STRONGLY CONTINUOUS BICOMPLEX SEMIGROUPS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we study the semigroups of operator algebras with bicomplex scalars and also investigate the ∗-derivation.


INTRODUCTION
Bicomplex numbers are a generalization of complex numbers which form a ring under the usual addition and multiplication. Moreover, this ring is a module over itself. The set of bicomplex numbers do not form a field because every non-zero bicomplex number does not possesses its multiplicative inverse. Zero divisors of bicomplex numbers play a significant role in the idempotent representation of bicomplex numbers. For basics of bicomplex numbers and their properties one can refer to [1], [8], [9] and references therein.
Hyperbolic numbers are a natural replacement for the real number system. However, it was seen that the hyperbolic numbers are subset of bicomplex numbers. The role of hyperbolic for bicomplex numbers is analogous to the role of real numbers for complex numbers.
In the last several years, the theory of bicomplex numbers and hyperbolic numbers has found many applications in different areas of mathematics and theoretical physics (cf. [3], [5], [6]), and references there in provide more information on these applications.
The work in this paper can be seen as a continuation of work in [11].

PRELIMINARIES
In this section, we summarize some known results about bicomplex numbers.
The set of bicomplex numbers is denoted by BC and is defined as the commutative ring whose elements are of the form where z 1 = x 1 + iy 1 ∈ C(i) and z 2 = x 2 + iy 2 ∈ C(i) are complex numbers with imaginary unit i and where i and j = i are commutating imaginary units i.e., i j = ji. Note that i 2 = j 2 = −1.
In equation (1), if z 1 = x is real and z 2 = iy is purely imaginary with i. j = k, then we obtain the ring of hyperbolic numbers. D = {x + ky : k 2 = 1 and x, y ∈ R with k / ∈ R}.
A hyperbolic number α can be written as We denote by D + , the set of all "positive" hyperbolic numbers. The set is called the set of positive hyperbolic numbers. (cf. [1], page 5). For P, Q ∈ D, (set of hyperbolic numbers) we define a relation on D by P Q ⇐⇒ Q − P ∈ D + .
This relation is reflexive, anti-symmetric as well as transitive and hence defines a partial order on D, (cf. [1]).

MAIN RESULTS
In this section, we study the semigroups on operator algebras with bicomplex scalars and also investigate the * -derivation.  Let µ 1 (p) and µ 2 (p) are operators with complex scalars defined by (e pA 1 ) p∈D and (e pA 2 ) p∈D respectively. And the differentation of the map The uniformly continuous group (U (p)) p∈D of * -automorphisms on L(H) given by the unitary group (e pA ) p∈D has generator Γ given by We now study uniformly continuous groups consisting of * -automorphism.
The proof of the following results is based on [[2], Chapter 1 ].
Finally, we show that the operator A ∈ L(H) satisfying the property AΦ − ΦA, for all Φ ∈ L(H) can be taken as a bicomplex skew-adjoint operator. Since D is a * -derivation, we have and so x ∈ Ω, the corresponding hyperbolic norm on B is defined as One can check that B is a BC Banach module under the hyperbolic norm . defined above.

STRONGLY CONTINUOUS SEMIGROUPS
In this section, we study strongly continuous semigroup or C 0 semigroup and also discuss some of its basic properties. The results of this section are essentially based on [2, pp. 36-39 ]. (2) strongly continuous i.e., Φ(p)x → x as p → 0, ∀x ∈ B.
If these properties hold for D instead of D + , we call (Φ(p)) p∈D a strongly continuous (one- Note that this norm is hyperbolic norm on Φ. The operator (BC -linear) Φ ∈ L(B) satisfying is called contraction, while isometry is defined by Besides the uniform operator topology on L(B), which is induced by the operator norm Φ D , we consider two more topologies on L(B). Let (Φ α ) α∈A be a bicomplex net of bicomplex linear operators on the BC Banach module B.
Consider that (Φ α ) α∈A converges to some operator Φ on B. This could have different meanings as follows: If Φ α − Φ D → 0, then we say that Φ α converges to Φ in the uniform operator topology.
, then we say that Φ α converges to Φ in the strong operator topology.
x ∈ B in the weak topology of B. In this case we say that Φ α converges to Φ in the weak operator topology. (a) (Φ(p)) p∈D + is strongly continuous.
(c) There exists δ 0, M 1, and a dense subset D ⊂ B such that Then Thus C φ p D 1.
Therefore {C φ p } p 0 is uniformly bounded. Hence Since the p-norms (for functions on bounded intervals) is weaker, we have are continuous images of a compact interval, hence compact and therefore bounded for each x ∈ B. So by the Uniform Boundedness Principle [4], each strongly continuous semigroup is uniformly bounded on each compact interval, a fact that implies exponential boundedness on D + .

CONCLUSION
In this paper, we have concluded that the classical results in semigroups of operator algebras and strongly continuous semigroups with real and complex scalars can be proved in bicomplex framework with idempotent decomposition.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.