analogs of locally convex spaces, that is, quantizations of locally convex spaces. Namely, one might consider linear spaces of unbounded Hilbert space operators or, more generally, projective limits of operator spaces. In recent years, Effros and Webster [4] started to develop this theory under the name "local operator spaces." A central, subtle result of their investigations is an operator version of the classical bipolar theorem [4, Proposition 4.1].The goal of the present note is to give an intrinsic description of local operator spaces similar to the above-mentioned characterization of operator spaces. We show that each local operator space can be realized as a subspace of unbounded operators on a Hilbert space. Moreover, if the given local operator space has a bounded locally convex topology, then it can be realized by bounded operators on a Hilbert space. This result generalizes Ruan's representation theorem for operator spaces. To restore the natural connection between local operator spaces and unital multinormed C * -algebras, as it is in the normed case, we introduce local operator systems motivated by the representation theorem for local operator spaces. The known class of unital multinormed C * -algebras called Op * -algebras ([7], [6, Sec. 3.2.3]) presents a bright example of local operator systems. The central role in local operator systems is played by the concept of local positivity. In terms of local positivity, we prove the Arveson...

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