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Compact sets in the free topology
Linear Algebra and its Applications 506 (2016) 6–9
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Compact sets in the free topology M. Augat a , S. Balasubramanian b,1 , Scott McCullough a,∗,2 a b
Department of Mathematics, University of Florida, Gainesville, United States Department of Mathematics, IIT Madras, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 2 May 2016 Accepted 12 May 2016 Available online xxxx Submitted by P. Semrl
a b s t r a c t Subsets of the set of g-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, the hull of a single point. © 2016 Elsevier Inc. All rights reserved.
MSC: primary 47A20, 47A13 secondary 46L07 Keywords: Dilation hull Free analysis Free polynomials
1. Introduction Given positive integers n, g, let Mn (C)g denote the set of g-tuples of n × n matrices. Let M (C)g denote the sequence (Mn (C)g )n . A subset E of M (C)g is a sequence (E(n)) where E(n) ⊂ Mn (C)g . The free topology [1] has as a basis sets of the form Gδ = (Gδ (n)) ⊂ M (C)g , where * Corresponding author. E-mail addresses: mlaugat@ufl.edu (M. Augat), [email protected] (S. Balasubramanian), [email protected]fl.edu (S. McCullough). 1 Supported by the New Faculty Initiative Grant (MAT/15-16/836/NFIG/SRIM) of IIT Madras. 2 Research supported by the NSF grant DMS-1361501. http://dx.doi.org/10.1016/j.laa.2016.05.010 0024-3795/© 2016 Elsevier Inc. All rights reserved.
M. Augat et al. / Linear Algebra and its Applications 506 (2016) 6–9
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Gδ (n) = {X ∈ Mn (C)g : δ(X) < 1}, and δ is a (matrix-valued) free polynomial. Agler and McCarthy [1] prove the remarkable result that a bounded free function on a basis set Gδ is uniformly approximable by polynomials on each smaller set of the form Gδ ⊃ Kδs = {X ∈ M (C)g : δ(X) ≤ s}, 0 ≤ s < 1. See [3] for a generalization of this result. For the definitive treatment of free function theory, see [6]. Sets E ⊂ M (C)g naturally arising in free analysis ([2,4,5,7–9,12] is a sampling of the references not already cited) are typically closed with respect to direct sums in the sense that if X ∈ E(n) and Y ∈ E(m), then Xg 0 X1 0 ,..., ∈ E(n + m). X ⊕Y = 0 Y1 0 Yg Theorem 1.1 below, characterizing free topology compact sets that are closed with respect to directs sums, is the main result of this article. A tuple Y ∈ Mn (C)g polynomially dilates to a tuple X ∈ MN (C)g if there is an isometry V : Cn → CN such that for all free polynomials p, p(Y ) = V ∗ p(X)V. An ampliation of X is a tuple of the form Ik ⊗ X, for some positive integer k. The polynomial dilation hull of X ∈ M (C)g is the set of all Y that dilate to an ampliation of X. Theorem 1.1. A nonempty subset E of M (C)g that is closed with respect to direct sums is compact if and only if it is the polynomial dilation hull of an X ∈ E. Corollary 1.2. If E ⊂ M (C)g is closed with respect to direct sums and is compact in the free topology, then there exists a free polynomial p such that E is a subset of the zero set of p; i.e., p(Y ) = 0 for all Y ∈ E. In particular, there is an N such that for n ≥ N the set E(n) has empty interior. Proof. By Theorem 1.1, there is an n and X ∈ E(n) such that each Y ∈ E polynomially dilates to an ampliation of X. Choose a nonzero (scalar) free polynomial p such that p(X) = 0 (using the fact that the span of {w(X) : w is a word} is a subset of the finite dimensional vector space Mn (C)). It follows that p(Y ) = 0 for all Y . Hence E is a subset of the zero set of p. It is well known (see for instance the Amistur–Levitzki Theorem [11]) that the zero set p in Mn (C)g must have empty interior for sufficiently large n. 2 The authors thank Igor Klep for a fruitful correspondence that led to this article. The proof of Theorem 1.1 occupies the remainder of this article.
