The purpose of this work is to present the quantum Hermite-Hadamard inequality through the Green function approach. While doing this, we deduce some novel quantum identities. Using these identities, we establish some new inequalities in this direction. We contemplate the possibility of expanding the method, outlined herein, to recast the proofs of some known inequalities in the literature.
Abstract.If one has a unitary representation p : 7r -+ U(H) of the fundamental group 7rl(M) of the manifold M, then one can do many useful things: 1) construct a natural vector bundle over M; 2) construct the cohomology groups with respect to the local system of coefficients; 3) construct the signature of manifold M with respect to the local system of coefficients; and others. In particular, one can write the Hirzebruch formula, which compares the signature with the characteristic classes of the manifold M, further, based on this, find the homotopy invariant characteristic classes (i.e., the Novikov conjecture). Taking into account that the family of known representations is not sufficiently large, it would be interesting to extend this family to some larger one. Using the ideas of P. de la Harpe and M. Karoubi
Abstract. The main purpose of this paper is to investigate some topological properties of the double multiplier algebra on a topological algebra. Let M d (A) be the double multiplier algebra on a topological algebra A, and let u and s be the uniform and strong operator topologies on M d (A), respectively. It is shown, under some additional hypotheses on A, In view of the applications of (nonnormed) topological algebras in other fields such as quantum mechanics and quantum statistics (see, e.g., Lassner [10,11]) and recent developments in the theory of topological algebras (see, for instance, the book of Mallios [13]), it is important to consider operators on more general classes of topological algebras. More recently, Phillips [14,15] has studied inner and approximately inner derivations on pro-C * -algebras (inverse limits of C * -algebras, also called LMC * -algebras) using multipliers, while Van Daele [20] has considered multipliers on Hopf algebras which provide a natural framework to study quantum groups. Therefore, it is important to develop the theory of multipliers for general topological algebras and, in particular, for metrizable topological algebras.In this paper, we are mainly concerned with the linear topological properties of
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