In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C 1 . Keywords: Functions of a hypercomplex variable, Quaternions, Octonions, Fundamental theorem of algebra, Cauchy integral formula Math. Subj. Class: 30C15, 30G35, 32A30, 17D05 * Work partially supported by MIUR (PRIN Project "Proprietà geometriche delle varietà reali e complesse") and GNSAGA of INdAMProposition 5. Let J, K ∈ S A with J − K invertible. Every slice function f ∈ S(Ω D ) is uniquely determined by its values on the two distinct half planes C + J and C + K . In particular, taking K = −J, we have that f is uniquely determined by its values on a complex plane C J .Proof. For f ∈ S(Ω D ), let f + J be the restrictionIt is sufficient to show that the stem function F such that I(F ) = f can be recovered from the restrictions f + J , f + K . This follows from the formulawhich holds for every α + iβ ∈ D with β ≥ 0. In particular, it implies the vanishing of F 2 when β = 0. Then F 2 is determined also for β < 0 by imposing oddness w.r.t. β. Moreover, the formula f + J (α + βJ) − JF 2 (α + iβ) = F 1 (α + iβ) defines the first component F 1 as an even function on D.

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.

In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative * -algebra A over R. These recently introduced function theories generalize to higher dimensions the classical theory of functions of a complex variable. Slice functions over A, which comprise all polynomials over A, form an alternative * -algebra themselves when endowed with appropriate operations. We presently study this algebraic structure in detail and we confront with questions about the existence of multiplicative inverses. This study leads us to a detailed investigation of the zero sets of slice functions and of slice regular functions, which are of course of independent interest.

We study two types of series over a real alternative * -algebra A. The first type are series of the form n (x − y) ·n an, where an and y belong to A and (x − y) ·n denotes the n-th power of x − y w.r.t. the usual product obtained by requiring commutativity of the indeterminate x with the elements of A. In the real and in the complex cases, the sums of power series define, respectively, the real analytic and the holomorphic functions. In the quaternionic case, a series of this type produces, in the interior of its set of convergence, a function belonging to the recently introduced class of slice regular functions. We show that also in the general setting of an alternative algebra A, the sum of a power series is a slice regular function. We consider also a second type of series, the spherical series, where the powers are replaced by a different sequence of slice regular polynomials. It is known that on the quaternions, the set of convergence of these series is an open set, a property not always valid in the case of power series. We characterize the sets of convergence of this type of series for an arbitrary alternative * -algebra A.In particular, we prove that these sets are always open in the quadratic cone of A. Moreover, we show that every slice regular function has a spherical series expansion at every point.2000 Mathematics Subject Classification. Primary 30G35; Secondary 30B10, 30G30, 32A30.

Abstract. The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann's foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has been started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A problem of quaternionic formulation is the description of composite quantum systems in absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A notion with this property, called intertwining quaternionic PVM, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover in particular the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator.