We give some new characterizations of semisimple alternative algebras among the pseudo-Euclidean alternative algebras. These charcterizations are based on the index of a pseudo Euclidean alternative algebra, operators of Casimir type and Representations of alternative algebras.

Recently in \cite{DvZa5,DvZa6}, new algebras, called weak pseudo EMV-algebras, wPEMV-algebras for short, were introduced generalizing pseudo MV-algebras, generalized Boolean algebras and pseudo EMV-algebras. For these algebras a top element is not assumed a priori. For this class of algebras, we define a state as a finitely additive mapping from a wPEMV-algebra into the real interval $[0,1]$ which preserves a partial addition of two non-interactive elements and attaining the value $1$ in some element. It can happen that some commutative wPEMV-algebras are stateless, e.g. cancellative ones.The paper is divided into two parts. Part I deals with basic properties of states and state-morphisms which are wPEMV-homomorphisms from a wPEMV-algebra into the real interval $[0,1]$ endowed as a commutative wPEMV-algebra. We show that there is a one-to-one correspondence between the set of state-morphisms and the set of maximal and normal ideals having a special property.In Part II, we present an analogue of the Krein-Mil'man theorem applied to the set of states. We characterize the space of the state-morphisms of a wPEMV-algebra without top element as a Hausdorff locally compact space in the weak topology of states and we present its Alexandroff's one-point compactification. Moreover, we give an integral representation of any (finitely additive) state by a unique regular Borel $\sigma$-additive probability measure.

In this article, the structure of the Clifford-Weyl superalgebras and their associated Lie superalgebras will be investigated. These superalgebras have a natural supersymmetric inner product which is invariant under their Lie superalgebra structures. The Clifford-Weyl superalgebras can be realized as tensor product of the algebra of alternating and symmetric tensors respectively, on the even and odd parts of their underlying superspace. For Physical applications in elementary particles, we add star structures to these algebras and investigate the basic relations. Ortho-symplectic Lie algebras are naturally present in these algebras and their Representations on these algebras can be described easily.

In the present paper for a large family of topological semigroups, namely foundation semigroups, for which topological groups and discrete semigroups are elementary examples, it is shown that R(S) is the dual of a function algebra.

Let G be a locally compact group or a discrete group and let w be a weight function on G. We define a new convolution on the Beurling algebra L1(G, w) and show that it is also a Banach algebra with this new convolution. We show that there is a one to one correspondence between the unitary Representations of G and the non-degenerate ?-Representations of L1(G, w). We show that L1(G, w) is amenable if and only if G is amenable. This improves earlier results which require w to be diagonally bounded.

We yield operators U and V on Hilbert spaces that are parameterized by the orbits of certain interval maps that exhibit chaotic behavior and obey the (deformed) Baumslag–Solitar relationUV=e2piaV Un,    aÎR,nÎNWe then prove that the scalar e2pia can be removed whilst retaining the isomorphism class of the C*-algebra generated by U and V. Finally, we simultaneously unitarize U and V by gluing pairs of orbits of the underlying noninvertible dynamical system and investigate these unitary Representations under distinct pairs of orbits.

Using fixed point methods, we prove the stability of orthogonally quintic functional equation on C ∗ -algebras for the functional equation$$Df(x, y) = f(3x + y) − 5f(2x + y) + f(2x − y) + 10f(x + y) − 5f(x − y) − 10f(y) − f(3x) + 3f(2x) + 27f(x).$$