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2. The proof of Theorem 1.1 Proposition 2.1. Suppose E ⊂ M (C)g is nonempty and closed with respect to direct sums. If for each X ∈ E there is a matrix-valued free polynomial δ and a Y ∈ E such that δ(X) < δ(Y ), then E is not compact in the free topology. Proof. By hypothesis, for each X ∈ E there is a matrix-valued polynomial δX and YX ∈ E such that δX (X) < 1 < δX (YX ). The collection G = {GδX : X ∈ E} is an open cover of E. Suppose S ⊂ E is a finite. Since E is closed with respect to direct sums, Z = ⊕X∈S YX ∈ E. On the other hand, for a fixed W ∈ S, δW (Z) ≥ δW (YW ) > 1. Thus Z ∈ / GδW and therefore Z ∈ E but Z ∈ / ∪X∈S GδX . Thus G admits no finite subcover of E and therefore E is not compact. 2 The following lemma is a standard result. Lemma 2.2. Suppose X, Y ∈ M (C)g . The tuple Y polynomially dilates to an ampliation of X if and only if δ(Y ) ≤ δ(X) for every free matrix-valued polynomial δ. Proof. Let P denote the set of scalar free polynomials in g variables. Given a tuple Z ∈ Mn (C)g , let S(Z) = {p(Z) : p ∈ P} ⊂ Mn (C). The set S(Z) is a unital operator algebra. Let m and n denote the sizes of Y and X respectively. The hypotheses thus imply that the unital homomorphism λ : S(X) → S(Y ) given by λ(p(X)) = p(Y ) is well defined and completely contractive. Thus by Corollary 7.6 of [10], there exists a completely positive map ϕ : Mn (C) → Mm (C) extending λ. By Choi’s Theorem [10], there exists an M and, for 1 ≤ j ≤ M , mappings Wj : Cm → Cn such that Wj∗ Wj = I and ϕ(T ) =
Wj∗ T Wj .
Let W denote the column matrix with entries Wi . With this notation, ϕ(T ) = W ∗ (IM ⊗ T )W . Since I = ϕ(I) = W ∗ W , the map W is an isometry. Moreover, for polynomials p, p(Y ) = ϕ(p(X)) = W ∗ (IM ⊗ p(X))W
M. Augat et al. / Linear Algebra and its Applications 506 (2016) 6–9
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and the proof of the reverse direction is complete. To prove the forward direction, suppose there is an X ∈ E, an N and an isometry V such that for all free scalar polynomials p, p(Y ) = V ∗ p(IN ⊗ X)V = V ∗ [IN ⊗ p(X)]V. Thus for all matrix free polynomials δ, say of size d × d (without loss of generality δ can be assumed square), δ(Y ) = [V ⊗ Id ]∗ [IN ⊗ δ(X)][V ⊗ Id ]. It follows that δ(Y ) ≤ δ(X). 2 Proof of Theorem 1.1. If for each X ∈ E there is a Y ∈ E that does not polynomially dilate to an ampliation of X, then, by Lemma 2.2, for each X ∈ E there is a Y ∈ E and a matrix-valued polynomial δX such that δX (X) < δX (Y ). An application of Proposition 2.1 shows E is not compact. To prove the converse, suppose there exists an X ∈ E such that every Y ∈ E polynomially dilates to an ampliation of X. Let G be an open cover of E. There is a G ∈ G such that X ∈ G. There is a δ such that X ∈ Gδ ⊂ G. Since Y dilates to X, it follows that δ(Y ) ≤ δ(X) < 1. Hence Y ∈ Gδ and therefore E ⊂ G. Thus E is compact. 2 References [1] J. Agler, J. McCarthy, Global holomorphic functions in several non-commuting variables, Canad. J. Math. 67 (2015) 241–285. [2] J. Agler, J. McCarthy, Pick interpolation for free holomorphic functions, Amer. J. Math. 137 (2015) 1685–1701. [3] J.A. Ball, G. Marx, V. Vinnikov, Interpolation and transfer-function realization for the noncommutative Schur–Agler class, preprint, http://arxiv.org/abs/1602.00762. [4] Sabine Burgdorf, Igor Klep, Janez Povh, Optimization of Polynomials in Non-commuting Variables, Springer Briefs Math., Springer-Verlag, 2016. [5] J. William Helton, Igor Klep, Christopher S. Nelson, Noncommutative polynomials nonnegative on a variety intersect a convex set, J. Funct. Anal. 266 (2014) 6684–6752. [6] D. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, Foundations of Free Noncommutative Function Theory, Math. Surveys Monogr., vol. 199, AMS, 2014. [7] Igor Klep, Jurij Volcic, Free loci of matrix pencils and domains of noncommutative rational functions, preprint, http://arxiv.org/abs/1512.02648. [8] I. Klep, Š. Špenko, Free function theory through matrix invariants, Canad. J. Math. (2016), http://dx.doi.org/10.4153/CJM-2015-055-7, in press. [9] J.E. Pascoe, The inverse function theorem and the Jacobian conjecture for free analysis, Math. Z. 278 (2014) 987–994. [10] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Univ. Press, 2002. [11] L.H. Rowen, Polynomials Identities in Ring Theory, Academic Press, New York, 1980. [12] D.-V. Voiculescu, Free analysis questions II: the Grassmannian completion and the series expansions at the origin, J. Reine Angew. Math. 645 (2010) 155–236.
