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The unitalization case $A=J^+$ is treated in Blackadar's $K$-theory for Operator Algebras. In Proposition 5.4.1 he directlyshowesshows that
$$
K_0(J^+,J)\cong\ker(K_0(J^+)\to K_0({J^+}/J)).
$$
The unitalization case $A=J^+$ is treated in Blackadar's $K$-theory for Operator Algebras. In Proposition 5.4.1 he directlyshowes that
$$
K_0(J^+,J)\cong\ker(K_0(J^+)\to K_0({J^+}/J)).
$$
The unitalization case $A=J^+$ is treated in Blackadar's $K$-theory for Operator Algebras. In Proposition 5.4.1 he directlyshows that
$$
K_0(J^+,J)\cong\ker(K_0(J^+)\to K_0({J^+}/J)).
$$
The unitalization case $A=J^+$ is treated in Blackadar's $K$-theory for Operator Algebras. In Proposition 5.4.1 he directly showes that
$$
K_0(J^+,J)\cong\ker(K_0(J^+)\to K_0({J^+}/J)).
$$
