God alone possesses immortality (1 Tim 6:16); we participate in immortality in union/communion with God, and insofar as "He is before all things, and in him all things hold together" (Col 1:17). Apart from God nothing holds together of itself.
"And this is the testimony: God has given us eternal life, and this life is in His Son." -1 Jn 5:11
"God, the blessed and only Ruler, the King of kings and Lord of lords,who alone possesses immortality [όνος ἔχων ἀθανασίαν],and who lives in unapproachable light, whom no one has seen or can see: to him be honor and might forever. Amen." -1 Tim 6:15-16
Endless life is a property not of Adam or the sons or daughters of Adam, but of Christ, and in Christ, of which we partake only in communion/union "in Christ." "...the gift of God is eternal life in Christ Jesus our Lord" Rom 6:23
Even those in hell will also be "held together" by the love of God: "I say that even those who are scourged in hell are tormented with the scourgings of love. The scourges that result from love -that is, the scourges of those who have become aware that they have sinned against love- are harder and more bitter than the torments which result from fear. The pain which gnaws the heart as the result of sinning against love is sharper than all other torments that there are. It is wrong to imagine that the sinners in hell are deprived of the love of God.... [but] the power of love works in two ways: it torments those who have sinned, just as happens among friends here on earth, but to those who have observed its duties, love gives delight." St. Isaac of Syria, Homily 27(28).
"Make ready, O Bethlehem, for Eden hath been opened for all. Prepare, O Ephratha, for the tree of life hath blossomed forth in the cave from the Virgin; for her womb did appear as a spiritual paradise in which is planted the divine Plant, whereof eating we shall live and not die as did Adam. Christ shall be born, raising the image that fell of old"
"And then, lest Adam stretch forth his hand to the tree of life and live forever, God dismissed him from paradise." St. Augustine, Two Books on Genesis Against the Manichaeans 2.22.34
God did not kill Adam, but prevented his continual access to the Tree of Life:
"Therefore, lest Adam and Eve, after having eaten of this tree [of the knowledge of good and evil] live forever and remain in eternal lives of suffering, God forbade them to eat, after they were clothed with a curse, that which he had prepared to give them before they incurred the curse and when they were still clothed with glory" -St. Ephrem of Syria, Commentary on Genesis 2.35.1.
"Blessed was He who was pierced and so removed the sword from the entry to paradise" -St. Ephrem the Syrian, Hymns On Paradise 2.1
"...disobedience was the cause of death; for that reason not God but man himself was the agent of his own death." St. Ambrose, Paradise, 7.35
"By punishing us with death, the lawgiver cut off the spread of sin. And yet through that very punishment he also demonstrated his love for us. He bound sin and death together when he gave the law placing the sinner under punishment of death. And yet he ordered things in such a way that the punishment might in itself serve the goal of salvation. For death brings about separation from this life and brings evil works to an end. It sets us free from labor, sweat, and pain, and ends the suffering of the body. Thus the Judge mixes his love for us with the punishment." -Theodoret, On the Incarnation of the Lord