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In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu_X)$ and $N=L^\infty(Y,\mu_Y)$ is given by a measure class $\mu$ on $X\times Y$ with marginal projections $\mathrm{Pr_X}(\mu)$, $\mathrm{Pr_Y}(\mu)$ absolutely continuous w.r.t. $\mu_X$, $\mu_Y$ and by a $\mu$-measurable function $n: X\times Y \rightarrow \mathbb{Z}$. There is no proof for this fact in the book, and it is not clear for me, why does $\mu$ appear to be a countably-additive measure, not just a finitely-additive one.

It follows from the definition that $N$-$M$-bimodule is a representation $\pi$ of maximal $C^\star$-tensor product $N\otimes_{max} M^{o}$ such that restrictions of $\pi$ on $N$ and $M^o$ are both normal. Functionals of the form $$ \mu_\pi: z\in N\otimes_{max} M^{o} \rightarrow \langle \pi(z)\xi,\xi\rangle $$ where $\xi$ is a cyclic vector, are exactly binormal states, i.e. such normed positive functionals on $N\otimes_{max} M^{o}$ that the maps $(f,g)\rightarrow \mu_\pi(f\otimes g)$ are normal separately in both arguments.

In the commutative ("measure-theoretic") case binormal states are in one-to-one correspondence with finitely-additive probability measures on $X\times Y$ having countably-additive marginal projections, absolutely continuous w.r.t. reference measures $\mu_X$, $\mu_Y$ respectively. It is not a priori clear, why they are countably-additive themselves.

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu_X)$ and $N=L^\infty(Y,\mu_Y)$ is given by a measure class $\mu$ on $X\times Y$ with marginal projections $\mathrm{Pr_X}(\mu)$, $\mathrm{Pr_Y}(\mu)$ absolutely continuous w.r.t. $\mu_X$, $\mu_Y$ and by a $\mu$-measurable function $n: X\times Y \rightarrow \mathbb{Z}$. There is no proof for this fact in the book, and it is not clear for me, why does $\mu$ appear to be a countably-additive measure, not just a finitely-additive one.

It follows from the definition that $N$-$M$-bimodule is a representation $\pi$ of maximal $C^\star$-tensor product $N\otimes_{max} M^{o}$ such that restrictions of $\pi$ on $N$ and $M^o$ are both normal. Functionals of the form $$ \mu_\pi: z\in N\otimes_{max} M^{o} \rightarrow \langle \pi(z)\xi,\xi\rangle $$ where $\xi$ is a cyclic vector, are exactly binormal states, i.e. such normed positive functionals on $N\otimes_{max} M^{o}$ that the maps $(f,g)\rightarrow \mu_\pi(f\otimes g)$ are normal separately in both arguments.

In the commutative ("measure-theoretical") case binormal states can be associated (via the appropriate form of Riesz representation theorem) with finitely-additive probability measures on $X\times Y$ having countably-additive marginal projections, absolutely continuous w.r.t. reference measures $\mu_X$, $\mu_Y$ respectively. It is not a priori clear, why they are countably-additive themselves.

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu)$ and $N=L^\infty(Y,\nu)$ is given by a measure class $\mu$ on $X\times Y$ with marginal projections $\mathrm{Pr_X}(\mu)$, $\mathrm{Pr_Y}(\mu)$ absolutely continuous w.r.t. $\mu_X$, $\mu_Y$ and by a $\mu$-measurable function $n: X\times Y \rightarrow \mathbb{Z}$. There is no proof for this fact in the book, and it is not clear for me, why does $\mu$ appear to be a countably-additive measure, not just a finitely-additive one.

It follows from the definition that $N$-$M$-bimodule is a representation $\pi$ of maximal $C^\star$-tensor product $N\otimes_{max} M^{o}$ such that restrictions of $\pi$ on $N$ and $M^o$ are both normal. Functionals of the form $$ \mu_\pi: z\in N\otimes_{max} M^{o} \rightarrow \langle \pi(z)\xi,\xi\rangle $$ where $\xi$ is a cyclic vector, are exactly binormal states, i.e. such normed positive functionals on $N\otimes_{max} M^{o}$ that the maps $(f,g)\rightarrow \mu_\pi(f\otimes g)$ are normal separately in both arguments.

In the commutative ("measure-theoretic") case binormal states are in one-to-one correspondence with finitely-additive probability measures on $X\times Y$ having countably-additive marginal projections, absolutely continuous w.r.t. reference measures $\mu$, $\nu$ on $X$, $Y$ respectively. It is not a priori clear, why they are countably-additive themselves.

