Motivated by the notion of P-functional, we introduce a notion of α-completely positive map between  ∗-algebras which is a Hermitian map satisfying a certain positivity condition, and then a α-completely positive map which is not completely positive is constructed. We establish the Kasparov-Stinespring-Gelfand-Naimark-Segal constructions of C∗-algebra and  ∗-algebra on Krein C∗-modules with α-completely positive maps.

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The temperature on Earth has been increased by 0.74 °C due to human activities over the past century. When the body temperature rises by 1 or 2 °C, a human feels the strangeness of health. Likewise, Earth temperature changes have resulted in various environmental abnormal phenomena, such as sea ice melting and desertification. Furthermore, temperature changes caused by global warming are causing extreme weather events, such as heat waves, large hurricanes, droughts, typhoons, heavy rains, and heavy snow in areas inhabited by humans. The main cause of global warming is carbon dioxide. Carbon dioxide is primarily emitted from fossil fuels used by humans. It is accumulated in the atmosphere and causes the greenhouse effect, thereby negatively affecting living organisms and humans' survival. The international community has recognized this problem, and members of the United Nations (UN) have agreed to the obligation of greenhouse gas (GHG) reduction at the 21st General Assembly of the Parties to the UN Framework Convention on Climate Change held in Paris in 2015. This UN obligation of GHG has been applied to all 195 UN member countries from January 2021. The convention presents the long-term goal of maintaining the global average temperature increase below 2 °C. Among anthropogenic GHG emissions, the shipping sector's contribution increased from 800 million tons in 2012 to 1 billion tons in 2018, constituting approximately 3% of the total global emissions. Hence, GHGs generated from the shipping industry must be reduced urgently. In April 2018, the International Maritime Organization (IMO) announced an early strategy to reduce GHGs emitted from ships by 50% compared with those emitted in 2008 and has continuously expressed the importance of reducing GHG emissions (Kim et al., 2018). To reduce GHGs in the shipping industry, the existing ships must be transformed into eco-friendly systems. To comply with environmental regulations, the global shipping industry changes ship fuels from heavy fuel oil (HFO) to liquefied natural gas (LNG) or marine diesel oil. And the additional emission reduction systems such as scrubber systems for SOx reduction and selective catalytic reduction systems for NOx reduction are installed on the ships (Lee, 2018). However, the solutions for GHG reduction in the shipping sector are still insufficient to achieve the IMO's GHG reduction goal. One of the well-known practical alternative solutions

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