Contents lists available at ScienceDirect
Linear Algebra and its Applications www.elsevier.com/locate/laa
Compact sets in the free topology M. Augat a , S. Balasubramanian b,1 , Scott McCullough a,∗,2 a b
Department of Mathematics, University of Florida, Gainesville, United States Department of Mathematics, IIT Madras, Chennai 600036, India
a r t i c l e
i n f o
Article history: Received 2 May 2016 Accepted 12 May 2016 Available online xxxx Submitted by P. Semrl
a b s t r a c t Subsets of the set of g-tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized. They are, in a dilation theoretic sense, the hull of a single point. © 2016 Elsevier Inc. All rights reserved.
MSC: primary 47A20, 47A13 secondary 46L07 Keywords: Dilation hull Free analysis Free polynomials
1. Introduction Given positive integers n, g, let Mn (C)g denote the set of g-tuples of n × n matrices. Let M (C)g denote the sequence (Mn (C)g )n . A subset E of M (C)g is a sequence (E(n)) where E(n) ⊂ Mn (C)g . The free topology [1] has as a basis sets of the form Gδ = (Gδ (n)) ⊂ M (C)g , where * Corresponding author. E-mail addresses: mlaugat@ufl.edu (M. Augat), [email protected] (S. Balasubramanian), [email protected]fl.edu (S. McCullough). 1 Supported by the New Faculty Initiative Grant (MAT/15-16/836/NFIG/SRIM) of IIT Madras. 2 Research supported by the NSF grant DMS-1361501. http://dx.doi.org/10.1016/j.laa.2016.05.010 0024-3795/© 2016 Elsevier Inc. All rights reserved.
M. Augat et al. / Linear Algebra and its Applications 506 (2016) 6–9
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Gδ (n) = {X ∈ Mn (C)g : δ(X) < 1}, and δ is a (matrix-valued) free polynomial. Agler and McCarthy [1] prove the remarkable result that a bounded free function on a basis set Gδ is uniformly approximable by polynomials on each smaller set of the form Gδ ⊃ Kδs = {X ∈ M (C)g : δ(X) ≤ s}, 0 ≤ s < 1. See [3] for a generalization of this result. For the definitive treatment of free function theory, see [6]. Sets E ⊂ M (C)g naturally arising in free analysis ([2,4,5,7–9,12] is a sampling of the references not already cited) are typically closed with respect to direct sums in the sense that if X ∈ E(n) and Y ∈ E(m), then Xg 0 X1 0 ,..., ∈ E(n + m). X ⊕Y = 0 Y1 0 Yg Theorem 1.1 below, characterizing free topology compact sets that are closed with respect to directs sums, is the main result of this article. A tuple Y ∈ Mn (C)g polynomially dilates to a tuple X ∈ MN (C)g if there is an isometry V : Cn → CN such that for all free polynomials p, p(Y ) = V ∗ p(X)V. An ampliation of X is a tuple of the form Ik ⊗ X, for some positive integer k. The polynomial dilation hull of X ∈ M (C)g is the set of all Y that dilate to an ampliation of X. Theorem 1.1. A nonempty subset E of M (C)g that is closed with respect to direct sums is compact if and only if it is the polynomial dilation hull of an X ∈ E. Corollary 1.2. If E ⊂ M (C)g is closed with respect to direct sums and is compact in the free topology, then there exists a free polynomial p such that E is a subset of the zero set of p; i.e., p(Y ) = 0 for all Y ∈ E. In particular, there is an N such that for n ≥ N the set E(n) has empty interior. Proof. By Theorem 1.1, there is an n and X ∈ E(n) such that each Y ∈ E polynomially dilates to an ampliation of X. Choose a nonzero (scalar) free polynomial p such that p(X) = 0 (using the fact that the span of {w(X) : w is a word} is a subset of the finite dimensional vector space Mn (C)). It follows that p(Y ) = 0 for all Y . Hence E is a subset of the zero set of p. It is well known (see for instance the Amistur–Levitzki Theorem [11]) that the zero set p in Mn (C)g must have empty interior for sufficiently large n. 2 The authors thank Igor Klep for a fruitful correspondence that led to this article. The proof of Theorem 1.1 occupies the remainder of this article.