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu_X)$ and $N=L^\infty(Y,\mu_Y)$ is given by a measure class $\mu$ on $X\times Y$ with marginal projections $\mathrm{Pr_X}(\mu)$, $\mathrm{Pr_Y}(\mu)$ absolutely continuous w.r.t. $\mu_X$, $\mu_Y$ and by a $\mu$-measurable function $n: X\times Y \rightarrow \mathbb{Z}$. There is no proof for this fact in the book, and it is not clear for me, why does $\mu$ appear to be a countably-additive measure, not just a finitely-additive one.

It follows from the definition that $N$-$M$-bimodule is a representation $\pi$ of maximal $C^\star$-tensor product $N\otimes_{max} M^{o}$ such that restrictions of $\pi$ on $N$ and $M^o$ are both normal. Functionals of the form $$ \mu_\pi: z\in N\otimes_{max} M^{o} \rightarrow \langle \pi(z)\xi,\xi\rangle $$ where $\xi$ is a cyclic vector, are exactly binormal states, i.e. such normed positive functionals on $N\otimes_{max} M^{o}$ that the maps $(f,g)\rightarrow \mu_\pi(f\otimes g)$ are normal separately in both arguments.

In the commutative ("measure-theoretic") case binormal states are in one-to-one correspondence with finitely-additive probability measures on $X\times Y$ having countably-additive marginal projections, absolutely continuous w.r.t. reference measures $\mu_X$, $\mu_Y$ respectively. It is not a priori clear, why they are countably-additive themselves.

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu)$ and $N=L^\infty(Y,\nu)$ is given by a measure class $\mu$ on $X\times Y$ with marginal projections $\mathrm{Pr_X}(\mu)$, $\mathrm{Pr_Y}(\mu)$ absolutely continuous w.r.t. $\mu_X$, $\mu_Y$ and by a $\mu$-measurable function $n(s,t): X\times Y \rightarrow \mathbb{Z}$. There is no proof for this fact in the book, and it is not clear for me, why does $\mu$ appear to be a countably-additive measure, not just a finitely-additive one.

It follows from the definition that $N$-$M$-bimodule is a representation $\pi$ of maximal $C^\star$-tensor product $N\otimes_{max} M^{o}$ such that restrictions of $\pi$ on $N$ and $M^o$ are both normal. Functionals of the form $$ \mu_\pi: z\in N\otimes_{max} M^{o} \rightarrow \langle \pi(z)\xi,\xi\rangle $$ where $\xi$ is a cyclic vector, are exactly binormal states, i.e. such normed positive functionals on $N\otimes_{max} M^{o}$ that the maps $(f,g)\rightarrow \mu_\pi(f\otimes g)$ are normal separately in both arguments.

In the commutative ("measure-theoretic") case binormal states are in one-to-one correspondence with finitely-additive probability measures on $X\times Y$ having countably-additive marginal projections, absolutely continuous w.r.t. reference measures $\mu$, $\nu$ on $X$, $Y$ respectively. It is not a priori clear, why they are countably-additive themselves.

In his "Noncommutative Geometry" book Connes asserts (on p. 539) that for two standard probability spaces $(X,\mu_X)$, $(Y,\nu_Y)$ an $N$-$M$-bimodule for $M=L^\infty(X,\mu)$ and $N=L^\infty(Y,\nu)$ is given by a measure class $\mu$ on $X\times Y$ with marginal projections $\mathrm{Pr_X}(\mu)$, $\mathrm{Pr_Y}(\mu)$ absolutely continuous w.r.t. $\mu_X$, $\mu_Y$ and by a $\mu$-measurable function $n: X\times Y \rightarrow \mathbb{Z}$. There is no proof for this fact in the book, and it is not clear for me, why does $\mu$ appear to be a countably-additive measure, not just a finitely-additive one.

It follows from the definition that $N$-$M$-bimodule is a representation $\pi$ of maximal $C^\star$-tensor product $N\otimes_{max} M^{o}$ such that restrictions of $\pi$ on $N$ and $M^o$ are both normal. Functionals of the form $$ \mu_\pi: z\in N\otimes_{max} M^{o} \rightarrow \langle \pi(z)\xi,\xi\rangle $$ where $\xi$ is a cyclic vector, are exactly binormal states, i.e. such normed positive functionals on $N\otimes_{max} M^{o}$ that the maps $(f,g)\rightarrow \mu_\pi(f\otimes g)$ are normal separately in both arguments.

In the commutative ("measure-theoretic") case binormal states are in one-to-one correspondence with finitely-additive probability measures on $X\times Y$ having countably-additive marginal projections, absolutely continuous w.r.t. reference measures $\mu$, $\nu$ on $X$, $Y$ respectively. It is not a priori clear, why they are countably-additive themselves.