arXiv:2005.09995v1 [math.FA] 17 May 2020 INTEGRAL FRAME IN HILBERT C * -MODULES MOHAMED ROSSAFI 1 * , FREJ CHOUCHENE 2 and SAMIR KABBAJ 3 Abstract. Frame theory is an exciting, dynamic and fast paced subject with applications in numerous fields of mathematics and engineering. In this paper we study Integral Frame and introduce Integral Frame with C * -valued bounds. Also, we establich some properties. 1. Introduction and preliminaries The concept of frames in Hilbert spaces has been introduced by Duffin and Schaeffer [6] in 1952 to study some deep problems in nonharmonic Fourier series, after the fundamental paper [5] by Daubechies, Grossman and Meyer, frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames [8]. Traditionally, frames have been used in signal processing, image processing, data compression and sampling in sampling theory. A discreet frame is a count- able family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into an expansion of the frame elements. The concept of a generalization of frames to a family indexed by some locally compact space endowed with a Radon measure was proposed by G. Kaiser [12] and independently by Ali, Antoine and Gazeau [1]. These frames are known as continuous frames. Gabardo and Han in [9] called these frames frames associated with measurable spaces, Askari-Hemmat, Dehghan and Radja- balipour in [2] called them generalized frames and in mathematical physics they are referred to as coherent states [1]. In this paper, we introduce the notions of Integral Frame on a Hilbert C * - Modules over a unital C * -algebra which is a generalization of discrete frames, the ∗-Integral Frame which are generalization of ∗-Frame in Hilbert C * -Modules introduced by A. Alijani, M. Dehghan [3] and we establish some new results. The paper is organized as follows, we continue this introductory section we briefly recall the definitions and basic properties of Hilbert C * -modules. In Sec- tion 2, we introduce the Integral Frame, the pre-Integral frame operator and the Integral frame operator. In Section 3, we introduce the ∗-Integral frame and the ∗-Integral frame operator. In Section 4, we discuss the stability problem for Integral Frame and ∗-Integral frame. Date : * Corresponding author. 2010 Mathematics Subject Classification. 41A58, 42C15. Key words and phrases. Integral Frame, ∗-Integral Frame, C * -algebra, Hilbert C * -modules. 1