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**A non-self-adjoint Lebesgue decomposition.**
*(English)*
Zbl 1302.47104

To formulate noncommutative generalizations of the classical function spaces \(H^2\) and \(H^\infty\), we begin with the the algebra of all noncommutative polynomials in \(d\) complex variables, \(Z_1,\dots ,Z_d\). This algebra can be completed in a natural inner product to form the noncommutative Hardy space, denoted \({F_d}^2\). Every element \(F\) of \({F_d}^2\) induces an action \(L_F\) on \({F_d}^2\) by left multiplication, with the caveat that \(L_F\) need not be bounded. This leads to the so-called noncommutative analytic Toeplitz algebra, denoted \({F_d}^\infty\), consisting of those \(F\) for which \(L_F\) is a bounded operator on \({F_d}^2\).

After giving suitable definitions for what it means for a bounded linear functional on \({F_d}^\infty\) to be absolutely continuous or singular, the authors provide a generalization of the Lebesgue decomposition for the space \({F_d}^\infty\). Specifically, Theorem 3.6 establishes that, if \(\phi\) is a bounded linear functional on \({F_d}^\infty\), then there are unique functionals \(\phi_a\) and \(\phi_s\) such that \(\phi_a\) is absolutely continuous, \(\phi_s\) is singular, and \(\phi=\phi_a + \phi_s\); moreover, \(\|\phi\|\leq \|\phi_a\|+\|\phi_s\|\leq \sqrt{2}\|\phi\|\). (For \(d=1\), the first inequality is an equality.) This leads to a version of the F. and M. Riesz theorem in Theorem 4.1, which asserts that, if \(J\) is a two-sided ideal in \({F_d}^\infty\) and if the bounded linear functional \(\phi=\phi_a+\phi_s\) vanishes on \(J\), then each summand \(\phi_a\) and \(\phi_s\) also vanishes on \(J\). These two results together lead to Theorem 5.2, a Lebesgue decomposition theorem for bounded linear functionals on quotient algebras \({F_d}^\infty/J\), again with the relationship \(\|\phi\|\leq \|\phi_a\|+\|\phi_s\|\leq \sqrt{2}\|\phi\|\) holding for the norms. Since the multiplier algebra of an irreducible Nevanlinna-Pick space may be viewed as the compression of \({F_d}^\infty\) to a coinvariant subspace, and, indeed, as a quotient of \({F_d}^\infty\) by a two-sided ideal, this generalized Lebesgue decomposition holds for bounded linear functionals on these multiplier algebras (Corollary 5.3). In the final section of the paper, the authors show that every quotient \({F_d}^\infty/J\) (and, hence, the multiplier algebra of a Nevanlinna-Pick space) has a strongly unique predual.

This work builds on earlier work of K. R. Davidson et al. [J. Funct. Anal. 224, No. 1, 160–191 (2005; Zbl 1084.46042)] and makes important use of the universal representation theory for \({F_d}^\infty\) put forward by D. P. Blecher and C. Le Merdy [Operator algebras and their modules – an operator space approach. Oxford University Press (2004; Zbl 1061.47002)].

After giving suitable definitions for what it means for a bounded linear functional on \({F_d}^\infty\) to be absolutely continuous or singular, the authors provide a generalization of the Lebesgue decomposition for the space \({F_d}^\infty\). Specifically, Theorem 3.6 establishes that, if \(\phi\) is a bounded linear functional on \({F_d}^\infty\), then there are unique functionals \(\phi_a\) and \(\phi_s\) such that \(\phi_a\) is absolutely continuous, \(\phi_s\) is singular, and \(\phi=\phi_a + \phi_s\); moreover, \(\|\phi\|\leq \|\phi_a\|+\|\phi_s\|\leq \sqrt{2}\|\phi\|\). (For \(d=1\), the first inequality is an equality.) This leads to a version of the F. and M. Riesz theorem in Theorem 4.1, which asserts that, if \(J\) is a two-sided ideal in \({F_d}^\infty\) and if the bounded linear functional \(\phi=\phi_a+\phi_s\) vanishes on \(J\), then each summand \(\phi_a\) and \(\phi_s\) also vanishes on \(J\). These two results together lead to Theorem 5.2, a Lebesgue decomposition theorem for bounded linear functionals on quotient algebras \({F_d}^\infty/J\), again with the relationship \(\|\phi\|\leq \|\phi_a\|+\|\phi_s\|\leq \sqrt{2}\|\phi\|\) holding for the norms. Since the multiplier algebra of an irreducible Nevanlinna-Pick space may be viewed as the compression of \({F_d}^\infty\) to a coinvariant subspace, and, indeed, as a quotient of \({F_d}^\infty\) by a two-sided ideal, this generalized Lebesgue decomposition holds for bounded linear functionals on these multiplier algebras (Corollary 5.3). In the final section of the paper, the authors show that every quotient \({F_d}^\infty/J\) (and, hence, the multiplier algebra of a Nevanlinna-Pick space) has a strongly unique predual.

This work builds on earlier work of K. R. Davidson et al. [J. Funct. Anal. 224, No. 1, 160–191 (2005; Zbl 1084.46042)] and makes important use of the universal representation theory for \({F_d}^\infty\) put forward by D. P. Blecher and C. Le Merdy [Operator algebras and their modules – an operator space approach. Oxford University Press (2004; Zbl 1061.47002)].

Reviewer: Timothy Feeman (Villanova)

### MSC:

47L50 | Dual spaces of operator algebras |

47L55 | Representations of (nonselfadjoint) operator algebras |

47B32 | Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) |

46B04 | Isometric theory of Banach spaces |

### Keywords:

Lebesgue decomposition; extended F. and M. Riesz theorem; unique predual; Nevanlinna-Pick space; noncommutative analytic Toeplitz algebra; Drury-Arveson space
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\textit{M. Kennedy} and \textit{D. Yang}, Anal. PDE 7, No. 2, 497--512 (2014; Zbl 1302.47104)

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