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2. The proof of Theorem 1.1 Proposition 2.1. Suppose E ⊂ M (C)g is nonempty and closed with respect to direct sums. If for each X ∈ E there is a matrix-valued free polynomial δ and a Y ∈ E such that δ(X) < δ(Y ), then E is not compact in the free topology. Proof. By hypothesis, for each X ∈ E there is a matrix-valued polynomial δX and YX ∈ E such that δX (X) < 1 < δX (YX ). The collection G = {GδX : X ∈ E} is an open cover of E. Suppose S ⊂ E is a finite. Since E is closed with respect to direct sums, Z = ⊕X∈S YX ∈ E. On the other hand, for a fixed W ∈ S, δW (Z) ≥ δW (YW ) > 1. Thus Z ∈ / GδW and therefore Z ∈ E but Z ∈ / ∪X∈S GδX . Thus G admits no finite subcover of E and therefore E is not compact. 2 The following lemma is a standard result. Lemma 2.2. Suppose X, Y ∈ M (C)g . The tuple Y polynomially dilates to an ampliation of X if and only if δ(Y ) ≤ δ(X) for every free matrix-valued polynomial δ. Proof. Let P denote the set of scalar free polynomials in g variables. Given a tuple Z ∈ Mn (C)g , let S(Z) = {p(Z) : p ∈ P} ⊂ Mn (C). The set S(Z) is a unital operator algebra. Let m and n denote the sizes of Y and X respectively. The hypotheses thus imply that the unital homomorphism λ : S(X) → S(Y ) given by λ(p(X)) = p(Y ) is well defined and completely contractive. Thus by Corollary 7.6 of [10], there exists a completely positive map ϕ : Mn (C) → Mm (C) extending λ. By Choi’s Theorem [10], there exists an M and, for 1 ≤ j ≤ M , mappings Wj : Cm → Cn such that Wj∗ Wj = I and ϕ(T ) =
Wj∗ T Wj .
Let W denote the column matrix with entries Wi . With this notation, ϕ(T ) = W ∗ (IM ⊗ T )W . Since I = ϕ(I) = W ∗ W , the map W is an isometry. Moreover, for polynomials p, p(Y ) = ϕ(p(X)) = W ∗ (IM ⊗ p(X))W
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and the proof of the reverse direction is complete. To prove the forward direction, suppose there is an X ∈ E, an N and an isometry V such that for all free scalar polynomials p, p(Y ) = V ∗ p(IN ⊗ X)V = V ∗ [IN ⊗ p(X)]V. Thus for all matrix free polynomials δ, say of size d × d (without loss of generality δ can be assumed square), δ(Y ) = [V ⊗ Id ]∗ [IN ⊗ δ(X)][V ⊗ Id ]. It follows that δ(Y ) ≤ δ(X). 2 Proof of Theorem 1.1. If for each X ∈ E there is a Y ∈ E that does not polynomially dilate to an ampliation of X, then, by Lemma 2.2, for each X ∈ E there is a Y ∈ E and a matrix-valued polynomial δX such that δX (X) < δX (Y ). An application of Proposition 2.1 shows E is not compact. To prove the converse, suppose there exists an X ∈ E such that every Y ∈ E polynomially dilates to an ampliation of X. Let G be an open cover of E. There is a G ∈ G such that X ∈ G. There is a δ such that X ∈ Gδ ⊂ G. Since Y dilates to X, it follows that δ(Y ) ≤ δ(X) < 1. Hence Y ∈ Gδ and therefore E ⊂ G. Thus E is compact. 2 References [1] J. Agler, J. McCarthy, Global holomorphic functions in several non-commuting variables, Canad. J. Math. 67 (2015) 241–285. [2] J. Agler, J. McCarthy, Pick interpolation for free holomorphic functions, Amer. J. Math. 137 (2015) 1685–1701. [3] J.A. Ball, G. Marx, V. Vinnikov, Interpolation and transfer-function realization for the noncommutative Schur–Agler class, preprint, http://arxiv.org/abs/1602.00762. [4] Sabine Burgdorf, Igor Klep, Janez Povh, Optimization of Polynomials in Non-commuting Variables, Springer Briefs Math., Springer-Verlag, 2016. [5] J. William Helton, Igor Klep, Christopher S. Nelson, Noncommutative polynomials nonnegative on a variety intersect a convex set, J. Funct. Anal. 266 (2014) 6684–6752. [6] D. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, Foundations of Free Noncommutative Function Theory, Math. Surveys Monogr., vol. 199, AMS, 2014. [7] Igor Klep, Jurij Volcic, Free loci of matrix pencils and domains of noncommutative rational functions, preprint, http://arxiv.org/abs/1512.02648. [8] I. Klep, Š. Špenko, Free function theory through matrix invariants, Canad. J. Math. (2016), http://dx.doi.org/10.4153/CJM-2015-055-7, in press. [9] J.E. Pascoe, The inverse function theorem and the Jacobian conjecture for free analysis, Math. Z. 278 (2014) 987–994. [10] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Univ. Press, 2002. [11] L.H. Rowen, Polynomials Identities in Ring Theory, Academic Press, New York, 1980. [12] D.-V. Voiculescu, Free analysis questions II: the Grassmannian completion and the series expansions at the origin, J. Reine Angew. Math. 645 (2010) 155–236.