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Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and he shows similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)

The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.

Added after the first two comments:

Kaplansky's Rings of operators has another approach. Since there is no Google preview, I'll outline what is done. Theorem 26 shows that if $A$ is a unital ring with involution $*$ such that $1+x^*x$ is invertible for all $x\in A$, then for each idempotent $f\in A$ there is a projection $e\in A$ such that $fA=eA$. (The projection is obtained just as in the previous two sources, $e=ff^*(1+(f-f^*)(f^*-f))^{-1}$.) A previous exercise (4 on page 24) shows that if $f$ and $e$ are idempotents in a unital ring $A$ and $fA=eA$, then $f$ and $e$ are similar.

Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and he shows similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)

The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.

Added after the first two comments:

Kaplansky's Rings of operators has another approach. Since there is no Google preview, I'll outline what is done. Theorem 26 shows that if $A$ is a unital ring with involution $*$ such that $1+x^*x$ is invertible for all $x\in A$, then for each idempotent $f\in A$ there is a projection $e\in A$ such that $fA=eA$. (The projection is obtained just as in the previous two sources, $e=ff^*(1+(f-f^*)(f^*-f))^{-1}$.) A previous exercise (4 on page 24) shows that if $f$ and $e$ are idempotents in a unital ring $A$ and $fA=eA$, then $f$ and $e$ are similar.

Here is yet another reference, older and with a different approach, but this time not from a standard text. It is Lemma 16 on page 856 of Kaplansky's "Modules over operator algebras", American Journal of Mathematics, Vol. 75, No. 4 (Oct., 1953), pp. 839-858. Of the references I have provided, this is the only one that sets out to prove precisely the result in question as opposed to something stronger or more general. The approach is to invoke the theory of polynomial identities to reduce to the case of 2-by-2 scalar matrices. This paper is famous among operator algebraists because of its introduction of what are now called Hilbert $C^*$-modules (but only over commutative $C^*$-algebras).

Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and they show similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)

The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.

Added after the first two comments:

Kaplansky's Rings of operators has another approach. Since there is no Google preview, I'll outline what is done. Theorem 26 shows that if $A$ is a unital ring with involution $*$ such that $1+x^*x$ is invertible for all $x\in A$, then for each idempotent $f\in A$ there is a projection $e\in A$ such that $fA=eA$. (The projection is obtained just as in the previous two sources.) A previous exercise (4 on page 24) shows that if $f$ and $e$ are idempotents in a unital ring $A$ and $fA=eA$, then $f$ and $e$ are similar.

Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and he shows similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)

The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.

Added after the first two comments:

Kaplansky's Rings of operators has another approach. Since there is no Google preview, I'll outline what is done. Theorem 26 shows that if $A$ is a unital ring with involution $*$ such that $1+x^*x$ is invertible for all $x\in A$, then for each idempotent $f\in A$ there is a projection $e\in A$ such that $fA=eA$. (The projection is obtained just as in the previous two sources, $e=ff^*(1+(f-f^*)(f^*-f))^{-1}$.) A previous exercise (4 on page 24) shows that if $f$ and $e$ are idempotents in a unital ring $A$ and $fA=eA$, then $f$ and $e$ are similar.

Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and they show similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)

The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.

Blackadar's K-Theory for operator algebras has it, although the way it is done there is perhaps overkill if this is all you need. The result is generalized to local $C^*$-algebras, and they show similarity by showing the stronger property of homotopy equivalence. It is Proposition 4.6.2 on page 23 of the 2nd edition (1998). (Proposition 4.3.3 shows that homotopy equivalence is stronger.)

The stronger equivalence (but just for $C^*$-algebras) is also shown in the K-theory book by Rørdam et al., Lemma 11.2.7, with a very similar proof.

Added after the first two comments:

Kaplansky's Rings of operators has another approach. Since there is no Google preview, I'll outline what is done. Theorem 26 shows that if $A$ is a unital ring with involution $*$ such that $1+x^*x$ is invertible for all $x\in A$, then for each idempotent $f\in A$ there is a projection $e\in A$ such that $fA=eA$. (The projection is obtained just as in the previous two sources.) A previous exercise (4 on page 24) shows that if $f$ and $e$ are idempotents in a unital ring $A$ and $fA=eA$, then $f$ and $e$ are similar.