Continuous functional calculus

Abstract: The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic $C^*$--algebras and to define, on each of these $C^*$--algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

Abstract: Two uniqueness results concerning Frechet module structures over algebras of holomorphic functions defined on some complex manifolds are pre- sented, containing as particular cases uniqueness theorems for J. L. Taylor's analytic functional calculi for commuting «-tuples of linear continuous operators on Frechet spaces (7), (9). Namely, the first statement says that the Spectral Mapping Theorem insures the unicity of the functional calculus and thus it improves Zame's unicity theorem (11, Theorem 1), while the second statement gives a unicity condition which is an analogue of the compatibility property (3, Theorem 1.4.1) in spectral theory of several variables in commutative Banach algebras. As a corollary the two functional calculi constructed in (7) and (9) by J. L. Taylor coincide. 0. Introduction. This note is a continuation of (5) and (6) and is presented in the spirit of and with the technique initiated by J. L. Taylor in (9). The general philosophy is to regard a commuting «-tuple a of linear continuous operators on a Frechet space M as a Frechet module structure of M over the algebra 0(C) of entire functions and then to use homological-topological methods. Proposition 1 essentially contains in its statement Frechet module structures over algebras of holomorphic functions on complex manifolds which are not canonically imbedded in a numerical space, and thus it is quite difficult to give it an interpreta- tion in terms of «-tuples of commuting operators. On the other hand, Theorem 2 has a clear specialization in the operator-theoretic language as follows: Theorem 1. For each open subset U of C" which contains the joint spectrum of an n-tuple a of linear continuous commuting operators on a Banach space, there is exactly one continuous functional calculus for a with analytic functions on U which satisfies the spectral mapping theorem. In fact this statement holds for Frechet spaces with the additional assumption that the «-tuple has a continuous functional calculus with entire functions. The corre- spondence with Theorem 1 of (11), which asserts that an «-tuple of elements of a Banach algebra has exactly one continuous functional calculus with germs of analytic functions in neighbourhoods of the joint spectrum and which commutes with the Gel'fand transformation, is now obvious. The statements below are presented in the spirit of (5) and (6), without dis- tinguished coordinates, and their form is imposed by the restriction that the open

Abstract: A generalized Hermitian (GH-) algebra is a generalization of the partially ordered Jordan algebra of all Hermitian operators on a Hilbert space. We introduce the notion of a gh-tribe, which is a commutative GH-algebra of functions on a nonempty set $X$ with pointwise partial order and operations, and we prove that every commutative GH-algebra is the image of a gh-tribe under a surjective GH-morphism. Using this result, we prove each element $a$ of a GH-algebra $A$ corresponds to a real observable $\xi_a$ on the $\sigma$-orthomodular lattice of projections in $A$ and that $\xi_a$ determines the spectral resolution of $a$. Also, if $f$ is a continuous function defined on the spectrum of $a$, we formulate a definition of $f(a)$, thus obtaining a continuous functional calculus for $A$.