## 1 Introduction  
The enzyme \(\beta\)-galactosidase (E.C 3.2.1.23) catalyzes the hydrolysis of the disaccharide lactose into the monosaccharides glucose and galactose. The enzyme may act in solutions and also in different matrices rich in lactose, being applied in different stages of the processes as a fundamental way to reduce the lactose content of dairy products intended for people with lactose intolerance [1, 2]. Around 70% of the adult population worldwide is lactose-intolerant, reaching 95-100% in Asians, for example. This makes it necessary for the food industry to apply technologies to reduce the lactose contents in products intended for these lactose-intolerant people in order to avoid symptoms such as diarrhea, flatulence, and abdominal cramps that appear with the consumption of lactose [3,4]. Moreover, hydrolysis with \(\beta\)-galactosidase contributes to the formation of galactooligosaccharides (GOS) [5, 6, 7], which are beneficial to humans [8] and improve the technological and sensory characteristics of foods [9, 10, 11]. However, for effective hydrolysis using enzymes, recovery techniques that do not compromise the enzymatic activity, and consequently the applications, viability, and economic potential, need to be applied [1].  
Ultrafiltration (UF) is a technique based on the macromolecule size-exclusion mechanism and can be applied to both the concentration and polishing of biological products [12]. The concentration and separation of the biological material using membranes does not require additives or phase changes and can be performed isothermally at low temperatures and pressures, with lower energy consumption compared with the thermal processes. Furthermore, these conditions ensure minimal denaturation of bioproducts that are highly unstable and are less expensive than other methods [12, 13, 14]. In ultrafiltration systems, the operating conditions strongly affect the concentration process, particularly the protein size of the bioproducts [15], which should be deeply studied.  
Some of the important applications of ultrafiltration techniques include fractionation and the recovery of numerous proteins [16, 17, 18], polysaccharides [19, 20], enzymes [1, 21], biosurfactants [22], oils [23, 24], and bioactive peptides [25, 26]. Most of the studies on \(\beta\)-galactosidaseases reported the use of only ultrafiltration when modeling the hydrolysis mechanism [27] and galactooligosaccharide synthesis [28, 29], and for the removal of impurities such as salts [30] and polymers in an aqueous purification system [31]. The application of these techniques often neglects the characteristic peculiarities of the bioproduct or does not consider the interactions between molecules, which in turn compromise the efficiency of the process and hinder the scale-up procedure, resulting in the infeasibility of the process [1].  
Ultrafiltration can be used as an alternative to other concentration techniques; therefore, it is considered an important process for applications in the food and pharmaceutics industries. This technique can also be applied for the removal of salts and polyethylene glycol used in the purification process when utilized in the diafiltration mode. In this case, ultrafiltration is achieved after precipitation or separation in an aqueous two-phase system of \(\beta\)-galactosidasease from _Kluyveromyces marxianus_ used in the production of GOS [5] or other products of industrial interest [32].  
Thus, we evaluated the influence of the pH, membrane molecular weight cut-off (MWCO), operating pressure, and ionic strength of the ultrafiltration process of \(\beta\)-galactosidase from _K. marxianus_ CCT 7082. Furthermore, the ultrafiltration process was applied in the diafiltration mode as an alternative for salt removal from the \(\beta\)-galactosidase after the purification process. To the best of our knowledge, no previous study has been performed in this regard.